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image | page_id
string | expressions
sequence |
|---|---|---|
0001213_page17 | {
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"\\begin {equation} S_{p-2}=-\\frac {V_{p-2}}{(2\\pi )^{p-2}\\tilde {g}}\\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p}\\left [ \\sqrt {\\frac {1+(au)^{7-p}}{(au)^{7-p}}}\\sqrt {\\tilde {f}}-1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {equation}",
"\\begin {equation} S_{p-2}= -\\frac {V_{p-2}}{(2\\pi )^{p-2}\\tilde {g}}\\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p}\\left [ \\sqrt { \\tilde {f}}-1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {equation}",
"\\begin {equation} F_{p-2}= -\\frac {V_{p-2}}{(2\\pi )^{p-2}\\tilde {g}} \\left (\\frac {u}{R}\\right )^{7-p}\\left [ \\sqrt { \\tilde {f}}-1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {equation}",
"\\begin {eqnarray} F_{p-2}|_{u=u_0} &=&-\\frac {V_{p-2}}{2(2\\pi )^{p-2}\\tilde {g}} \\left (\\frac {u_0}{R} \\right )^{7-p} \\\\ &=& -\\frac {V_{p-2}}{2(2\\pi )^{p-2}\\tilde {g}} \\left (\\frac {4\\pi RT}{7-p}\\right )^{\\frac {2(7-p)}{5-p}}. \\end {eqnarray}",
"\\begin {equation} \\label {3e18} \\frac {\\delta N_{p-2}}{\\delta N_p}= \\frac {\\tilde {V}_2}{(2\\pi )^2 \\tilde {b}}, \\end {equation}",
"\\begin {equation} F_{p-2}|_{u=u_0}=\\frac {dF}{dN_{p-2}}\\delta N_{p-2}, \\end {equation}"
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"\\begin{equation*} S _ { p - 2 } = - \\frac { V _ { p - 2 } } { ( 2 \\pi ) ^ { p - 2 } \\widetilde { g } } \\int d \\tau { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\frac { 1 + ( a u ) ^ { 7 - p } } { ( a u ) ^ { 7 - p } } } \\sqrt { \\widetilde { f } } - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] . \\end{equation*}",
"\\begin{equation*} S _ { p - 2 } = - \\frac { V _ { p - 2 } } { ( 2 \\pi ) ^ { p - 2 } \\widetilde { g } } \\int d \\tau { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\widetilde { f } } - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] . \\end{equation*}",
"\\begin{equation*} F _ { p - 2 } = - \\frac { V _ { p - 2 } } { ( 2 \\pi ) ^ { p - 2 } \\widetilde { g } } { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\widetilde { f } } - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] . \\end{equation*}",
"\\begin{align*} F _ { p - 2 } \\vert _ { u = u _ { 0 } } & = & - \\frac { V _ { p - 2 } } { 2 ( 2 \\pi ) ^ { p - 2 } \\widetilde { g } } { ( \\frac { u _ { 0 } } { R } ) } ^ { 7 - p } \\\\ & = & - \\frac { V _ { p - 2 } } { 2 ( 2 \\pi ) ^ { p - 2 } \\widetilde { g } } { ( \\frac { 4 \\pi R T } { 7 - p } ) } ^ { \\frac { 2 ( 7 - p ) } { 5 - p } } . \\end{align*}",
"\\begin{equation*} \\frac { \\delta N _ { p - 2 } } { \\delta N _ { p } } = \\frac { \\widetilde { V } _ { 2 } } { ( 2 \\pi ) ^ { 2 } \\widetilde { b } } , \\end{equation*}",
"\\begin{equation*} F _ { p - 2 } \\vert _ { u = u _ { 0 } } = \\frac { d F } { d N _ { p - 2 } } \\delta N _ { p - 2 } , \\end{equation*}"
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"\\begin{equation*} \\mitS _ { \\mitp - 2 } = - \\frac { \\mitV _ { \\mitp - 2 } } { ( 2 \\mitpi ) ^ { \\mitp - 2 } \\tilde { \\mitg } } \\int \\mitd \\mittau { \\left( \\frac { \\mitu } { \\mitR } \\right) } ^ { 7 - \\mitp } \\left[ \\sqrt { \\tilde { \\mitf } } - 1 + \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } \\right] . \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\mitp - 2 } = - \\frac { \\mitV _ { \\mitp - 2 } } { ( 2 \\mitpi ) ^ { \\mitp - 2 } \\tilde { \\mitg } } { \\left( \\frac { \\mitu } { \\mitR } \\right) } ^ { 7 - \\mitp } \\left[ \\sqrt { \\tilde { \\mitf } } - 1 + \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } \\right] . \\end{equation*}",
"\\begin{align*} \\mitF _ { \\mitp - 2 } \\vert _ { \\mitu = \\mitu _ { 0 } } & = & - \\frac { \\mitV _ { \\mitp - 2 } } { 2 ( 2 \\mitpi ) ^ { \\mitp - 2 } \\tilde { \\mitg } } { \\left( \\frac { \\mitu _ { 0 } } { \\mitR } \\right) } ^ { 7 - \\mitp } \\\\ & = & - \\frac { \\mitV _ { \\mitp - 2 } } { 2 ( 2 \\mitpi ) ^ { \\mitp - 2 } \\tilde { \\mitg } } { \\left( \\frac { 4 \\mitpi \\mitR \\mitT } { 7 - \\mitp } \\right) } ^ { \\frac { 2 ( 7 - \\mitp ) } { 5 - \\mitp } } . \\end{align*}",
"\\begin{equation*} \\frac { \\mitdelta \\mitN _ { \\mitp - 2 } } { \\mitdelta \\mitN _ { \\mitp } } = \\frac { \\tilde { \\mitV } _ { 2 } } { ( 2 \\mitpi ) ^ { 2 } \\tilde { \\mitb } } , \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\mitp - 2 } \\vert _ { \\mitu = \\mitu _ { 0 } } = \\frac { \\mitd \\mitF } { \\mitd \\mitN _ { \\mitp - 2 } } \\mitdelta \\mitN _ { \\mitp - 2 } , \\end{equation*}"
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|
0001213_page18 | {
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"\\begin {equation} F_{p-2}|_{u=u_0}\\approx F(N_{p-2}+1)-F(N_{p-2}). \\end {equation}",
"\\begin {equation} \\label {4e1} S_p=-\\frac {T_pV_p}{g\\cos \\theta } \\int d\\tau H^{-1}[\\sqrt {f-H(f^{-1}\\dot {r}^2 + r^2 \\dot {\\Omega }_{8-p}^2)}-1 + H_0 - H], \\end {equation}",
"\\begin {eqnarray} S_p =-m_p\\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p}\\left [ \\sqrt {\\tilde {f} -\\left (\\frac {R}{u}\\right )^{7-p}\\left (\\tilde {f}^{-1} \\dot {u}^2 +u^2 \\dot {\\Omega }^2_{8-p}\\right )}-1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ], \\end {eqnarray}",
"\\begin {equation} \\tilde {f}^{-1} du^2 +u^2 d\\Omega _{8-p}^2 = u^2 {\\rho }^{-2} (d{\\rho }^2 +{\\rho }^2 d\\Omega ^2_{8-p}), \\end {equation}"
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"\\begin{equation*} F _ { p - 2 } \\vert _ { u = u _ { 0 } } \\approx F ( N _ { p - 2 } + 1 ) - F ( N _ { p - 2 } ) . \\end{equation*}",
"\\begin{equation*} S _ { p } = - \\frac { T _ { p } V _ { p } } { g \\operatorname { c o s } \\theta } \\int d \\tau H ^ { - 1 } [ \\sqrt { f - H ( f ^ { - 1 } \\dot { r } ^ { 2 } + r ^ { 2 } \\dot { \\Omega } _ { 8 - p } ^ { 2 } ) } - 1 + H _ { 0 } - H ] , \\end{equation*}",
"\\begin{equation*} S _ { p } = - m _ { p } \\int d \\tau { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\widetilde { f } - { ( \\frac { R } { u } ) } ^ { 7 - p } ( \\widetilde { f } ^ { - 1 } \\dot { u } ^ { 2 } + u ^ { 2 } \\dot { \\Omega } _ { 8 - p } ^ { 2 } ) } - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] , \\end{equation*}",
"\\begin{equation*} \\widetilde { f } ^ { - 1 } d u ^ { 2 } + u ^ { 2 } d \\Omega _ { 8 - p } ^ { 2 } = u ^ { 2 } \\rho ^ { - 2 } ( d \\rho ^ { 2 } + \\rho ^ { 2 } d \\Omega _ { 8 - p } ^ { 2 } ) , \\end{equation*}"
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"\\begin{equation*} \\mitS _ { \\mitp } = - \\frac { \\mitT _ { \\mitp } \\mitV _ { \\mitp } } { \\mitg \\operatorname { c o s } \\mittheta } \\int \\mitd \\mittau \\mitH ^ { - 1 } [ \\sqrt { \\mitf - \\mitH ( \\mitf ^ { - 1 } \\dot { \\mitr } ^ { 2 } + \\mitr ^ { 2 } \\dot { \\mupOmega } _ { 8 - \\mitp } ^ { 2 } ) } - 1 + \\mitH _ { 0 } - \\mitH ] , \\end{equation*}",
"\\begin{equation*} \\mitS _ { \\mitp } = - \\mitm _ { \\mitp } \\int \\mitd \\mittau { \\left( \\frac { \\mitu } { \\mitR } \\right) } ^ { 7 - \\mitp } \\left[ \\sqrt { \\tilde { \\mitf } - { \\left( \\frac { \\mitR } { \\mitu } \\right) } ^ { 7 - \\mitp } \\left( \\tilde { \\mitf } ^ { - 1 } \\dot { \\mitu } ^ { 2 } + \\mitu ^ { 2 } \\dot { \\mupOmega } _ { 8 - \\mitp } ^ { 2 } \\right) } - 1 + \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } \\right] , \\end{equation*}",
"\\begin{equation*} \\tilde { \\mitf } ^ { - 1 } \\mitd \\mitu ^ { 2 } + \\mitu ^ { 2 } \\mitd \\mupOmega _ { 8 - \\mitp } ^ { 2 } = \\mitu ^ { 2 } \\mitrho ^ { - 2 } ( \\mitd \\mitrho ^ { 2 } + \\mitrho ^ { 2 } \\mitd \\mupOmega _ { 8 - \\mitp } ^ { 2 } ) , \\end{equation*}"
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0001213_page19 | {
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"\\begin {equation} \\tilde {f}^{-1}\\dot {u}^2 + u^2 \\dot {\\Omega }_{8-p}^2 \\equiv u^2{\\rho }^{-2}v^2. \\end {equation}",
"\\begin {equation} S_p =\\int d\\tau [\\frac {1}{2}m_p v^2 -{\\cal V}(\\rho ,v) +{\\cal O}(1/\\rho ^{2(7-p)})], \\end {equation}",
"\\begin {equation} \\label {4e7} {\\cal V}(\\rho , v)=-m_p\\frac {u_0^{7-p}}{\\rho ^{7-p}} \\left \\{ \\frac {9-p}{4(7-p)} v^2 +\\frac {1}{8}\\left [\\left (\\frac {u_0}{R}\\right )^{7-p} + \\left (\\frac {R}{u_0}\\right )^{7-p}v^4 \\right ] \\right \\} \\end {equation}",
"\\begin {equation} \\delta (\\rho ,v) =-\\int ^{\\infty }_0 d\\tau {\\cal V}[\\rho (\\tau ),v], \\ \\ \\ \\ \\rho ^2(\\tau ) =\\rho ^2 +v^2 \\tau ^2, \\end {equation}",
"\\begin {equation} \\delta (\\rho ,v) = -\\frac {B\\left (\\frac {1}{2}, \\frac {6-p}{2}\\right )} {2v \\rho ^{6-p}} \\lambda (v), \\end {equation}",
"\\begin {equation} L=\\frac {m_p u^2 \\dot {\\phi }}{ \\sqrt {\\tilde {f} -\\left (\\frac {R}{u}\\right )^{7-p} \\left (\\tilde {f}^{-1}\\dot {u}^2 +u^2\\dot {\\phi }^2\\right )} }. \\end {equation}"
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"\\begin{equation*} \\widetilde { f } ^ { - 1 } \\dot { u } ^ { 2 } + u ^ { 2 } \\dot { \\Omega } _ { 8 - p } ^ { 2 } \\equiv u ^ { 2 } \\rho ^ { - 2 } v ^ { 2 } . \\end{equation*}",
"\\begin{equation*} S _ { p } = \\int d \\tau [ \\frac { 1 } { 2 } m _ { p } v ^ { 2 } - V ( \\rho , v ) + O ( 1 \\slash \\rho ^ { 2 ( 7 - p ) } ) ] , \\end{equation*}",
"\\begin{equation*} V ( \\rho , v ) = - m _ { p } \\frac { u _ { 0 } ^ { 7 - p } } { \\rho ^ { 7 - p } } \\{ \\frac { 9 - p } { 4 ( 7 - p ) } v ^ { 2 } + \\frac { 1 } { 8 } [ { ( \\frac { u _ { 0 } } { R } ) } ^ { 7 - p } + { ( \\frac { R } { u _ { 0 } } ) } ^ { 7 - p } v ^ { 4 } ] \\} \\end{equation*}",
"\\begin{equation*} \\delta ( \\rho , v ) = - \\int _ { 0 } ^ { \\infty } d \\tau V [ \\rho ( \\tau ) , v ] , ~ ~ ~ ~ \\rho ^ { 2 } ( \\tau ) = \\rho ^ { 2 } + v ^ { 2 } \\tau ^ { 2 } , \\end{equation*}",
"\\begin{equation*} \\delta ( \\rho , v ) = - \\frac { B ( \\frac { 1 } { 2 } , \\frac { 6 - p } { 2 } ) } { 2 v \\rho ^ { 6 - p } } \\lambda ( v ) , \\end{equation*}",
"\\begin{equation*} L = \\frac { m _ { p } u ^ { 2 } \\dot { \\phi } } { \\sqrt { \\widetilde { f } - { ( \\frac { R } { u } ) } ^ { 7 - p } ( \\widetilde { f } ^ { - 1 } \\dot { u } ^ { 2 } + u ^ { 2 } \\dot { \\phi } ^ { 2 } ) } } . \\end{equation*}"
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"\\begin{equation*} \\mitS _ { \\mitp } = \\int \\mitd \\mittau [ \\frac { 1 } { 2 } \\mitm _ { \\mitp } \\mitv ^ { 2 } - \\mitV ( \\mitrho , \\mitv ) + \\mitO ( 1 \\slash \\mitrho ^ { 2 ( 7 - \\mitp ) } ) ] , \\end{equation*}",
"\\begin{equation*} \\mitV ( \\mitrho , \\mitv ) = - \\mitm _ { \\mitp } \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { \\mitrho ^ { 7 - \\mitp } } \\left\\{ \\frac { 9 - \\mitp } { 4 ( 7 - \\mitp ) } \\mitv ^ { 2 } + \\frac { 1 } { 8 } \\left[ { \\left( \\frac { \\mitu _ { 0 } } { \\mitR } \\right) } ^ { 7 - \\mitp } + { \\left( \\frac { \\mitR } { \\mitu _ { 0 } } \\right) } ^ { 7 - \\mitp } \\mitv ^ { 4 } \\right] \\right\\} \\end{equation*}",
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0001213_page20 | {
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"\\begin {equation} E= \\frac {1}{2}m_p \\dot {u}^2 + V(u), \\end {equation}",
"\\begin {equation} \\label {4e14} V(u)=E\\left [1 -\\frac {m_p \\tilde {f}^2}{2E}\\left (\\frac {u}{R}\\right )^{7-p} \\left (1-\\frac {\\tilde {f}}{{\\cal A}^2}\\right )\\right ] +\\frac {L^2 \\tilde {f}^3}{2m_p u^2 {\\cal A}^2}. \\end {equation}",
"\\begin {equation} {\\cal A}= 1-\\frac {u_0^{7-p}}{2u^{7-p}}+\\frac {E}{m_p} \\left (\\frac {R}{u}\\right )^{7-p}. \\end {equation}",
"\\begin {equation} V(u)=E\\left \\{1-\\frac {1}{2}\\left (\\frac {u}{u_*}\\right )^{7-p} \\left [1-\\frac {1}{\\left (1+(u_*/u)^{7-p}\\right )^2}\\right ] \\right \\} +\\frac {Eu_{**}^2}{2u^2 \\left (1+(u_*/u)^{7-p}\\right )^2}, \\end {equation}",
"\\begin {equation} u_*=R\\left (\\frac {E}{m_p}\\right )^{1/(7-p)}, \\ \\ \\^^Mu_{**} = L\\left (\\frac {1}{m_pE}\\right )^{1/2}. \\end {equation}",
"\\begin {equation} 2 +\\left (\\frac {u_*}{u_c}\\right )^{7-p} =\\left (\\frac {u_{**}}{u_c}\\right )^2. \\end {equation}",
"\\begin {equation} u_c= \\left (\\frac {u_*^{7-p}}{u_{**}^2}\\right )^{1/(5-p)}. \\end {equation}"
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"\\begin{equation*} E = \\frac { 1 } { 2 } m _ { p } \\dot { u } ^ { 2 } + V ( u ) , \\end{equation*}",
"\\begin{equation*} V ( u ) = E [ 1 - \\frac { m _ { p } \\widetilde { f } ^ { 2 } } { 2 E } { ( \\frac { u } { R } ) } ^ { 7 - p } ( 1 - \\frac { \\widetilde { f } } { A ^ { 2 } } ) ] + \\frac { L ^ { 2 } \\widetilde { f } ^ { 3 } } { 2 m _ { p } u ^ { 2 } A ^ { 2 } } . \\end{equation*}",
"\\begin{equation*} A = 1 - \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } + \\frac { E } { m _ { p } } { ( \\frac { R } { u } ) } ^ { 7 - p } . \\end{equation*}",
"\\begin{equation*} V ( u ) = E \\{ 1 - \\frac { 1 } { 2 } { ( \\frac { u } { u _ { \\ast } } ) } ^ { 7 - p } [ 1 - \\frac { 1 } { { ( 1 + ( u _ { \\ast } \\slash u ) ^ { 7 - p } ) } ^ { 2 } } ] \\} + \\frac { E u _ { \\ast \\ast } ^ { 2 } } { 2 u ^ { 2 } { ( 1 + ( u _ { \\ast } \\slash u ) ^ { 7 - p } ) } ^ { 2 } } , \\end{equation*}",
"\\begin{equation*} u _ { \\ast } = R { ( \\frac { E } { m _ { p } } ) } ^ { 1 \\slash ( 7 - p ) } , ~ ~ ~ u _ { \\ast \\ast } = L { ( \\frac { 1 } { m _ { p } E } ) } ^ { 1 \\slash 2 } . \\end{equation*}",
"\\begin{equation*} 2 + { ( \\frac { u _ { \\ast } } { u _ { c } } ) } ^ { 7 - p } = { ( \\frac { u _ { \\ast \\ast } } { u _ { c } } ) } ^ { 2 } . \\end{equation*}",
"\\begin{equation*} u _ { c } = { ( \\frac { u _ { \\ast } ^ { 7 - p } } { u _ { \\ast \\ast } ^ { 2 } } ) } ^ { 1 \\slash ( 5 - p ) } . \\end{equation*}"
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"\\begin{equation*} \\mitE = \\frac { 1 } { 2 } \\mitm _ { \\mitp } \\dot { \\mitu } ^ { 2 } + \\mitV ( \\mitu ) , \\end{equation*}",
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"\\begin{equation*} \\mitA = 1 - \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } + \\frac { \\mitE } { \\mitm _ { \\mitp } } { \\left( \\frac { \\mitR } { \\mitu } \\right) } ^ { 7 - \\mitp } . \\end{equation*}",
"\\begin{equation*} \\mitV ( \\mitu ) = \\mitE \\left\\{ 1 - \\frac { 1 } { 2 } { \\left( \\frac { \\mitu } { \\mitu _ { \\ast } } \\right) } ^ { 7 - \\mitp } \\left[ 1 - \\frac { 1 } { { \\left( 1 + ( \\mitu _ { \\ast } \\slash \\mitu ) ^ { 7 - \\mitp } \\right) } ^ { 2 } } \\right] \\right\\} + \\frac { \\mitE \\mitu _ { \\ast \\ast } ^ { 2 } } { 2 \\mitu ^ { 2 } { \\left( 1 + ( \\mitu _ { \\ast } \\slash \\mitu ) ^ { 7 - \\mitp } \\right) } ^ { 2 } } , \\end{equation*}",
"\\begin{equation*} \\mitu _ { \\ast } = \\mitR { \\left( \\frac { \\mitE } { \\mitm _ { \\mitp } } \\right) } ^ { 1 \\slash ( 7 - \\mitp ) } , ~ ~ ~ \\mitu _ { \\ast \\ast } = \\mitL { \\left( \\frac { 1 } { \\mitm _ { \\mitp } \\mitE } \\right) } ^ { 1 \\slash 2 } . \\end{equation*}",
"\\begin{equation*} 2 + { \\left( \\frac { \\mitu _ { \\ast } } { \\mitu _ { \\mitc } } \\right) } ^ { 7 - \\mitp } = { \\left( \\frac { \\mitu _ { \\ast \\ast } } { \\mitu _ { \\mitc } } \\right) } ^ { 2 } . \\end{equation*}",
"\\begin{equation*} \\mitu _ { \\mitc } = { \\left( \\frac { \\mitu _ { \\ast } ^ { 7 - \\mitp } } { \\mitu _ { \\ast \\ast } ^ { 2 } } \\right) } ^ { 1 \\slash ( 5 - \\mitp ) } . \\end{equation*}"
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|
0001213_page21 | {
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"\\begin {eqnarray} S_{p-2} &=& - \\frac {T_{p-2}V_{p-2}}{g}\\int d\\tau H^{-1}\\left [(Hh^{-1})^{1/2} \\sqrt {f -h(\\dot {x}_{p-1}^2 +\\dot {x}_p^2) -H(f^{-1}\\dot {r}^2 + r^2 \\dot {\\Omega }_{8-p}^2)} \\right . \\\\ && \\hspace {4cm} - \\left . (1-H_0)\\sin \\theta -H\\right ], \\end {eqnarray}",
"\\begin {eqnarray} S_{p-2} &=& - \\; m_{p-2} \\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p} \\left [\\sqrt {\\frac {1}{(au)^{7-p}\\tilde {h} }} \\sqrt {\\tilde {f} -\\tilde {h}\\left (\\dot {\\tilde x}_{p-1}^2 +\\dot {\\tilde x}_p^2 \\right ) -\\left (\\frac {R}{u}\\right )^{7-p} \\left (\\tilde {f}^{-1} \\dot {u}^2 +u^2\\dot {\\Omega }_{8-p}^2 \\right )} \\right . \\\\ && \\hspace {4cm} - \\left . 1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {eqnarray}",
"\\begin {eqnarray} S_{p-2} &=& - m_{p-2} \\int d\\tau \\left (\\frac {u}{R}\\right )^{7-p} \\left [ \\sqrt {\\tilde {f} -\\frac {1}{(au)^{7-p}}\\left (\\dot {\\tilde x}_{p-1}^2 +\\dot {\\tilde x}_p^2 \\right ) -\\left (\\frac {R}{u}\\right )^{7-p} \\left (\\tilde {f}^{-1} \\dot {u}^2 +u^2\\dot {\\Omega }_{8-p}^2 \\right )} \\right . \\\\ && \\hspace {4cm} - \\left . 1 +\\frac {u_0^{7-p}}{2u^{7-p}}\\right ]. \\end {eqnarray}"
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"\\begin{align*} S _ { p - 2 } & = & - \\; m _ { p - 2 } \\int d \\tau { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\frac { 1 } { ( a u ) ^ { 7 - p } \\widetilde { h } } } \\sqrt { \\widetilde { f } - \\widetilde { h } ( \\dot { \\widetilde { x } } _ { p - 1 } ^ { 2 } + \\dot { \\widetilde { x } } _ { p } ^ { 2 } ) - { ( \\frac { R } { u } ) } ^ { 7 - p } ( \\widetilde { f } ^ { - 1 } \\dot { u } ^ { 2 } + u ^ { 2 } \\dot { \\Omega } _ { 8 - p } ^ { 2 } ) } \\\\ & & \\hspace{113.81pt} - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] . \\end{align*}",
"\\begin{align*} S _ { p - 2 } & = & - m _ { p - 2 } \\int d \\tau { ( \\frac { u } { R } ) } ^ { 7 - p } [ \\sqrt { \\widetilde { f } - \\frac { 1 } { ( a u ) ^ { 7 - p } } ( \\dot { \\widetilde { x } } _ { p - 1 } ^ { 2 } + \\dot { \\widetilde { x } } _ { p } ^ { 2 } ) - { ( \\frac { R } { u } ) } ^ { 7 - p } ( \\widetilde { f } ^ { - 1 } \\dot { u } ^ { 2 } + u ^ { 2 } \\dot { \\Omega } _ { 8 - p } ^ { 2 } ) } \\\\ & & \\hspace{113.81pt} - 1 + \\frac { u _ { 0 } ^ { 7 - p } } { 2 u ^ { 7 - p } } ] . \\end{align*}"
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"\\begin{align*} \\mitS _ { \\mitp - 2 } & = & - \\mitm _ { \\mitp - 2 } \\int \\mitd \\mittau { \\left( \\frac { \\mitu } { \\mitR } \\right) } ^ { 7 - \\mitp } \\left[ \\sqrt { \\tilde { \\mitf } - \\frac { 1 } { ( \\mita \\mitu ) ^ { 7 - \\mitp } } \\left( \\dot { \\tilde { \\mitx } } _ { \\mitp - 1 } ^ { 2 } + \\dot { \\tilde { \\mitx } } _ { \\mitp } ^ { 2 } \\right) - { \\left( \\frac { \\mitR } { \\mitu } \\right) } ^ { 7 - \\mitp } \\left( \\tilde { \\mitf } ^ { - 1 } \\dot { \\mitu } ^ { 2 } + \\mitu ^ { 2 } \\dot { \\mupOmega } _ { 8 - \\mitp } ^ { 2 } \\right) } \\right. \\\\ & & \\hspace{113.81pt} - \\left. 1 + \\frac { \\mitu _ { 0 } ^ { 7 - \\mitp } } { 2 \\mitu ^ { 7 - \\mitp } } \\right] . \\end{align*}"
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0001213_page22 | {
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0001213_page23 | {
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"\\begin{equation*} \\mitF _ { \\mitn + 1 } ( \\mitz \\mitg ) = \\left\\{ \\sum _ { \\mitr = 1 } ^ { \\mitn } \\mitg _ { \\mitr + 1 } \\frac { \\mitpartial } { \\mitpartial \\mitg _ { \\mitr } } + \\mitz \\mitg _ { 1 } \\right\\} \\mitF _ { \\mitn } ( \\mitz \\mitg ) . \\end{equation*}",
"\\begin{equation*} \\mitB _ { \\mitn \\mitj } = \\mitB _ { \\mitn \\mitj } [ \\mitg ] = \\mitB _ { \\mitn \\mitj } ( \\mitg _ { 1 } , \\cdots , \\mitg _ { \\mitn - \\mitj + 1 } ) \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\mitn } ( \\mitz \\mitg _ { 1 } , \\cdots , \\mitz \\mitg _ { \\mitn } ) = \\sum _ { \\mitj \\geq 0 } \\mitz ^ { \\mitj } \\mitB _ { \\mitn \\mitj } ( \\mitg _ { 1 } , \\cdots , \\mitg _ { \\mitn - \\mitj + 1 } ) . \\end{equation*}",
"\\begin{equation*} \\mitB _ { \\mitn 0 } = \\mitdelta _ { \\mitn 0 } , ~ \\mitB _ { \\mitn \\mitj } = 0 ~ ( \\mitn < \\mitj ) . \\end{equation*}",
"\\begin{equation*} \\mitB _ { \\mitj \\mitk } [ \\mitf ( \\mitg ( \\mitx ) ) ] = \\sum _ { \\mitn = \\mitk } ^ { \\mitj } \\mitB _ { \\mitj \\mitn } [ \\mitg ] \\mitB _ { \\mitn \\mitk } [ \\mitf ] \\end{equation*}",
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|
0002003_page04 | {
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"$k=1,\\cdots , p-i$",
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"$(p,i)$",
"$(p,i)=(2,0);$",
"${\\mathbf {C}}P^1$",
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"$\\cal {P}_{\\mbox {B}}$",
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"\\begin {equation} \\sum _{\\mu }{}'\\partial _{\\mu }^{p-i}(u^k) \\partial _{\\mu }^i(\\bar {u}^l)=0 \\label {eqn:4-4} \\end {equation}",
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"\\begin {equation} \\sum _{\\mu }{}'\\partial _{\\mu }^2 u \\partial _{\\mu } \\bar {u}=0, \\quad \\sum _{\\mu }{}'\\partial _{\\mu }^2 (u^2) \\partial _{\\mu } \\bar {u}=0. \\label {eqn:3-1sub} \\end {equation}",
"\\begin {equation} \\Phi : \\cal {P}_{\\mbox {B}} \\rightarrow \\mathbf {C}[\\xi , \\bar {\\xi }], \\end {equation}",
"\\begin {equation} \\Phi (F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g}))=\\xi ^n \\bar {\\xi }^m. \\end {equation}",
"$$ F_n(zg)=\\mbox {e}^{-zg(x)} \\partial _x^n \\mbox {e}^{zg(x)} \\longmapsto \\mbox {e}^{-zg(x)} {\\xi }^n \\mbox {e}^{zg(x)}=\\xi ^n . $$",
"\\begin {equation} \\partial \\equiv \\sum _{r=1}^{\\infty } \\left ( g_{r+1} \\frac {\\partial }{\\partial g_r} + \\bar {g}_{r+1} \\frac {\\partial }{\\partial \\bar {g}_r} \\right ) +zg_1+\\bar {z}\\bar {g}_1 \\end {equation}"
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"\\begin{equation*} \\sum _ { \\mu } { } ^ { \\prime } \\partial _ { \\mu } ^ { 2 } u \\partial _ { \\mu } \\bar { u } = 0 , \\quad \\sum _ { \\mu } { } ^ { \\prime } \\partial _ { \\mu } ^ { 2 } ( u ^ { 2 } ) \\partial _ { \\mu } \\bar { u } = 0 . \\end{equation*}",
"\\begin{equation*} \\Phi : P _ { B } \\rightarrow C [ \\xi , \\bar { \\xi } ] , \\end{equation*}",
"\\begin{equation*} \\Phi ( F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) ) = \\xi ^ { n } \\bar { \\xi } ^ { m } . \\end{equation*}",
"\\begin{equation*} F _ { n } ( z g ) = e ^ { - z g ( x ) } \\partial _ { x } ^ { n } e ^ { z g ( x ) } \\longmapsto e ^ { - z g ( x ) } \\xi ^ { n } e ^ { z g ( x ) } = \\xi ^ { n } . \\end{equation*}",
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"\\begin{equation*} \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 2 } \\mitu = 0 , \\quad \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 2 } ( \\mitu ^ { 2 } ) = 0 . \\end{equation*}",
"\\begin{equation*} \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 3 } \\mitu = 0 , \\quad \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 3 } ( \\mitu ^ { 2 } ) = 0 , \\quad \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 3 } ( \\mitu ^ { 3 } ) = 0 , \\end{equation*}",
"\\begin{equation*} \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 2 } \\mitu \\mitpartial _ { \\mitmu } \\bar { \\mitu } = 0 , \\quad \\sum _ { \\mitmu } { } ^ { \\prime } \\mitpartial _ { \\mitmu } ^ { 2 } ( \\mitu ^ { 2 } ) \\mitpartial _ { \\mitmu } \\bar { \\mitu } = 0 . \\end{equation*}",
"\\begin{equation*} \\mupPhi : \\mitP _ { \\mathrm { B } } \\rightarrow \\mbfC [ \\mitxi , \\bar { \\mitxi } ] , \\end{equation*}",
"\\begin{equation*} \\mupPhi ( \\mitF _ { \\mitn } ( \\mitz \\mitg ) \\bar { \\mitF } _ { \\mitm } ( \\bar { \\mitz } \\bar { \\mitg } ) ) = \\mitxi ^ { \\mitn } \\bar { \\mitxi } ^ { \\mitm } . \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\mitn } ( \\mitz \\mitg ) = \\mathrm { e } ^ { - \\mitz \\mitg ( \\mitx ) } \\mitpartial _ { \\mitx } ^ { \\mitn } \\mathrm { e } ^ { \\mitz \\mitg ( \\mitx ) } \\longmapsto \\mathrm { e } ^ { - \\mitz \\mitg ( \\mitx ) } \\mitxi ^ { \\mitn } \\mathrm { e } ^ { \\mitz \\mitg ( \\mitx ) } = \\mitxi ^ { \\mitn } . \\end{equation*}",
"\\begin{equation*} \\mitpartial \\equiv \\sum _ { \\mitr = 1 } ^ { \\infty } \\left( \\mitg _ { \\mitr + 1 } \\frac { \\mitpartial } { \\mitpartial \\mitg _ { \\mitr } } + \\bar { \\mitg } _ { \\mitr + 1 } \\frac { \\mitpartial } { \\mitpartial \\bar { \\mitg } _ { \\mitr } } \\right) + \\mitz \\mitg _ { 1 } + \\bar { \\mitz } \\bar { \\mitg } _ { 1 } \\end{equation*}"
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|
0002003_page05 | {
"latex": [
"$\\cal {P}_{\\mbox {B}}$",
"$F_n$",
"$n$",
"$\\Phi $",
"$\\mu \\in \\{ 0,\\cdots ,n \\}$",
"$x=x_{\\mu }$",
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"$g_r=\\partial _{\\mu }^r u$",
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"$\\bar {F}_{n,\\, \\mu }$",
"$F_{n,\\, \\mu }$",
"$: \\ :$",
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"\\begin {equation} \\partial (F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g})) = F_{n+1}(zg)\\bar {F}_m(\\bar {z}\\bar {g}) +F_n(zg)\\bar {F}_{m+1}(\\bar {z}\\bar {g}). \\label {eqn:der} \\end {equation}",
"\\begin {eqnarray} && \\hspace {-5mm} \\partial (F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g})) \\\\ &=& \\left \\{ \\sum _{r=1}^{n} g_{r+1} \\frac {\\partial }{\\partial g_r} +\\sum _{r=1}^{m} \\bar {g}_{r+1} \\frac {\\partial }{\\partial \\bar {g}_r} +zg_1+\\bar {z}\\bar {g}_1 \\right \\} F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g}) \\\\ &=& \\sum _{r=1}^{n} g_{r+1} \\frac {\\partial F_n(zg)}{\\partial g_r}\\bar {F}_m(\\bar {z}\\bar {g}) +zg_1 F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g}) \\\\ && \\quad +F_n(zg) \\sum _{r=1}^{m} \\bar {g}_{r+1} \\frac {\\partial \\bar {F}_m(\\bar {z}\\bar {g})}{\\partial \\bar {g}_r} +\\bar {z}\\bar {g}_1 F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g}) \\\\ &=& F_{n+1}(zg)\\bar {F}_m(\\bar {z}\\bar {g}) +F_n(zg)\\bar {F}_{m+1}(\\bar {z}\\bar {g}). \\end {eqnarray}",
"\\begin {equation} \\Phi \\circ \\partial \\circ \\Phi ^{-1} = (\\xi + \\bar {\\xi } ), \\label {eqn:opes} \\end {equation}",
"\\begin {equation} F_n \\bar {F}_m \\quad \\mbox {with} \\quad \\xi ^n \\bar {\\xi }^m \\quad \\mbox {and} \\quad \\partial \\quad \\mbox {with} \\quad (\\xi +\\bar {\\xi }) \\ . \\label {eqn:iden} \\end {equation}",
"\\begin {eqnarray} F_{n,\\, \\mu } &\\equiv & :F_n(zg_1, \\cdots ,zg_n)|_{z=\\frac {\\partial }{\\partial u}}:\\\\ &=& :F_n(\\partial _{\\mu }u \\frac {\\partial }{\\partial u}, \\partial _{\\mu }^2 u \\frac {\\partial }{\\partial u}, \\cdots , \\partial _{\\mu }^n u \\frac {\\partial }{\\partial u}):\\\\ &=& \\hspace {-10mm} \\sum _{{\\scriptstyle k_1+2k_2+ \\cdots +nk_n=n}\\atop {\\scriptstyle k_1 \\geq 0,\\cdots ,k_n \\geq 0}} \\hspace {-1mm} \\frac {n!}{k_1! \\cdots k_n!} \\left ( \\frac {\\partial _{\\mu }u}{1!} \\right )^{k_1} \\hspace {-2mm} \\left ( \\frac {\\partial _{\\mu }^2 u}{2!} \\right )^{k_2} \\hspace {-3mm} \\cdots \\left ( \\frac {\\partial _{\\mu }^n u}{n!} \\right )^{k_n} \\hspace {-2mm} \\left ( \\frac {\\partial }{\\partial u} \\right )^{k_1+k_2+ \\cdots +k_n} \\\\ && \\end {eqnarray}",
"\\begin {equation} \\partial _{\\mu } :F_{n,\\, \\mu }\\bar {F}_{m,\\, \\mu }:f(u,\\bar {u}) \\ =\\^^M:\\partial (F_n(zg)\\bar {F}_m(\\bar {z}\\bar {g})) |_{z=\\frac {\\partial }{\\partial u}}:f(u,\\bar {u}). \\end {equation}"
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"\\begin{align*} & & \\hspace{-14.23pt} \\partial ( F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) ) \\\\ & = & \\{ \\sum _ { r = 1 } ^ { n } g _ { r + 1 } \\frac { \\partial } { \\partial g _ { r } } + \\sum _ { r = 1 } ^ { m } \\bar { g } _ { r + 1 } \\frac { \\partial } { \\partial \\bar { g } _ { r } } + z g _ { 1 } + \\bar { z } \\bar { g } _ { 1 } \\} F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) \\\\ & = & \\sum _ { r = 1 } ^ { n } g _ { r + 1 } \\frac { \\partial F _ { n } ( z g ) } { \\partial g _ { r } } \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) + z g _ { 1 } F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) \\\\ & & \\quad + F _ { n } ( z g ) \\sum _ { r = 1 } ^ { m } \\bar { g } _ { r + 1 } \\frac { \\partial \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) } { \\partial \\bar { g } _ { r } } + \\bar { z } \\bar { g } _ { 1 } F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) \\\\ & = & F _ { n + 1 } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) + F _ { n } ( z g ) \\bar { F } _ { m + 1 } ( \\bar { z } \\bar { g } ) . \\end{align*}",
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"\\begin{equation*} \\partial _ { \\mu } : F _ { n , \\, \\mu } \\bar { F } _ { m , \\, \\mu } : f ( u , \\bar { u } ) ~ = ~ : \\partial ( F _ { n } ( z g ) \\bar { F } _ { m } ( \\bar { z } \\bar { g } ) ) \\vert _ { z = \\frac { \\partial } { \\partial u } } : f ( u , \\bar { u } ) . \\end{equation*}"
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0002003_page06 | {
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"$\\partial _{\\mu }$",
"$g_r=\\partial _{\\mu }^r u$",
"$z=\\frac {\\partial }{\\partial u}$",
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"\\begin {equation} h(u, \\partial _{\\mu } u, \\cdots , \\partial _{\\mu }^n u;\\^^M\\bar {u}, \\partial _{\\mu } \\bar {u}, \\cdots , \\partial _{\\mu }^m \\bar {u}), \\end {equation}",
"\\begin {equation} \\partial _{\\mu } = \\sum _{r=1}^n \\partial _{\\mu }^{r+1}u \\frac {\\partial }{\\partial (\\partial _{\\mu }^r u)} +\\sum _{r=1}^m \\partial _{\\mu }^{r+1}\\bar {u} \\frac {\\partial }{\\partial (\\partial _{\\mu }^r \\bar {u})} +\\partial _{\\mu }u \\frac {\\partial }{\\partial u} +\\partial _{\\mu }\\bar {u} \\frac {\\partial }{\\partial \\bar {u}}. \\end {equation}",
"\\begin {equation} \\sum _{\\mu }{}':F_{p-i,\\, \\mu }\\bar {F}_{i,\\, \\mu }:=0. \\end {equation}",
"\\begin {eqnarray*} && \\sum _{\\mu }{}'\\ \\partial _{\\mu }^{p-i}(u^k)\\partial _{\\mu }^i(\\bar {u}^l) \\\\ &=& \\sum _{\\mu }{}'\\ \\sum _{j_1=1}^{{}p-i} B_{p-i,j_1}(g_1,\\cdots ,g_{p-i-j_1+1}) \\left ( \\frac {\\partial }{\\partial u} \\right )^{j_1} (u^k) \\\\ && \\qquad \\times \\sum _{j_2=0}^{i} B_{i,j_2}(\\bar {g}_1,\\cdots ,\\bar {g}_{i-j_2+1}) \\left ( \\frac {\\partial }{\\partial \\bar {u}} \\right )^{j_2} (\\bar {u}^l) \\\\ &=& \\sum _{j_1=1}^{{}p-i} \\sum _{j_2=0}^{i} j_1! j_2! \\left ( \\begin {array}{cc} k \\\\ j_1 \\end {array} \\right ) \\left ( \\begin {array}{cc} l \\\\ j_2 \\end {array} \\right ) \\\\ && \\qquad \\times \\sum _{\\mu }{}'\\ B_{p-i,j_1}(g_1,\\cdots ,g_{p-i-j_1+1}) B_{i,j_2}(\\bar {g}_1,\\cdots ,\\bar {g}_{i-j_2+1}) u^{k-j_1} \\bar {u}^{l-j_2} \\\\ && \\quad \\mbox {for} \\quad k=1,\\cdots , p-i, \\ l=0,\\cdots ,i. \\end {eqnarray*}",
"\\begin {eqnarray*} && \\sum _{\\mu }{}'\\ B_{p-i,j_1}(g_1,\\cdots ,g_{p-i-j_1+1}) B_{i,j_2}(\\bar {g}_1,\\cdots ,\\bar {g}_{i-j_2+1})=0 \\\\ && \\quad \\mbox {for} \\quad j_1=1,\\cdots , p-i, \\ j_2=0,\\cdots ,i, \\end {eqnarray*}",
"$$ \\hspace {45mm} \\sum _{\\mu }{}':F_{p-i,\\, \\mu }\\bar {F}_{i,\\, \\mu }:=0. \\hspace {45mm} \\qed $$"
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"\\begin{equation*} \\partial _ { \\mu } = \\sum _ { r = 1 } ^ { n } \\partial _ { \\mu } ^ { r + 1 } u \\frac { \\partial } { \\partial ( \\partial _ { \\mu } ^ { r } u ) } + \\sum _ { r = 1 } ^ { m } \\partial _ { \\mu } ^ { r + 1 } \\bar { u } \\frac { \\partial } { \\partial ( \\partial _ { \\mu } ^ { r } \\bar { u } ) } + \\partial _ { \\mu } u \\frac { \\partial } { \\partial u } + \\partial _ { \\mu } \\bar { u } \\frac { \\partial } { \\partial \\bar { u } } . \\end{equation*}",
"\\begin{equation*} \\sum _ { \\mu } { } ^ { \\prime } : F _ { p - i , \\, \\mu } \\bar { F } _ { i , \\, \\mu } : = 0 . \\end{equation*}",
"\\begin{align*} & & \\sum _ { \\mu } { } ^ { \\prime } ~ \\partial _ { \\mu } ^ { p - i } ( u ^ { k } ) \\partial _ { \\mu } ^ { i } ( \\bar { u } ^ { l } ) \\\\ & = & \\sum _ { \\mu } { } ^ { \\prime } ~ \\sum _ { j _ { 1 } = 1 } ^ { p - i } B _ { p - i , j _ { 1 } } ( g _ { 1 } , \\cdots , g _ { p - i - j _ { 1 } + 1 } ) { ( \\frac { \\partial } { \\partial u } ) } ^ { j _ { 1 } } ( u ^ { k } ) \\\\ & & \\qquad \\times \\sum _ { j _ { 2 } = 0 } ^ { i } B _ { i , j _ { 2 } } ( \\bar { g } _ { 1 } , \\cdots , \\bar { g } _ { i - j _ { 2 } + 1 } ) { ( \\frac { \\partial } { \\partial \\bar { u } } ) } ^ { j _ { 2 } } ( \\bar { u } ^ { l } ) \\\\ & = \\\\ \\\\ \\\\ \\\\ & & \\qquad \\times \\sum _ { \\mu } { } ^ { \\prime } ~ B _ { p - i , j _ { 1 } } ( g _ { 1 } , \\cdots , g _ { p - i - j _ { 1 } + 1 } ) B _ { i , j _ { 2 } } ( \\bar { g } _ { 1 } , \\cdots , \\bar { g } _ { i - j _ { 2 } + 1 } ) u ^ { k - j _ { 1 } } \\bar { u } ^ { l - j _ { 2 } } \\\\ & & \\quad f o r \\quad k = 1 , \\cdots , p - i , ~ l = 0 , \\cdots , i . \\end{align*}",
"\\begin{align*} & & \\sum _ { \\mu } { } ^ { \\prime } ~ B _ { p - i , j _ { 1 } } ( g _ { 1 } , \\cdots , g _ { p - i - j _ { 1 } + 1 } ) B _ { i , j _ { 2 } } ( \\bar { g } _ { 1 } , \\cdots , \\bar { g } _ { i - j _ { 2 } + 1 } ) = 0 \\\\ & & \\quad f o r \\quad j _ { 1 } = 1 , \\cdots , p - i , ~ j _ { 2 } = 0 , \\cdots , i , \\end{align*}",
"\\begin{equation*} \\hspace{128.04pt} \\sum _ { \\mu } { } ^ { \\prime } : F _ { p - i , \\, \\mu } \\bar { F } _ { i , \\, \\mu } : = 0 . \\hspace{128.04pt} \\quad \\square \\end{equation*}"
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"\\begin{equation*} \\mitpartial _ { \\mitmu } = \\sum _ { \\mitr = 1 } ^ { \\mitn } \\mitpartial _ { \\mitmu } ^ { \\mitr + 1 } \\mitu \\frac { \\mitpartial } { \\mitpartial ( \\mitpartial _ { \\mitmu } ^ { \\mitr } \\mitu ) } + \\sum _ { \\mitr = 1 } ^ { \\mitm } \\mitpartial _ { \\mitmu } ^ { \\mitr + 1 } \\bar { \\mitu } \\frac { \\mitpartial } { \\mitpartial ( \\mitpartial _ { \\mitmu } ^ { \\mitr } \\bar { \\mitu } ) } + \\mitpartial _ { \\mitmu } \\mitu \\frac { \\mitpartial } { \\mitpartial \\mitu } + \\mitpartial _ { \\mitmu } \\bar { \\mitu } \\frac { \\mitpartial } { \\mitpartial \\bar { \\mitu } } . \\end{equation*}",
"\\begin{equation*} \\sum _ { \\mitmu } { } ^ { \\prime } : \\mitF _ { \\mitp - \\miti , \\, \\mitmu } \\bar { \\mitF } _ { \\miti , \\, \\mitmu } : = 0 . \\end{equation*}",
"\\begin{align*} & & \\sum _ { \\mitmu } { } ^ { \\prime } ~ \\mitpartial _ { \\mitmu } ^ { \\mitp - \\miti } ( \\mitu ^ { \\mitk } ) \\mitpartial _ { \\mitmu } ^ { \\miti } ( \\bar { \\mitu } ^ { \\mitl } ) \\\\ & = & \\sum _ { \\mitmu } { } ^ { \\prime } ~ \\sum _ { \\mitj _ { 1 } = 1 } ^ { \\mitp - \\miti } \\mitB _ { \\mitp - \\miti , \\mitj _ { 1 } } ( \\mitg _ { 1 } , \\cdots , \\mitg _ { \\mitp - \\miti - \\mitj _ { 1 } + 1 } ) { \\left( \\frac { \\mitpartial } { \\mitpartial \\mitu } \\right) } ^ { \\mitj _ { 1 } } ( \\mitu ^ { \\mitk } ) \\\\ & & \\qquad \\times \\sum _ { \\mitj _ { 2 } = 0 } ^ { \\miti } \\mitB _ { \\miti , \\mitj _ { 2 } } ( \\bar { \\mitg } _ { 1 } , \\cdots , \\bar { \\mitg } _ { \\miti - \\mitj _ { 2 } + 1 } ) { \\left( \\frac { \\mitpartial } { \\mitpartial \\bar { \\mitu } } \\right) } ^ { \\mitj _ { 2 } } ( \\bar { \\mitu } ^ { \\mitl } ) \\\\ & = \\\\ \\\\ \\\\ \\\\ & & \\qquad \\times \\sum _ { \\mitmu } { } ^ { \\prime } ~ \\mitB _ { \\mitp - \\miti , \\mitj _ { 1 } } ( \\mitg _ { 1 } , \\cdots , \\mitg _ { \\mitp - \\miti - \\mitj _ { 1 } + 1 } ) \\mitB _ { \\miti , \\mitj _ { 2 } } ( \\bar { \\mitg } _ { 1 } , \\cdots , \\bar { \\mitg } _ { \\miti - \\mitj _ { 2 } + 1 } ) \\mitu ^ { \\mitk - \\mitj _ { 1 } } \\bar { \\mitu } ^ { \\mitl - \\mitj _ { 2 } } \\\\ & & \\quad \\mathrm { f o r } \\quad \\mitk = 1 , \\cdots , \\mitp - \\miti , ~ \\mitl = 0 , \\cdots , \\miti . \\end{align*}",
"\\begin{align*} & & \\sum _ { \\mitmu } { } ^ { \\prime } ~ \\mitB _ { \\mitp - \\miti , \\mitj _ { 1 } } ( \\mitg _ { 1 } , \\cdots , \\mitg _ { \\mitp - \\miti - \\mitj _ { 1 } + 1 } ) \\mitB _ { \\miti , \\mitj _ { 2 } } ( \\bar { \\mitg } _ { 1 } , \\cdots , \\bar { \\mitg } _ { \\miti - \\mitj _ { 2 } + 1 } ) = 0 \\\\ & & \\quad \\mathrm { f o r } \\quad \\mitj _ { 1 } = 1 , \\cdots , \\mitp - \\miti , ~ \\mitj _ { 2 } = 0 , \\cdots , \\miti , \\end{align*}",
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0002003_page07 | {
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"\\begin {eqnarray} V_{(3,0),\\, \\mu }(f) &=& F_{2,\\, \\mu }(f) -:F_{1,\\, \\mu }\\bar {F}_{1,\\, \\mu }:(f) +\\bar {F}_{2,\\, \\mu }(f) \\\\ &=& \\partial _{\\mu }^2 u \\frac {\\partial f}{\\partial u} +(\\partial _{\\mu }u)^2 \\frac {\\partial ^2 f}{\\partial u^2} -\\partial _{\\mu }u \\partial _{\\mu }\\bar {u} \\frac {\\partial ^2 f}{\\partial u \\partial \\bar {u}} +\\partial _{\\mu }^2 \\bar {u} \\frac {\\partial f}{\\partial \\bar {u}} +(\\partial _{\\mu }\\bar {u})^2 \\frac {\\partial ^2 f}{\\partial \\bar {u}^2}, \\\\ && \\end {eqnarray}",
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0002003_page08 | {
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0002003_page09 | {
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"$ \\mupLambda ( \\mitR ^ { \\mitq } ) = \\sum _ { \\miti = 0 } ^ { \\mitq } \\mupLambda ^ { \\miti } ( \\mitR ^ { \\mitq } ) $",
"$ \\mitq $",
"$ \\mittheta ^ { 1 } , \\mittheta ^ { 2 } , \\ldots \\mittheta ^ { \\mitq } $",
"$ \\mittheta ^ { \\miti } $",
"$ \\mita _ { \\miti _ { 1 } \\miti _ { 2 } \\ldots \\miti _ { \\mitj } } $",
"$ \\mitp _ { \\mita } $",
"$ \\mitp _ { \\mitb } $",
"$ \\mitm = 0 , 1 $",
"$ \\mitpartial ^ { \\mitL } : \\mitV \\mapsto \\mitV $",
"\\begin{equation*} \\mitp ( \\mita \\cdot \\mitb ) = \\mitp ( \\mita ) + \\mitp ( \\mitb ) . \\end{equation*}",
"\\begin{equation*} [ \\mitX , \\mitY ] = - ( - 1 ) ^ { \\mitp _ { \\mitX } \\mitp _ { \\mitY } } [ \\mitY , \\mitX ] , \\end{equation*}",
"\\begin{equation*} [ \\mitX , [ \\mitY , \\mitZ ] ] + ( - 1 ) ^ { \\mitp _ { \\mitZ } ( \\mitp _ { \\mitX } + \\mitp _ { \\mitY } ) } [ \\mitZ , [ \\mitX , \\mitY ] ] + ( - 1 ) ^ { \\mitp _ { \\mitX } ( \\mitp _ { \\mitY } + \\mitp _ { \\mitZ } ) } [ \\mitY , [ \\mitZ , \\mitX ] ] . \\end{equation*}",
"\\begin{equation*} \\mita ( \\mitx , \\mittheta ) = \\mita _ { 0 } ( \\mitx ) + \\mita _ { \\miti } ( \\mitx ) \\mittheta ^ { \\miti } + \\mita _ { \\miti _ { 1 } \\miti _ { 2 } } \\mittheta ^ { \\miti _ { 1 } } \\wedge \\mittheta ^ { \\miti _ { 2 } } + \\cdots + \\mita _ { \\miti _ { 1 } \\miti _ { 2 } \\ldots \\miti _ { \\mitq } } \\mittheta ^ { \\miti _ { 1 } } \\wedge \\mittheta ^ { \\miti _ { 2 } } \\cdots \\wedge \\mittheta ^ { \\miti _ { \\mitq } } , \\end{equation*}",
"\\begin{equation*} \\mita \\cdot \\mitb = ( - 1 ) ^ { \\mitp _ { \\mita } \\mitp _ { \\mitb } } \\mitb \\cdot \\mita \\end{equation*}",
"\\begin{equation*} \\mitpartial ^ { \\mitL } ( \\mita \\cdot \\mitb ) = \\mitpartial ^ { \\mitL } ( \\mita ) \\cdot \\mitb + ( - 1 ) ^ { \\mitm \\mitp _ { \\mita } } \\mita \\cdot \\mitpartial ^ { \\mitL } ( \\mitb ) . \\end{equation*}"
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0002084_page05 | {
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"$\\Z _2$",
"$L$",
"$L=L_0+L_1$",
"$L_0$",
"$L_1$",
"$\\{\\;,\\;\\}$",
"$a$",
"$\\{a,\\,\\cdot \\,\\}$",
"$\\mathcal {S}^{p,2}$",
"$\\partial _i$",
"$\\partial _\\alpha ^L$",
"$\\partial ^R_\\alpha =\\partial ^L_\\alpha $",
"$\\partial ^R_\\alpha =-\\partial ^L_\\alpha $",
"$[\\partial _\\alpha ^R,\\partial _\\beta ^L]_-=0$",
"$$ \\partial ^R(a\\cdot b)=(-1)^{mp_b}\\partial ^R(a)\\cdot b +a\\cdot \\partial ^R(b). $$",
"$$ \\{a,b\\cdot c\\}=\\{a,b\\}\\cdot c+(-1)^{p_ap_b}a\\cdot \\{b,c\\}, $$",
"$$ \\{b\\cdot c,a\\}=b\\cdot \\{c,a\\}+(-1)^{p_ap_c}\\{b,a\\}\\cdot c. $$",
"$$ \\Phi (x,\\theta )=\\Phi _0(x)+\\Phi _{\\alpha }(x)\\theta _\\alpha + \\Phi _{\\alpha \\beta }(x)\\theta _\\alpha \\wedge \\theta _\\beta . $$",
"$$ \\partial _i\\Phi (x,\\theta )=\\partial _i\\Phi _0(x)+\\partial _i\\Phi _{\\alpha }(x) \\theta _\\alpha + \\partial _i\\Phi _{\\alpha \\beta }(x)\\theta _\\alpha \\wedge \\theta _\\beta $$",
"$$ \\partial ^L_\\alpha \\Phi (x,\\theta )=\\Phi _{\\alpha }(x)+ 2\\Phi _{\\alpha \\beta }(x)\\theta _\\beta $$",
"$$ \\partial ^R_\\alpha \\Phi (x,\\theta )=\\Phi _{\\alpha }(x)+ 2\\Phi _{\\beta \\alpha }(x)\\theta _\\beta $$"
],
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"$ L $",
"$ L = L _ { 0 } + L _ { 1 } $",
"$ L _ { 0 } $",
"$ L _ { 1 } $",
"$ \\{ \\; , \\; \\} $",
"$ a $",
"$ \\{ a , \\, \\cdot \\, \\} $",
"$ S ^ { p , 2 } $",
"$ \\partial _ { i } $",
"$ \\partial _ { \\alpha } ^ { L } $",
"$ \\partial _ { \\alpha } ^ { R } = \\partial _ { \\alpha } ^ { L } $",
"$ \\partial _ { \\alpha } ^ { R } = - \\partial _ { \\alpha } ^ { L } $",
"$ [ \\partial _ { \\alpha } ^ { R } , \\partial _ { \\beta } ^ { L } ] _ { - } = 0 $",
"\\begin{equation*} \\partial ^ { R } ( a \\cdot b ) = ( - 1 ) ^ { m p _ { b } } \\partial ^ { R } ( a ) \\cdot b + a \\cdot \\partial ^ { R } ( b ) . \\end{equation*}",
"\\begin{equation*} \\{ a , b \\cdot c \\} = \\{ a , b \\} \\cdot c + ( - 1 ) ^ { p _ { a } p _ { b } } a \\cdot \\{ b , c \\} , \\end{equation*}",
"\\begin{equation*} \\{ b \\cdot c , a \\} = b \\cdot \\{ c , a \\} + ( - 1 ) ^ { p _ { a } p _ { c } } \\{ b , a \\} \\cdot c . \\end{equation*}",
"\\begin{equation*} \\Phi ( x , \\theta ) = \\Phi _ { 0 } ( x ) + \\Phi _ { \\alpha } ( x ) \\theta _ { \\alpha } + \\Phi _ { \\alpha \\beta } ( x ) \\theta _ { \\alpha } \\wedge \\theta _ { \\beta } . \\end{equation*}",
"\\begin{equation*} \\partial _ { i } \\Phi ( x , \\theta ) = \\partial _ { i } \\Phi _ { 0 } ( x ) + \\partial _ { i } \\Phi _ { \\alpha } ( x ) \\theta _ { \\alpha } + \\partial _ { i } \\Phi _ { \\alpha \\beta } ( x ) \\theta _ { \\alpha } \\wedge \\theta _ { \\beta } \\end{equation*}",
"\\begin{equation*} \\partial _ { \\alpha } ^ { L } \\Phi ( x , \\theta ) = \\Phi _ { \\alpha } ( x ) + 2 \\Phi _ { \\alpha \\beta } ( x ) \\theta _ { \\beta } \\end{equation*}",
"\\begin{equation*} \\partial _ { \\alpha } ^ { R } \\Phi ( x , \\theta ) = \\Phi _ { \\alpha } ( x ) + 2 \\Phi _ { \\beta \\alpha } ( x ) \\theta _ { \\beta } \\end{equation*}"
],
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"$ \\mitZ _ { 2 } $",
"$ \\mitL $",
"$ \\mitL = \\mitL _ { 0 } + \\mitL _ { 1 } $",
"$ \\mitL _ { 0 } $",
"$ \\mitL _ { 1 } $",
"$ \\{ \\; , \\; \\} $",
"$ \\mita $",
"$ \\{ \\mita , \\, \\cdot \\, \\} $",
"$ \\mscrS ^ { \\mitp , 2 } $",
"$ \\mitpartial _ { \\miti } $",
"$ \\mitpartial _ { \\mitalpha } ^ { \\mitL } $",
"$ \\mitpartial _ { \\mitalpha } ^ { \\mitR } = \\mitpartial _ { \\mitalpha } ^ { \\mitL } $",
"$ \\mitpartial _ { \\mitalpha } ^ { \\mitR } = - \\mitpartial _ { \\mitalpha } ^ { \\mitL } $",
"$ [ \\mitpartial _ { \\mitalpha } ^ { \\mitR } , \\mitpartial _ { \\mitbeta } ^ { \\mitL } ] _ { - } = 0 $",
"\\begin{equation*} \\mitpartial ^ { \\mitR } ( \\mita \\cdot \\mitb ) = ( - 1 ) ^ { \\mitm \\mitp _ { \\mitb } } \\mitpartial ^ { \\mitR } ( \\mita ) \\cdot \\mitb + \\mita \\cdot \\mitpartial ^ { \\mitR } ( \\mitb ) . \\end{equation*}",
"\\begin{equation*} \\{ \\mita , \\mitb \\cdot \\mitc \\} = \\{ \\mita , \\mitb \\} \\cdot \\mitc + ( - 1 ) ^ { \\mitp _ { \\mita } \\mitp _ { \\mitb } } \\mita \\cdot \\{ \\mitb , \\mitc \\} , \\end{equation*}",
"\\begin{equation*} \\{ \\mitb \\cdot \\mitc , \\mita \\} = \\mitb \\cdot \\{ \\mitc , \\mita \\} + ( - 1 ) ^ { \\mitp _ { \\mita } \\mitp _ { \\mitc } } \\{ \\mitb , \\mita \\} \\cdot \\mitc . \\end{equation*}",
"\\begin{equation*} \\mupPhi ( \\mitx , \\mittheta ) = \\mupPhi _ { 0 } ( \\mitx ) + \\mupPhi _ { \\mitalpha } ( \\mitx ) \\mittheta _ { \\mitalpha } + \\mupPhi _ { \\mitalpha \\mitbeta } ( \\mitx ) \\mittheta _ { \\mitalpha } \\wedge \\mittheta _ { \\mitbeta } . \\end{equation*}",
"\\begin{equation*} \\mitpartial _ { \\miti } \\mupPhi ( \\mitx , \\mittheta ) = \\mitpartial _ { \\miti } \\mupPhi _ { 0 } ( \\mitx ) + \\mitpartial _ { \\miti } \\mupPhi _ { \\mitalpha } ( \\mitx ) \\mittheta _ { \\mitalpha } + \\mitpartial _ { \\miti } \\mupPhi _ { \\mitalpha \\mitbeta } ( \\mitx ) \\mittheta _ { \\mitalpha } \\wedge \\mittheta _ { \\mitbeta } \\end{equation*}",
"\\begin{equation*} \\mitpartial _ { \\mitalpha } ^ { \\mitL } \\mupPhi ( \\mitx , \\mittheta ) = \\mupPhi _ { \\mitalpha } ( \\mitx ) + 2 \\mupPhi _ { \\mitalpha \\mitbeta } ( \\mitx ) \\mittheta _ { \\mitbeta } \\end{equation*}",
"\\begin{equation*} \\mitpartial _ { \\mitalpha } ^ { \\mitR } \\mupPhi ( \\mitx , \\mittheta ) = \\mupPhi _ { \\mitalpha } ( \\mitx ) + 2 \\mupPhi _ { \\mitbeta \\mitalpha } ( \\mitx ) \\mittheta _ { \\mitbeta } \\end{equation*}"
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|
0002084_page06 | {
"latex": [
"$P$",
"$C^\\infty (\\R ^n)$",
"$\\mathcal {S}^{p,q}$",
"$C^\\infty (\\R ^p)$",
"$h$",
"$C^\\infty (\\R ^n)$",
"$C^\\infty (\\R ^n)[[h]]=\\R [[h]]\\otimes C^\\infty (\\R ^n)$",
"$h$",
"$P(f\\otimes g)=\\{f,g\\}$",
"$\\mathcal {S}^{p,q}$",
"$P^{AB}$",
"$q=0$",
"\\begin {equation} \\{\\Phi ,\\Psi \\}=P^{ab}\\partial _a\\Phi \\partial _b\\Psi + P^{\\alpha \\beta }\\partial ^R_\\alpha \\Phi \\partial ^L_\\beta \\Psi = P^{AB}\\partial ^R_A\\Phi \\partial ^L_B\\Psi . \\label {spb} \\end {equation}",
"$$ P^{ab}=-P^{ba},\\quad P^{\\alpha \\beta }=P^{\\beta \\alpha }. $$",
"\\begin {eqnarray} \\star :\\mathcal {S}^{p,q}[[h]]\\otimes \\mathcal {S}^{p,q}[[h]]&\\longrightarrow &\\mathcal {S}^{p,q}[[h]]\\\\ f\\otimes g&\\mapsto & e^{hP}(f\\otimes g)\\end {eqnarray}",
"$$ e^{hP}=\\sum _{n=0}^\\infty \\frac {h^n}{n!}P^n $$",
"$$ P^n(f\\otimes g)=P^{A_1B_1}P^{A_2B_2}\\cdots P^{A_nB_n}(\\partial ^R_{A_1}\\partial ^R_{A_2}\\dots \\partial ^R_{A_n})f\\cdot (\\partial ^L_{B_1}\\partial ^L_{B_2} \\dots \\partial ^L_{B_n}g). $$"
],
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"$ P $",
"$ C ^ { \\infty } ( R ^ { n } ) $",
"$ S ^ { p , q } $",
"$ C ^ { \\infty } ( R ^ { p } ) $",
"$ h $",
"$ C ^ { \\infty } ( R ^ { n } ) $",
"$ C ^ { \\infty } ( R ^ { n } ) [ [ h ] ] = R [ [ h ] ] \\otimes C ^ { \\infty } ( R ^ { n } ) $",
"$ h $",
"$ P ( f \\otimes g ) = \\{ f , g \\} $",
"$ S ^ { p , q } $",
"$ P ^ { A B } $",
"$ q = 0 $",
"\\begin{equation*} \\{ \\Phi , \\Psi \\} = P ^ { a b } \\partial _ { a } \\Phi \\partial _ { b } \\Psi + P ^ { \\alpha \\beta } \\partial _ { \\alpha } ^ { R } \\Phi \\partial _ { \\beta } ^ { L } \\Psi = P ^ { A B } \\partial _ { A } ^ { R } \\Phi \\partial _ { B } ^ { L } \\Psi . \\end{equation*}",
"\\begin{equation*} P ^ { a b } = - P ^ { b a } , \\quad P ^ { \\alpha \\beta } = P ^ { \\beta \\alpha } . \\end{equation*}",
"\\begin{align*} \\star : S ^ { p , q } [ [ h ] ] \\otimes S ^ { p , q } [ [ h ] ] & \\longrightarrow & S ^ { p , q } [ [ h ] ] \\\\ f \\otimes g & \\mapsto & e ^ { h P } ( f \\otimes g ) \\end{align*}",
"\\begin{equation*} e ^ { h P } = \\sum _ { n = 0 } ^ { \\infty } \\frac { h ^ { n } } { n ! } P ^ { n } \\end{equation*}",
"\\begin{equation*} P ^ { n } ( f \\otimes g ) = P ^ { A _ { 1 } B _ { 1 } } P ^ { A _ { 2 } B _ { 2 } } \\cdots P ^ { A _ { n } B _ { n } } ( \\partial _ { A _ { 1 } } ^ { R } \\partial _ { A _ { 2 } } ^ { R } \\ldots \\partial _ { A _ { n } } ^ { R } ) f \\cdot ( \\partial _ { B _ { 1 } } ^ { L } \\partial _ { B _ { 2 } } ^ { L } \\ldots \\partial _ { B _ { n } } ^ { L } g ) . \\end{equation*}"
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"$ \\mitP $",
"$ \\mitC ^ { \\infty } ( \\mitR ^ { \\mitn } ) $",
"$ \\mscrS ^ { \\mitp , \\mitq } $",
"$ \\mitC ^ { \\infty } ( \\mitR ^ { \\mitp } ) $",
"$ \\Planckconst $",
"$ \\mitC ^ { \\infty } ( \\mitR ^ { \\mitn } ) $",
"$ \\mitC ^ { \\infty } ( \\mitR ^ { \\mitn } ) [ [ \\Planckconst ] ] = \\mitR [ [ \\Planckconst ] ] \\otimes \\mitC ^ { \\infty } ( \\mitR ^ { \\mitn } ) $",
"$ \\Planckconst $",
"$ \\mitP ( \\mitf \\otimes \\mitg ) = \\{ \\mitf , \\mitg \\} $",
"$ \\mscrS ^ { \\mitp , \\mitq } $",
"$ \\mitP ^ { \\mitA \\mitB } $",
"$ \\mitq = 0 $",
"\\begin{equation*} \\{ \\mupPhi , \\mupPsi \\} = \\mitP ^ { \\mita \\mitb } \\mitpartial _ { \\mita } \\mupPhi \\mitpartial _ { \\mitb } \\mupPsi + \\mitP ^ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitalpha } ^ { \\mitR } \\mupPhi \\mitpartial _ { \\mitbeta } ^ { \\mitL } \\mupPsi = \\mitP ^ { \\mitA \\mitB } \\mitpartial _ { \\mitA } ^ { \\mitR } \\mupPhi \\mitpartial _ { \\mitB } ^ { \\mitL } \\mupPsi . \\end{equation*}",
"\\begin{equation*} \\mitP ^ { \\mita \\mitb } = - \\mitP ^ { \\mitb \\mita } , \\quad \\mitP ^ { \\mitalpha \\mitbeta } = \\mitP ^ { \\mitbeta \\mitalpha } . \\end{equation*}",
"\\begin{align*} \\star : \\mscrS ^ { \\mitp , \\mitq } [ [ \\Planckconst ] ] \\otimes \\mscrS ^ { \\mitp , \\mitq } [ [ \\Planckconst ] ] & \\longrightarrow & \\mscrS ^ { \\mitp , \\mitq } [ [ \\Planckconst ] ] \\\\ \\mitf \\otimes \\mitg & \\mapsto & \\mite ^ { \\Planckconst \\mitP } ( \\mitf \\otimes \\mitg ) \\end{align*}",
"\\begin{equation*} \\mite ^ { \\Planckconst \\mitP } = \\sum _ { \\mitn = 0 } ^ { \\infty } \\frac { \\Planckconst ^ { \\mitn } } { \\mitn ! } \\mitP ^ { \\mitn } \\end{equation*}",
"\\begin{equation*} \\mitP ^ { \\mitn } ( \\mitf \\otimes \\mitg ) = \\mitP ^ { \\mitA _ { 1 } \\mitB _ { 1 } } \\mitP ^ { \\mitA _ { 2 } \\mitB _ { 2 } } \\cdots \\mitP ^ { \\mitA _ { \\mitn } \\mitB _ { \\mitn } } ( \\mitpartial _ { \\mitA _ { 1 } } ^ { \\mitR } \\mitpartial _ { \\mitA _ { 2 } } ^ { \\mitR } \\ldots \\mitpartial _ { \\mitA _ { \\mitn } } ^ { \\mitR } ) \\mitf \\cdot ( \\mitpartial _ { \\mitB _ { 1 } } ^ { \\mitL } \\mitpartial _ { \\mitB _ { 2 } } ^ { \\mitL } \\ldots \\mitpartial _ { \\mitB _ { \\mitn } } ^ { \\mitL } \\mitg ) . \\end{equation*}"
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0002084_page07 | {
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"$\\R [[h]]$",
"$\\mathcal {A}^{p,q}$",
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"$\\Theta ^1,\\dots \\Theta ^q$",
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"$X$",
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"\\begin {eqnarray} &&[X^a,X^b]_{-}=hP^{ab}, \\\\ &&[\\Theta ^\\alpha ,\\Theta ^\\beta ]_+=hP^{\\alpha \\beta }. \\end {eqnarray}",
"\\begin {equation} x^{i_1}\\cdot x^{i_2}\\cdots x^{i_n},\\qquad i_1\\leq i_2\\leq \\cdots \\leq i_n. \\label {b} \\end {equation}",
"\\begin {eqnarray*} &&\\mbox {Sym}(x^{i_1} x^{i_2}\\cdots x^{i_n})=\\frac {1}{n}\\sum \\limits _{\\sigma \\in S_n} X^{\\sigma (i_1)}\\cdot X^{\\sigma (i_2)}\\cdots X^{\\sigma (i_n)}=\\\\ &&\\mbox {exp}({X^i\\partial _i})(x^{i_1} x^{i_2}\\cdots x^{i_n})|_{x^{i_k}=0}, \\end {eqnarray*}",
"\\begin {equation} \\mbox {Sym}^{-1}(\\mbox {Sym}(f)\\cdot \\mbox {Sym}(g)) \\label {iso} \\end {equation}",
"\\begin {equation} \\theta _{i_1}\\wedge \\theta _{i_2}\\wedge \\cdots \\wedge \\theta _{i_n}, \\qquad i_1\\leq i_2\\leq \\cdots \\leq i_n. \\label {sb} \\end {equation}",
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"\\begin{equation*} x ^ { i _ { 1 } } \\cdot x ^ { i _ { 2 } } \\cdots x ^ { i _ { n } } , \\qquad i _ { 1 } \\leq i _ { 2 } \\leq \\cdots \\leq i _ { n } . \\end{equation*}",
"\\begin{align*} & & S y m ( x ^ { i _ { 1 } } x ^ { i _ { 2 } } \\cdots x ^ { i _ { n } } ) = \\frac { 1 } { n } \\sum _ { \\sigma \\in S _ { n } } X ^ { \\sigma ( i _ { 1 } ) } \\cdot X ^ { \\sigma ( i _ { 2 } ) } \\cdots X ^ { \\sigma ( i _ { n } ) } = \\\\ & & e x p ( X ^ { i } \\partial _ { i } ) ( x ^ { i _ { 1 } } x ^ { i _ { 2 } } \\cdots x ^ { i _ { n } } ) \\vert _ { x ^ { i _ { k } } = 0 } , \\end{align*}",
"\\begin{equation*} S y m ^ { - 1 } ( S y m ( f ) \\cdot S y m ( g ) ) \\end{equation*}",
"\\begin{equation*} \\theta _ { i _ { 1 } } \\wedge \\theta _ { i _ { 2 } } \\wedge \\cdots \\wedge \\theta _ { i _ { n } } , \\qquad i _ { 1 } \\leq i _ { 2 } \\leq \\cdots \\leq i _ { n } . \\end{equation*}",
"\\begin{equation*} S y m ( \\theta _ { i _ { 1 } } \\wedge \\theta _ { i _ { 2 } } \\wedge \\cdots \\wedge \\theta _ { i _ { n } } ) = \\Theta _ { i _ { 1 } } \\Theta _ { i _ { 2 } } \\cdots \\Theta _ { i _ { n } } \\end{equation*}"
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"\\begin{equation*} \\mitx ^ { \\miti _ { 1 } } \\cdot \\mitx ^ { \\miti _ { 2 } } \\cdots \\mitx ^ { \\miti _ { \\mitn } } , \\qquad \\miti _ { 1 } \\leq \\miti _ { 2 } \\leq \\cdots \\leq \\miti _ { \\mitn } . \\end{equation*}",
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"\\begin{equation*} \\mathrm { S y m } ^ { - 1 } ( \\mathrm { S y m } ( \\mitf ) \\cdot \\mathrm { S y m } ( \\mitg ) ) \\end{equation*}",
"\\begin{equation*} \\mittheta _ { \\miti _ { 1 } } \\wedge \\mittheta _ { \\miti _ { 2 } } \\wedge \\cdots \\wedge \\mittheta _ { \\miti _ { \\mitn } } , \\qquad \\miti _ { 1 } \\leq \\miti _ { 2 } \\leq \\cdots \\leq \\miti _ { \\mitn } . \\end{equation*}",
"\\begin{equation*} \\mathrm { S y m } ( \\mittheta _ { \\miti _ { 1 } } \\wedge \\mittheta _ { \\miti _ { 2 } } \\wedge \\cdots \\wedge \\mittheta _ { \\miti _ { \\mitn } } ) = \\mupTheta _ { \\miti _ { 1 } } \\mupTheta _ { \\miti _ { 2 } } \\cdots \\mupTheta _ { \\miti _ { \\mitn } } \\end{equation*}"
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0002084_page08 | {
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"\\begin{align*} . P ^ { \\alpha \\beta } = ( \\begin{array}{cccccc} 0 & 1 & 0 & 0 & \\dots & 0 \\\\ 1 & 0 & 0 & 0 & \\dots & 0 \\\\ 0 & 0 & 0 & 1 & \\dots & 0 \\\\ 0 & 0 & 1 & 0 & \\dots & 0 \\\\ \\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\ 0 & 0 & 0 & 0 & \\dots & 0 \\end{array} ) . \\end{align*}"
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"\\begin{align*} . \\mitP ^ { \\mita \\mitb } = \\left( \\begin{array}{ccc} 0 & \\mitI & 0 \\\\ - \\mitI & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right) , \\qquad \\mitP ^ { \\mitalpha \\mitbeta } = \\left( \\begin{array}{cccc} \\miteta _ { 1 } & 0 & \\dots & 0 \\\\ 0 & \\miteta _ { 2 } & \\dots & 0 \\\\ \\hdotsfor {4} \\\\ 0 & 0 & \\dots & \\miteta _ { \\mitq } \\end{array} \\right) \\end{align*}",
"\\begin{align*} . \\mitP ^ { \\mitalpha \\mitbeta } = \\left( \\begin{array}{cccccc} 0 & 1 & 0 & 0 & \\dots & 0 \\\\ 1 & 0 & 0 & 0 & \\dots & 0 \\\\ 0 & 0 & 0 & 1 & \\dots & 0 \\\\ 0 & 0 & 1 & 0 & \\dots & 0 \\\\ \\hdotsfor {6} \\\\ 0 & 0 & 0 & 0 & \\dots & 0 \\end{array} \\right) . \\end{align*}"
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0002084_page09 | {
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"$\\R ^p$",
"$\\Lambda (\\epsilon _1,\\dots ,\\epsilon _n)$",
"$\\Gamma ^q(\\R ^p)$",
"$\\Phi ^{p,q}_n=\\Gamma ^n(\\R ^p)\\otimes \\Lambda ^n $",
"$\\Phi ^{p,q}_n$",
"$\\Gamma ^n(\\R ^p)$",
"$n$",
"$n=\\mbox {dim}\\Lambda ^q=2^q$",
"$\\Phi ^{p,q}_n$",
"$\\Phi _{i_1i_2\\dots i_k}$",
"$\\Gamma ^q(\\R ^p)$",
"${i_1i_2\\dots i_k}$",
"$\\Phi ^{p,q}_n\\otimes \\R [[h]]$",
"$(\\Phi ^{p,q}_n[[h]],\\star )$",
"$\\{x^\\mu ,\\theta ^{\\alpha i}, \\bar \\theta ^{\\dot \\alpha }_i\\}$",
"$\\{x^\\mu \\}$",
"$\\mathcal {M}$",
"$\\{\\theta ^{\\alpha i}, \\bar \\theta ^{\\dot \\alpha }_i\\}$",
"$\\epsilon ^{\\alpha i},\\bar \\epsilon ^{\\dot \\alpha }_i$",
"$\\wedge $",
"$\\otimes $",
"$g(\\epsilon )$",
"$g$",
"$(g^{-1}\\Phi )(x,\\theta ,\\bar \\theta )=\\Phi (x',\\theta ',\\bar \\theta ')$",
"$$ (a\\otimes \\Psi _1)(b\\otimes \\Psi _2)=(-1)^{p_1p_b}ab\\otimes \\Psi _1\\Psi _2\\qquad a,b\\in \\Gamma ^n(\\R ^p), \\quad f,g\\in \\Lambda ^n, $$",
"$$ b(a\\otimes \\Psi )=(ba\\otimes \\Psi ),\\qquad (a\\otimes \\Psi )b=(-1)^{p_bp_\\Psi }(ab\\otimes \\Psi ). $$",
"$$ \\Phi (x,\\theta )=\\Phi _0(x) +\\theta _i\\otimes \\Phi _i(x) +\\theta _j\\wedge \\theta _j\\otimes \\Phi _{ij}+\\cdots $$",
"\\begin {eqnarray} x^\\mu &\\mapsto & {x'}^\\mu =x^\\mu +i(\\theta ^{\\alpha i}(\\sigma ^\\mu )_{\\alpha \\dot \\alpha } \\bar \\epsilon ^{\\dot \\alpha }_i-\\epsilon ^{\\alpha i}(\\sigma ^\\mu )_{\\alpha \\dot \\alpha } \\bar \\theta ^{\\dot \\alpha }_i)\\\\ \\theta ^{\\alpha i}&\\mapsto & {\\theta '}^{\\alpha i}=\\theta ^{\\alpha i}+\\epsilon ^{\\alpha i} \\\\ \\bar \\theta ^{\\dot \\alpha }_i&\\mapsto &{\\bar {\\theta '}}^{\\dot \\alpha }_i=\\bar \\theta ^{\\dot \\alpha }_i+\\bar \\epsilon ^{\\dot \\alpha }_i. \\end {eqnarray}",
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"$ \\Phi _ { n } ^ { p , q } = \\Gamma ^ { n } ( R ^ { p } ) \\otimes \\Lambda ^ { n } $",
"$ \\Phi _ { n } ^ { p , q } $",
"$ \\Gamma ^ { n } ( R ^ { p } ) $",
"$ n $",
"$ n = d i m \\Lambda ^ { q } = 2 ^ { q } $",
"$ \\Phi _ { n } ^ { p , q } $",
"$ \\Phi _ { i _ { 1 } i _ { 2 } \\ldots i _ { k } } $",
"$ \\Gamma ^ { q } ( R ^ { p } ) $",
"$ i _ { 1 } i _ { 2 } \\ldots i _ { k } $",
"$ \\Phi _ { n } ^ { p , q } \\otimes R [ [ h ] ] $",
"$ ( \\Phi _ { n } ^ { p , q } [ [ h ] ] , \\star ) $",
"$ \\{ x ^ { \\mu } , \\theta ^ { \\alpha i } , \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } \\} $",
"$ \\{ x ^ { \\mu } \\} $",
"$ M $",
"$ \\{ \\theta ^ { \\alpha i } , \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } \\} $",
"$ \\epsilon ^ { \\alpha i } , \\bar { \\epsilon } _ { i } ^ { \\dot { \\alpha } } $",
"$ \\wedge $",
"$ \\otimes $",
"$ g ( \\epsilon ) $",
"$ g $",
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"\\begin{equation*} ( a \\otimes \\Psi _ { 1 } ) ( b \\otimes \\Psi _ { 2 } ) = ( - 1 ) ^ { p _ { 1 } p _ { b } } a b \\otimes \\Psi _ { 1 } \\Psi _ { 2 } \\qquad a , b \\in \\Gamma ^ { n } ( R ^ { p } ) , \\quad f , g \\in \\Lambda ^ { n } , \\end{equation*}",
"\\begin{equation*} b ( a \\otimes \\Psi ) = ( b a \\otimes \\Psi ) , \\qquad ( a \\otimes \\Psi ) b = ( - 1 ) ^ { p _ { b } p _ { \\Psi } } ( a b \\otimes \\Psi ) . \\end{equation*}",
"\\begin{equation*} \\Phi ( x , \\theta ) = \\Phi _ { 0 } ( x ) + \\theta _ { i } \\otimes \\Phi _ { i } ( x ) + \\theta _ { j } \\wedge \\theta _ { j } \\otimes \\Phi _ { i j } + \\cdots \\end{equation*}",
"\\begin{align*} x ^ { \\mu } & \\mapsto & { x ^ { \\prime } } ^ { \\mu } = x ^ { \\mu } + i ( \\theta ^ { \\alpha i } ( \\sigma ^ { \\mu } ) _ { \\alpha \\dot { \\alpha } } \\bar { \\epsilon } _ { i } ^ { \\dot { \\alpha } } - \\epsilon ^ { \\alpha i } ( \\sigma ^ { \\mu } ) _ { \\alpha \\dot { \\alpha } } \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } ) \\\\ \\theta ^ { \\alpha i } & \\mapsto & { \\theta ^ { \\prime } } ^ { \\alpha i } = \\theta ^ { \\alpha i } + \\epsilon ^ { \\alpha i } \\\\ \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } & \\mapsto & \\bar { \\theta ^ { \\prime } } _ { i } ^ { \\dot { \\alpha } } = \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } + \\bar { \\epsilon } _ { i } ^ { \\dot { \\alpha } } . \\end{align*}",
"\\begin{equation*} \\Phi ( x ^ { \\mu } , \\theta ^ { \\alpha i } , \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } ) = \\Phi _ { 0 } ( x ) + \\theta ^ { \\alpha i } \\Psi _ { \\alpha } ^ { i } + \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } \\bar { \\Sigma } _ { \\dot { \\alpha } i } + \\theta ^ { \\alpha i } \\theta ^ { \\beta j } \\Psi _ { \\alpha \\beta } ^ { i j } + \\cdots , \\end{equation*}"
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"$ ( \\mupPhi _ { \\mitn } ^ { \\mitp , \\mitq } [ [ \\Planckconst ] ] , \\star ) $",
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"$ \\{ \\mitx ^ { \\mitmu } \\} $",
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"$ \\mitepsilon ^ { \\mitalpha \\miti } , \\bar { \\mitepsilon } _ { \\miti } ^ { \\dot { \\mitalpha } } $",
"$ \\wedge $",
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"\\begin{equation*} \\mupPhi ( \\mitx ^ { \\mitmu } , \\mittheta ^ { \\mitalpha \\miti } , \\bar { \\mittheta } _ { \\miti } ^ { \\dot { \\mitalpha } } ) = \\mupPhi _ { 0 } ( \\mitx ) + \\mittheta ^ { \\mitalpha \\miti } \\mupPsi _ { \\mitalpha } ^ { \\miti } + \\bar { \\mittheta } _ { \\miti } ^ { \\dot { \\mitalpha } } \\bar { \\mupSigma } _ { \\dot { \\mitalpha } \\miti } + \\mittheta ^ { \\mitalpha \\miti } \\mittheta ^ { \\mitbeta \\mitj } \\mupPsi _ { \\mitalpha \\mitbeta } ^ { \\miti \\mitj } + \\cdots , \\end{equation*}"
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0002084_page10 | {
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"$(g^{-1}\\Phi )(x,\\theta ,\\bar \\theta )=\\Phi (x',\\theta ',\\bar \\theta ')$",
"${ {\\bar D}^{R,Li} }_{\\dot \\alpha }$",
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"$$ {D^{R,L}}_{\\alpha i}(g\\Phi )=g({D^{R,L}}_{\\alpha i}\\Phi ). $$",
"\\begin {equation} \\{\\Phi ,\\Psi \\}=P^{\\mu \\nu }\\partial _\\mu \\Phi \\partial _\\nu \\Psi +P^{\\alpha i\\beta j} {D^{R}}_{\\alpha i}\\Phi {D^{L}}_{\\beta j}\\Psi . \\label {spb2} \\end {equation}",
"\\begin {equation} C^\\infty (\\mathcal {M}\\times \\hbox {SU}(N)/U(1)^{N-1})\\otimes \\Lambda ^{4N}. \\label {hs} \\end {equation}"
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"\\begin{align*} { D ^ { R , L } } _ { \\alpha i } & = & { \\partial ^ { R , L } } _ { \\alpha i } + ( i \\sigma _ { \\alpha \\dot { \\alpha } } ^ { \\mu } \\bar { \\theta } _ { i } ^ { \\dot { \\alpha } } ) \\partial _ { \\mu } ) ^ { R , L } , \\\\ \\bar { D } _ { \\dot { \\alpha } } ^ { R , L i } & = & - \\bar { \\partial } _ { \\dot { \\alpha } } ^ { R , L i } - ( i \\theta ^ { \\alpha i } \\sigma _ { \\alpha \\dot { \\alpha } } ^ { \\mu } \\partial _ { \\mu } ) ^ { R , L } . \\end{align*}",
"\\begin{equation*} { D ^ { R , L } } _ { \\alpha i } ( g \\Phi ) = g ( { D ^ { R , L } } _ { \\alpha i } \\Phi ) . \\end{equation*}",
"\\begin{equation*} \\{ \\Phi , \\Psi \\} = P ^ { \\mu \\nu } \\partial _ { \\mu } \\Phi \\partial _ { \\nu } \\Psi + P ^ { \\alpha i \\beta j } { D ^ { R } } _ { \\alpha i } \\Phi { D ^ { L } } _ { \\beta j } \\Psi . \\end{equation*}",
"\\begin{equation*} C ^ { \\infty } ( M \\times S U ( N ) \\slash U ( 1 ) ^ { N - 1 } ) \\otimes \\Lambda ^ { 4 N } . \\end{equation*}"
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"$ \\bar { \\mitD } _ { \\dot { \\mitalpha } } ^ { \\mitR , \\mitL \\miti } $",
"$ [ { \\mitD ^ { \\mitR } } _ { \\mitalpha \\miti } , { \\mitD ^ { \\mitL } } _ { \\mitbeta \\mitj } ] _ { - } = 0 $",
"$ \\mitD _ { \\mitalpha \\miti } ^ { \\mitR , \\mitL } \\mitP ^ { \\mitA \\mitB } = 0 $",
"$ \\mitA = \\mitmu , \\{ \\mitalpha \\, \\miti \\} $",
"$ \\mitN - \\mitk $",
"$ \\mitD $",
"$ \\mitk $",
"$ \\bar { \\mitD } $",
"$ \\mitk = 1 , \\ldots \\mitN $",
"$ \\bar { \\mitD } _ { \\dot { \\mitalpha } } ^ { \\mitR , \\mitL \\miti } \\mupPhi = 0 $",
"$ \\mupPhi ( \\mitx , \\mittheta ) $",
"$ \\mitP ^ { \\mitalpha \\miti \\mitbeta \\mitj } = 0 $",
"$ \\mitN $",
"$ \\mitN > 1 $",
"$ \\mitN $",
"$ { } ^ { \\mitN - 1 } $",
"$ 4 \\mitN $",
"$ ( \\mscrL \\otimes \\mathrm { S U } ( \\mitN ) ) \\otimes _ { \\mits } \\mscrS \\mscrT \\slash \\mscrL \\otimes \\mathrm { U } ( 1 ) ^ { \\mitN - 1 } $",
"$ ( \\mscrL \\otimes \\mathrm { S U } ( \\mitN ) ) \\otimes _ { \\mits } \\mscrS \\mscrT \\slash \\mscrL \\otimes \\mathrm { U } ( 1 ) ^ { \\mitN - 1 } $",
"$ \\mscrL $",
"$ \\mscrS \\mscrT $",
"\\begin{equation*} \\mitg ( \\mupPhi _ { 1 } \\star \\mupPhi _ { 2 } ) = ( \\mitg \\mupPhi _ { 1 } ) \\star ( \\mitg \\mupPhi _ { 2 } ) . \\end{equation*}",
"\\begin{align*} { \\mitD ^ { \\mitR , \\mitL } } _ { \\mitalpha \\miti } & = & { \\mitpartial ^ { \\mitR , \\mitL } } _ { \\mitalpha \\miti } + ( \\miti \\mitsigma _ { \\mitalpha \\dot { \\mitalpha } } ^ { \\mitmu } \\bar { \\mittheta } _ { \\miti } ^ { \\dot { \\mitalpha } } ) \\mitpartial _ { \\mitmu } ) ^ { \\mitR , \\mitL } , \\\\ \\bar { \\mitD } _ { \\dot { \\mitalpha } } ^ { \\mitR , \\mitL \\miti } & = & - \\bar { \\mitpartial } _ { \\dot { \\mitalpha } } ^ { \\mitR , \\mitL \\miti } - ( \\miti \\mittheta ^ { \\mitalpha \\miti } \\mitsigma _ { \\mitalpha \\dot { \\mitalpha } } ^ { \\mitmu } \\mitpartial _ { \\mitmu } ) ^ { \\mitR , \\mitL } . \\end{align*}",
"\\begin{equation*} { \\mitD ^ { \\mitR , \\mitL } } _ { \\mitalpha \\miti } ( \\mitg \\mupPhi ) = \\mitg ( { \\mitD ^ { \\mitR , \\mitL } } _ { \\mitalpha \\miti } \\mupPhi ) . \\end{equation*}",
"\\begin{equation*} \\{ \\mupPhi , \\mupPsi \\} = \\mitP ^ { \\mitmu \\mitnu } \\mitpartial _ { \\mitmu } \\mupPhi \\mitpartial _ { \\mitnu } \\mupPsi + \\mitP ^ { \\mitalpha \\miti \\mitbeta \\mitj } { \\mitD ^ { \\mitR } } _ { \\mitalpha \\miti } \\mupPhi { \\mitD ^ { \\mitL } } _ { \\mitbeta \\mitj } \\mupPsi . \\end{equation*}",
"\\begin{equation*} \\mitC ^ { \\infty } ( \\mscrM \\times \\mathrm { S U } ( \\mitN ) \\slash \\mitU ( 1 ) ^ { \\mitN - 1 } ) \\otimes \\mupLambda ^ { 4 \\mitN } . \\end{equation*}"
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0002084_page11 | {
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"$\\{x^\\mu , u,\\theta ^{\\alpha i}, \\bar \\theta ^{\\dot \\alpha }_i\\}$",
"$u$",
"$N$",
"$^{N-1}$",
"$\\Phi (x,u,\\theta , \\bar \\theta )$",
"$k$",
"$\\theta $",
"$N-k$",
"$\\bar \\theta $",
"$k=0,N$",
"$\\mathcal {M}$",
"$P^{\\mu \\nu }$",
"$\\Phi _i(x,u,\\theta )$",
"$$ \\mathcal {D}_{\\alpha I }=u_I^iD_{\\alpha i}, \\qquad \\bar {\\mathcal {D}}_{\\dot \\alpha }^I=u_I^i\\bar {D}_{\\dot \\alpha }^i. $$",
"$$ \\mathcal {D}_{\\alpha 1}\\Phi =\\cdots =\\mathcal {D}_{\\alpha k}\\Phi =0= \\bar {\\mathcal {D}}_{\\dot \\alpha }^{k+1}\\Phi =\\dots =\\bar {\\mathcal {D}}_{\\alpha }^N\\Phi . $$",
"$$iP=iP^{\\mu \\nu }\\frac {\\partial }{ \\partial x^\\mu }\\otimes \\frac {\\partial }{ \\partial x^\\nu } $$",
"$$ \\{\\Phi _1,\\Phi _2\\}=iP^{\\mu \\nu }\\frac {\\partial \\Phi _1}{\\partial x^\\mu } \\frac {\\partial \\Phi _2}{\\partial x^\\nu } $$",
"\\begin {equation} \\Phi _1\\star \\Phi _2= \\exp (iP)(\\Phi _1\\otimes \\Phi _2). \\label {gas}\\end {equation}",
"$$ \\int d^4xd^\\theta d^2\\bar \\theta \\; \\Phi \\bar \\Phi +\\int d^4x\\;(\\int d^2\\theta \\; (\\frac {m}{ 2}\\Phi ^2 + \\frac {g}{ 3}\\Phi ^3) +\\mbox {c. c.}), $$"
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"$ u $",
"$ N $",
"$ { } ^ { N - 1 } $",
"$ \\Phi ( x , u , \\theta , \\bar { \\theta } ) $",
"$ k $",
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"$ N - k $",
"$ \\bar { \\theta } $",
"$ k = 0 , N $",
"$ M $",
"$ P ^ { \\mu \\nu } $",
"$ \\Phi _ { i } ( x , u , \\theta ) $",
"\\begin{equation*} D _ { \\alpha I } = u _ { I } ^ { i } D _ { \\alpha i } , \\qquad \\bar { D } _ { \\dot { \\alpha } } ^ { I } = u _ { I } ^ { i } \\bar { D } _ { \\dot { \\alpha } } ^ { i } . \\end{equation*}",
"\\begin{equation*} D _ { \\alpha 1 } \\Phi = \\cdots = D _ { \\alpha k } \\Phi = 0 = \\bar { D } _ { \\dot { \\alpha } } ^ { k + 1 } \\Phi = \\cdots = \\bar { D } _ { \\alpha } ^ { N } \\Phi . \\end{equation*}",
"\\begin{equation*} i P = i P ^ { \\mu \\nu } \\frac { \\partial } { \\partial x ^ { \\mu } } \\otimes \\frac { \\partial } { \\partial x ^ { \\nu } } \\end{equation*}",
"\\begin{equation*} \\{ \\Phi _ { 1 } , \\Phi _ { 2 } \\} = i P ^ { \\mu \\nu } \\frac { \\partial \\Phi _ { 1 } } { \\partial x ^ { \\mu } } \\frac { \\partial \\Phi _ { 2 } } { \\partial x ^ { \\nu } } \\end{equation*}",
"\\begin{equation*} \\Phi _ { 1 } \\star \\Phi _ { 2 } = \\operatorname { e x p } ( i P ) ( \\Phi _ { 1 } \\otimes \\Phi _ { 2 } ) . \\end{equation*}",
"\\begin{equation*} \\int d ^ { 4 } x d ^ { \\theta } d ^ { 2 } \\bar { \\theta } \\; \\Phi \\bar { \\Phi } + \\int d ^ { 4 } x \\; ( \\int d ^ { 2 } \\theta \\; ( \\frac { m } { 2 } \\Phi ^ { 2 } + \\frac { g } { 3 } \\Phi ^ { 3 } ) + c . ~ c . ) , \\end{equation*}"
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"$ \\{ \\mitx ^ { \\mitmu } , \\mitu , \\mittheta ^ { \\mitalpha \\miti } , \\bar { \\mittheta } _ { \\miti } ^ { \\dot { \\mitalpha } } \\} $",
"$ \\mitu $",
"$ \\mitN $",
"$ { } ^ { \\mitN - 1 } $",
"$ \\mupPhi ( \\mitx , \\mitu , \\mittheta , \\bar { \\mittheta } ) $",
"$ \\mitk $",
"$ \\mittheta $",
"$ \\mitN - \\mitk $",
"$ \\bar { \\mittheta } $",
"$ \\mitk = 0 , \\mitN $",
"$ \\mscrM $",
"$ \\mitP ^ { \\mitmu \\mitnu } $",
"$ \\mupPhi _ { \\miti } ( \\mitx , \\mitu , \\mittheta ) $",
"\\begin{equation*} \\mscrD _ { \\mitalpha \\mitI } = \\mitu _ { \\mitI } ^ { \\miti } \\mitD _ { \\mitalpha \\miti } , \\qquad \\bar { \\mscrD } _ { \\dot { \\mitalpha } } ^ { \\mitI } = \\mitu _ { \\mitI } ^ { \\miti } \\bar { \\mitD } _ { \\dot { \\mitalpha } } ^ { \\miti } . \\end{equation*}",
"\\begin{equation*} \\mscrD _ { \\mitalpha 1 } \\mupPhi = \\cdots = \\mscrD _ { \\mitalpha \\mitk } \\mupPhi = 0 = \\bar { \\mscrD } _ { \\dot { \\mitalpha } } ^ { \\mitk + 1 } \\mupPhi = \\cdots = \\bar { \\mscrD } _ { \\mitalpha } ^ { \\mitN } \\mupPhi . \\end{equation*}",
"\\begin{equation*} \\miti \\mitP = \\miti \\mitP ^ { \\mitmu \\mitnu } \\frac { \\mitpartial } { \\mitpartial \\mitx ^ { \\mitmu } } \\otimes \\frac { \\mitpartial } { \\mitpartial \\mitx ^ { \\mitnu } } \\end{equation*}",
"\\begin{equation*} \\{ \\mupPhi _ { 1 } , \\mupPhi _ { 2 } \\} = \\miti \\mitP ^ { \\mitmu \\mitnu } \\frac { \\mitpartial \\mupPhi _ { 1 } } { \\mitpartial \\mitx ^ { \\mitmu } } \\frac { \\mitpartial \\mupPhi _ { 2 } } { \\mitpartial \\mitx ^ { \\mitnu } } \\end{equation*}",
"\\begin{equation*} \\mupPhi _ { 1 } \\star \\mupPhi _ { 2 } = \\operatorname { e x p } ( \\miti \\mitP ) ( \\mupPhi _ { 1 } \\otimes \\mupPhi _ { 2 } ) . \\end{equation*}",
"\\begin{equation*} \\int \\mitd ^ { 4 } \\mitx \\mitd ^ { \\mittheta } \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\mupPhi \\bar { \\mupPhi } + \\int \\mitd ^ { 4 } \\mitx \\; ( \\int \\mitd ^ { 2 } \\mittheta \\; ( \\frac { \\mitm } { 2 } \\mupPhi ^ { 2 } + \\frac { \\mitg } { 3 } \\mupPhi ^ { 3 } ) + \\mathrm { c } . ~ \\mathrm { c } . ) , \\end{equation*}"
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0002084_page12 | {
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"$\\Phi $",
"$y=x+i\\theta \\sigma \\bar \\theta $",
"$\\Phi ^{\\star n}=\\Phi \\star \\Phi \\cdots (n)\\cdots \\star \\Phi $",
"$\\sigma $",
"$(1,\\dots , n)$",
"$F$",
"$\\phi ^4$",
"$$ \\Phi =A(y)+\\sqrt {2}\\theta \\psi (y) +\\theta \\theta F(y), $$",
"\\begin {equation} \\int d^4xd^2\\theta d^2\\bar \\theta \\; \\Phi \\bar \\Phi +\\int d^4x\\;(\\int d^2\\theta \\;(\\frac {m}{2}\\Phi ^2 + \\frac {g}{3}\\Phi ^{\\star 3}) +\\mbox {c. c.}). \\label {incw} \\end {equation}",
"\\begin {equation} \\int d^4x\\;A\\star B=\\int d^4x\\;A B=\\int d^4x\\;B\\star A \\label {ip1} \\end {equation}",
"\\begin {equation} \\int d^4x\\;A_1\\star \\cdots \\star A_n=\\int d^4x\\;A_{\\sigma (1)}\\star \\cdots \\star A_{\\sigma (n)} \\label {ip2} \\end {equation}",
"$$ \\overline {A\\star B} =\\bar B\\star \\bar A. $$",
"$$ F=-m\\bar A-g\\bar A\\star \\bar A $$",
"\\begin {eqnarray} && i\\partial _\\mu \\bar \\psi \\bar \\sigma ^\\mu \\psi +\\bar A\\partial _\\mu \\partial ^\\mu A-\\frac {1}{ 2}m(\\psi \\psi +\\bar \\psi \\bar \\psi ) -m^2\\bar A A -g(A(\\psi \\star \\psi ) +\\\\&&\\bar A(\\bar \\psi \\star \\bar \\psi )) -mg(A(\\bar A\\star \\bar A )+ \\bar A(A\\star A)) -g^2(A\\star A)(\\bar A\\star \\bar A). \\end {eqnarray}"
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"$ \\Phi ^ { \\star n } = \\Phi \\star \\Phi \\cdots ( n ) \\cdots \\star \\Phi $",
"$ \\sigma $",
"$ ( 1 , \\ldots , n ) $",
"$ F $",
"$ \\phi ^ { 4 } $",
"\\begin{equation*} \\Phi = A ( y ) + \\sqrt { 2 } \\theta \\psi ( y ) + \\theta \\theta F ( y ) , \\end{equation*}",
"\\begin{equation*} \\int d ^ { 4 } x d ^ { 2 } \\theta d ^ { 2 } \\bar { \\theta } \\; \\Phi \\bar { \\Phi } + \\int d ^ { 4 } x \\; ( \\int d ^ { 2 } \\theta \\; ( \\frac { m } { 2 } \\Phi ^ { 2 } + \\frac { g } { 3 } \\Phi ^ { \\star 3 } ) + c . ~ c . ) . \\end{equation*}",
"\\begin{equation*} \\int d ^ { 4 } x \\; A \\star B = \\int d ^ { 4 } x \\; A B = \\int d ^ { 4 } x \\; B \\star A \\end{equation*}",
"\\begin{equation*} \\int d ^ { 4 } x \\; A _ { 1 } \\star \\cdots \\star A _ { n } = \\int d ^ { 4 } x \\; A _ { \\sigma ( 1 ) } \\star \\cdots \\star A _ { \\sigma ( n ) } \\end{equation*}",
"\\begin{equation*} \\overline { A \\star B } = \\bar { B } \\star \\bar { A } . \\end{equation*}",
"\\begin{equation*} F = - m \\bar { A } - g \\bar { A } \\star \\bar { A } \\end{equation*}",
"\\begin{align*} & & i \\partial _ { \\mu } \\bar { \\psi } \\bar { \\sigma } ^ { \\mu } \\psi + \\bar { A } \\partial _ { \\mu } \\partial ^ { \\mu } A - \\frac { 1 } { 2 } m ( \\psi \\psi + \\bar { \\psi } \\bar { \\psi } ) - m ^ { 2 } \\bar { A } A - g ( A ( \\psi \\star \\psi ) + \\\\ & & \\bar { A } ( \\bar { \\psi } \\star \\bar { \\psi } ) ) - m g ( A ( \\bar { A } \\star \\bar { A } ) + \\bar { A } ( A \\star A ) ) - g ^ { 2 } ( A \\star A ) ( \\bar { A } \\star \\bar { A } ) . \\end{align*}"
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"$ \\mupPhi $",
"$ \\mity = \\mitx + \\miti \\mittheta \\mitsigma \\bar { \\mittheta } $",
"$ \\mupPhi ^ { \\star \\mitn } = \\mupPhi \\star \\mupPhi \\cdots ( \\mitn ) \\cdots \\star \\mupPhi $",
"$ \\mitsigma $",
"$ ( 1 , \\ldots , \\mitn ) $",
"$ \\mitF $",
"$ \\mitphi ^ { 4 } $",
"\\begin{equation*} \\mupPhi = \\mitA ( \\mity ) + \\sqrt { 2 } \\mittheta \\mitpsi ( \\mity ) + \\mittheta \\mittheta \\mitF ( \\mity ) , \\end{equation*}",
"\\begin{equation*} \\int \\mitd ^ { 4 } \\mitx \\mitd ^ { 2 } \\mittheta \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\mupPhi \\bar { \\mupPhi } + \\int \\mitd ^ { 4 } \\mitx \\; ( \\int \\mitd ^ { 2 } \\mittheta \\; ( \\frac { \\mitm } { 2 } \\mupPhi ^ { 2 } + \\frac { \\mitg } { 3 } \\mupPhi ^ { \\star 3 } ) + \\mathrm { c } . ~ \\mathrm { c } . ) . \\end{equation*}",
"\\begin{equation*} \\int \\mitd ^ { 4 } \\mitx \\; \\mitA \\star \\mitB = \\int \\mitd ^ { 4 } \\mitx \\; \\mitA \\mitB = \\int \\mitd ^ { 4 } \\mitx \\; \\mitB \\star \\mitA \\end{equation*}",
"\\begin{equation*} \\int \\mitd ^ { 4 } \\mitx \\; \\mitA _ { 1 } \\star \\cdots \\star \\mitA _ { \\mitn } = \\int \\mitd ^ { 4 } \\mitx \\; \\mitA _ { \\mitsigma ( 1 ) } \\star \\cdots \\star \\mitA _ { \\mitsigma ( \\mitn ) } \\end{equation*}",
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0002084_page15 | {
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"\\begin {eqnarray*} P^{\\alpha \\beta }&=&(\\sigma _{\\mu \\nu })^{\\alpha \\beta }P^{\\mu \\nu }, \\qquad (\\mbox {symmetric in }(\\alpha ,\\beta )),\\\\ W_\\alpha &=&{\\bar D}^2D_\\alpha V,\\\\ \\nabla _\\nu &=&(\\sigma _\\nu )^{\\alpha \\dot \\alpha }[D_\\alpha ,D_{\\dot \\alpha }]. \\end {eqnarray*}"
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"\\begin{align*} P ^ { \\alpha \\beta } & = & ( \\sigma _ { \\mu \\nu } ) ^ { \\alpha \\beta } P ^ { \\mu \\nu } , \\qquad ( s y m m e t r i c ~ i n ~ ( \\alpha , \\beta ) ) , \\\\ W _ { \\alpha } & = & \\bar { D } ^ { 2 } D _ { \\alpha } V , \\\\ \\nabla _ { \\nu } & = & ( \\sigma _ { \\nu } ) ^ { \\alpha \\dot { \\alpha } } [ D _ { \\alpha } , D _ { \\dot { \\alpha } } ] . \\end{align*}"
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"\\begin{align*} \\mitP ^ { \\mitalpha \\mitbeta } & = & ( \\mitsigma _ { \\mitmu \\mitnu } ) ^ { \\mitalpha \\mitbeta } \\mitP ^ { \\mitmu \\mitnu } , \\qquad ( \\mathrm { s y m m e t r i c } ~ \\mathrm { i n } ~ ( \\mitalpha , \\mitbeta ) ) , \\\\ \\mitW _ { \\mitalpha } & = & \\bar { \\mitD } ^ { 2 } \\mitD _ { \\mitalpha } \\mitV , \\\\ \\nabla _ { \\mitnu } & = & ( \\mitsigma _ { \\mitnu } ) ^ { \\mitalpha \\dot { \\mitalpha } } [ \\mitD _ { \\mitalpha } , \\mitD _ { \\dot { \\mitalpha } } ] . \\end{align*}"
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0002084_page17 | {
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"$$ \\sqrt {X^2-Y}=X+ Y\\frac {\\sqrt {X^2-Y}-X}{Y}=X-\\frac {Y}{\\sqrt {X^2-Y}+X}, $$",
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"$$ T=\\bar D\\bar D{\\bar W}^2=-\\frac {1}{2} F_-^2 +\\cdots $$"
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"\\begin{equation*} X = \\epsilon + \\frac { 1 } { 4 } F ^ { 2 } , \\qquad Y = \\frac { 1 } { 1 6 } ( ( F ^ { 2 } ) ^ { 2 } - ( F \\widetilde { F } ) ^ { 2 } ) . \\end{equation*}",
"\\begin{equation*} F _ { \\pm } = \\frac { 1 } { 2 } ( F \\pm \\widetilde { F } ) \\end{equation*}",
"\\begin{equation*} F _ { \\pm } ^ { 2 } = \\frac { 1 } { 2 } ( F ^ { 2 } \\pm F \\widetilde { F } ) , \\qquad F _ { + } ^ { 2 } F _ { - } ^ { 2 } = \\frac { 1 } { 4 } ( ( F ^ { 2 } ) ^ { 2 } - ( F \\widetilde { F } ) ^ { 2 } ) . \\end{equation*}",
"\\begin{equation*} T = \\bar { D } \\bar { D } \\bar { W } ^ { 2 } = - \\frac { 1 } { 2 } F _ { - } ^ { 2 } + \\cdots \\end{equation*}"
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"\\begin{equation*} \\sqrt { \\mitX ^ { 2 } - \\mitY } = \\mitX + \\mitY \\frac { \\sqrt { \\mitX ^ { 2 } - \\mitY } - \\mitX } { \\mitY } = \\mitX - \\frac { \\mitY } { \\sqrt { \\mitX ^ { 2 } - \\mitY } + \\mitX } , \\end{equation*}",
"\\begin{equation*} \\mitX = \\mitepsilon + \\frac { 1 } { 4 } \\mitF ^ { 2 } , \\qquad \\mitY = \\frac { 1 } { 1 6 } \\big ( ( \\mitF ^ { 2 } ) ^ { 2 } - ( \\mitF \\tilde { \\mitF } ) ^ { 2 } \\big ) . \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\pm } = \\frac { 1 } { 2 } ( \\mitF \\pm \\tilde { \\mitF } ) \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\pm } ^ { 2 } = \\frac { 1 } { 2 } ( \\mitF ^ { 2 } \\pm \\mitF \\tilde { \\mitF } ) , \\qquad \\mitF _ { + } ^ { 2 } \\mitF _ { - } ^ { 2 } = \\frac { 1 } { 4 } \\big ( ( \\mitF ^ { 2 } ) ^ { 2 } - ( \\mitF \\tilde { \\mitF } ) ^ { 2 } \\big ) . \\end{equation*}",
"\\begin{equation*} \\mitT = \\bar { \\mitD } \\bar { \\mitD } \\bar { \\mitW } ^ { 2 } = - \\frac { 1 } { 2 } \\mitF _ { - } ^ { 2 } + \\cdots \\end{equation*}"
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0002084_page18 | {
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"$$ \\sqrt {X^2-Y}=\\sqrt {\\epsilon ^2-\\epsilon (T+\\bar T) +\\frac {1}{4}(T-\\bar T)^2}. $$",
"\\begin {equation} \\sqrt {\\frac {1}{4}(T-\\bar T)^2}=\\pm \\frac {1}{2}(T-\\bar T) \\label {pms} \\end {equation}",
"$$ \\frac {1}{2}(T-\\bar T)|_{\\theta =0}=\\frac {1}{4}F\\tilde F $$",
"$$ \\mp \\frac {1}{2}(\\int d^2\\theta W^2-\\int d^2\\bar \\theta {\\bar W}^2)=\\frac {1}{4}|F\\tilde F|+\\cdots $$",
"$$ 2\\epsilon \\int d^2\\theta d^2\\bar \\theta \\; \\frac {W^2{\\bar W}^2}{D^2W^2 (D^2W^2-{\\bar D}^2{\\bar W}^2)}. $$",
"\\begin {eqnarray} &&\\pm \\epsilon \\int d^2\\theta d^2\\bar \\theta \\; W^2{\\bar W}^2(\\frac {1}{D^2W}+ \\frac {1}{{\\bar D}^2{\\bar W}^2})\\frac {1}{(D^2W^2-{\\bar D}^2{\\bar W}^2)}-\\\\&& \\epsilon \\int d^2\\theta d^2\\bar \\theta \\; \\frac {W^2{\\bar W}^2}{(D^2W^2{\\bar D}^2{\\bar W}^2)} = \\epsilon (\\frac {F^2}{|F\\tilde F|}-1) +\\cdots . \\end {eqnarray}",
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"\\begin{equation*} 2 \\epsilon \\int d ^ { 2 } \\theta d ^ { 2 } \\bar { \\theta } \\; \\frac { W ^ { 2 } \\bar { W } ^ { 2 } } { D ^ { 2 } W ^ { 2 } ( D ^ { 2 } W ^ { 2 } - \\bar { D } ^ { 2 } \\bar { W } ^ { 2 } ) } . \\end{equation*}",
"\\begin{align*} & & \\pm \\epsilon \\int d ^ { 2 } \\theta d ^ { 2 } \\bar { \\theta } \\; W ^ { 2 } \\bar { W } ^ { 2 } ( \\frac { 1 } { D ^ { 2 } W } + \\frac { 1 } { \\bar { D } ^ { 2 } \\bar { W } ^ { 2 } } ) \\frac { 1 } { ( D ^ { 2 } W ^ { 2 } - \\bar { D } ^ { 2 } \\bar { W } ^ { 2 } ) } - \\\\ & & \\epsilon \\int d ^ { 2 } \\theta d ^ { 2 } \\bar { \\theta } \\; \\frac { W ^ { 2 } \\bar { W } ^ { 2 } } { ( D ^ { 2 } W ^ { 2 } \\bar { D } ^ { 2 } \\bar { W } ^ { 2 } ) } = \\epsilon ( \\frac { F ^ { 2 } } { \\vert F \\widetilde { F } \\vert } - 1 ) + \\cdots . \\end{align*}",
"\\begin{equation*} L = \\sqrt { \\operatorname { d e t } ( \\epsilon ^ { \\frac { 1 } { 2 } } + F ) } - \\epsilon . \\end{equation*}"
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"$ \\mitY $",
"$ \\mitepsilon $",
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"$ \\mscrL _ { \\mitS \\mitB \\mitI } $",
"$ \\mitepsilon $",
"$ \\mitW $",
"$ \\bar { \\mitW } $",
"$ \\mitD $",
"$ \\bar { \\mitD } $",
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"$ \\mscrL _ { \\mitS \\mitB \\mitI } $",
"$ \\mitepsilon $",
"\\begin{equation*} \\mitX = \\mitepsilon - \\frac { 1 } { 2 } ( \\mitT + \\bar { \\mitT } ) , \\quad \\mitY = \\mitT \\bar { \\mitT } . \\end{equation*}",
"\\begin{align*} \\mscrL _ { \\mitS \\mitB \\mitI } & = & - \\frac { 1 } { 2 } \\int \\mitd ^ { 2 } \\mittheta \\; \\mitW ^ { 2 } - \\frac { 1 } { 2 } \\int \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\bar { \\mitW } ^ { 2 } - \\int \\mitd ^ { 2 } \\mittheta \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\frac { \\bar { \\mitW } ^ { 2 } \\mitW ^ { 2 } } { \\sqrt { \\mitX ^ { 2 } - \\mitY } + \\mitX } . \\end{align*}",
"\\begin{equation*} \\sqrt { \\mitX ^ { 2 } - \\mitY } = \\sqrt { \\mitepsilon ^ { 2 } - \\mitepsilon ( \\mitT + \\bar { \\mitT } ) + \\frac { 1 } { 4 } ( \\mitT - \\bar { \\mitT } ) ^ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\sqrt { \\frac { 1 } { 4 } ( \\mitT - \\bar { \\mitT } ) ^ { 2 } } = \\pm \\frac { 1 } { 2 } ( \\mitT - \\bar { \\mitT } ) \\end{equation*}",
"\\begin{equation*} \\frac { 1 } { 2 } ( \\mitT - \\bar { \\mitT } ) \\vert _ { \\mittheta = 0 } = \\frac { 1 } { 4 } \\mitF \\tilde { \\mitF } \\end{equation*}",
"\\begin{equation*} \\mp \\frac { 1 } { 2 } ( \\int \\mitd ^ { 2 } \\mittheta \\mitW ^ { 2 } - \\int \\mitd ^ { 2 } \\bar { \\mittheta } \\bar { \\mitW } ^ { 2 } ) = \\frac { 1 } { 4 } \\vert \\mitF \\tilde { \\mitF } \\vert + \\cdots \\end{equation*}",
"\\begin{equation*} 2 \\mitepsilon \\int \\mitd ^ { 2 } \\mittheta \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\frac { \\mitW ^ { 2 } \\bar { \\mitW } ^ { 2 } } { \\mitD ^ { 2 } \\mitW ^ { 2 } ( \\mitD ^ { 2 } \\mitW ^ { 2 } - \\bar { \\mitD } ^ { 2 } \\bar { \\mitW } ^ { 2 } ) } . \\end{equation*}",
"\\begin{align*} & & \\pm \\mitepsilon \\int \\mitd ^ { 2 } \\mittheta \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\mitW ^ { 2 } \\bar { \\mitW } ^ { 2 } ( \\frac { 1 } { \\mitD ^ { 2 } \\mitW } + \\frac { 1 } { \\bar { \\mitD } ^ { 2 } \\bar { \\mitW } ^ { 2 } } ) \\frac { 1 } { ( \\mitD ^ { 2 } \\mitW ^ { 2 } - \\bar { \\mitD } ^ { 2 } \\bar { \\mitW } ^ { 2 } ) } - \\\\ & & \\mitepsilon \\int \\mitd ^ { 2 } \\mittheta \\mitd ^ { 2 } \\bar { \\mittheta } \\; \\frac { \\mitW ^ { 2 } \\bar { \\mitW } ^ { 2 } } { ( \\mitD ^ { 2 } \\mitW ^ { 2 } \\bar { \\mitD } ^ { 2 } \\bar { \\mitW } ^ { 2 } ) } = \\mitepsilon ( \\frac { \\mitF ^ { 2 } } { \\vert \\mitF \\tilde { \\mitF } \\vert } - 1 ) + \\cdots . \\end{align*}",
"\\begin{equation*} \\mscrL = \\sqrt { \\operatorname { d e t } ( \\mitepsilon ^ { \\frac { 1 } { 2 } } + \\mitF ) } - \\mitepsilon . \\end{equation*}"
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0002084_page19 | {
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"$B$",
"$F\\mapsto F+B$",
"$W_\\alpha =\\bar D\\bar DD_\\alpha V$",
"$V$",
"$W_\\alpha -L_\\alpha $",
"$L_\\alpha $",
"$B$",
"$\\theta $",
"$W_\\alpha -L_\\alpha $",
"$U$",
"$\\epsilon \\mapsto 0$",
"$B$",
"$\\phi =\\chi _\\alpha =0$",
"$W_\\alpha $",
"$W_\\alpha -\\theta _\\beta \\sigma ^{\\mu \\nu }_{\\alpha \\beta }B_{\\mu \\nu }$",
"$\\mathcal {O}(\\epsilon )$",
"$N=2$",
"$W$",
"$N=2$",
"$\\alpha '\\mapsto 0$",
"$N=2$",
"$N=4$",
"$N=4$",
"$W^{ij}=-W^{ji}$",
"$$ L_\\alpha =\\theta ^\\beta (\\sigma ^{\\mu \\nu }_{\\alpha \\beta }B_{\\mu \\nu } +\\epsilon _{\\alpha \\beta }\\phi ) + \\theta ^2\\chi _\\alpha , $$",
"\\begin {eqnarray*} V&\\mapsto & V+U\\\\ L_\\alpha &\\mapsto &L_\\alpha +{\\bar D}^2D_\\alpha U. \\end {eqnarray*}",
"$$ \\frac {F^2}{|F\\tilde F|} $$",
"$$ \\mathcal {L}_{SBI}(N=2)=\\int d^4\\theta d^4\\bar \\theta \\;\\frac {W^2{\\bar W}^2}{D^4W^2-{\\bar D}^4{\\bar W}^2}( \\frac {1}{D^4W^2}+\\frac {1}{{\\bar D}^4{\\bar W}^2}) $$",
"$$ \\mathcal {L}_{SBI}(N=4)=\\int d^8\\theta d^8\\bar \\theta \\; \\frac {W^{4(0,4,0)}W^{4(0,4,0)}|_{\\mbox {singlet}}} {F_+^2-F_-^2}(\\frac {1}{F_+^4F_-^2} +\\frac {1}{F_-^4F_+^2}), $$",
"\\begin {eqnarray*} &&W^{ij}=\\frac {1}{2}\\epsilon ^{ijkl}{\\bar W}^{kl}\\\\ &&{\\bar D}_{i\\dot \\alpha }W^{jk}=\\frac {1}{3}(\\delta _i^jW^{lk}-\\delta _i^k{\\bar D}_{l\\dot \\alpha }W^{lj}),\\\\ &&D_\\alpha ^iW^{jk} +D_\\alpha ^jW^{ik}=0, \\end {eqnarray*}"
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"$ L _ { \\alpha } $",
"$ B $",
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"$ U $",
"$ \\epsilon \\mapsto 0 $",
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"$ W _ { \\alpha } $",
"$ W _ { \\alpha } - \\theta _ { \\beta } \\sigma _ { \\alpha \\beta } ^ { \\mu \\nu } B _ { \\mu \\nu } $",
"$ O ( \\epsilon ) $",
"$ N = 2 $",
"$ W $",
"$ N = 2 $",
"$ \\alpha ^ { \\prime } \\mapsto 0 $",
"$ N = 2 $",
"$ N = 4 $",
"$ N = 4 $",
"$ W ^ { i j } = - W ^ { j i } $",
"\\begin{equation*} L _ { \\alpha } = \\theta ^ { \\beta } ( \\sigma _ { \\alpha \\beta } ^ { \\mu \\nu } B _ { \\mu \\nu } + \\epsilon _ { \\alpha \\beta } \\phi ) + \\theta ^ { 2 } \\chi _ { \\alpha } , \\end{equation*}",
"\\begin{align*} V & \\mapsto & V + U \\\\ L _ { \\alpha } & \\mapsto & L _ { \\alpha } + \\bar { D } ^ { 2 } D _ { \\alpha } U . \\end{align*}",
"\\begin{equation*} \\frac { F ^ { 2 } } { \\vert F \\widetilde { F } \\vert } \\end{equation*}",
"\\begin{equation*} L _ { S B I } ( N = 2 ) = \\int d ^ { 4 } \\theta d ^ { 4 } \\bar { \\theta } \\; \\frac { W ^ { 2 } \\bar { W } ^ { 2 } } { D ^ { 4 } W ^ { 2 } - \\bar { D } ^ { 4 } \\bar { W } ^ { 2 } } ( \\frac { 1 } { D ^ { 4 } W ^ { 2 } } + \\frac { 1 } { \\bar { D } ^ { 4 } \\bar { W } ^ { 2 } } ) \\end{equation*}",
"\\begin{equation*} L _ { S B I } ( N = 4 ) = \\int d ^ { 8 } \\theta d ^ { 8 } \\bar { \\theta } \\; \\frac { W ^ { 4 ( 0 , 4 , 0 ) } W ^ { 4 ( 0 , 4 , 0 ) } \\vert _ { s i n g l e t } } { F _ { + } ^ { 2 } - F _ { - } ^ { 2 } } ( \\frac { 1 } { F _ { + } ^ { 4 } F _ { - } ^ { 2 } } + \\frac { 1 } { F _ { - } ^ { 4 } F _ { + } ^ { 2 } } ) , \\end{equation*}",
"\\begin{align*} & & W ^ { i j } = \\frac { 1 } { 2 } \\epsilon ^ { i j k l } \\bar { W } ^ { k l } \\\\ & & \\bar { D } _ { i \\dot { \\alpha } } W ^ { j k } = \\frac { 1 } { 3 } ( \\delta _ { i } ^ { j } W ^ { l k } - \\delta _ { i } ^ { k } \\bar { D } _ { l \\dot { \\alpha } } W ^ { l j } ) , \\\\ & & D _ { \\alpha } ^ { i } W ^ { j k } + D _ { \\alpha } ^ { j } W ^ { i k } = 0 , \\end{align*}"
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"$ \\mitW _ { \\mitalpha } = \\bar { \\mitD } \\bar { \\mitD } \\mitD _ { \\mitalpha } \\mitV $",
"$ \\mitV $",
"$ \\mitW _ { \\mitalpha } - \\mitL _ { \\mitalpha } $",
"$ \\mitL _ { \\mitalpha } $",
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"$ \\mittheta $",
"$ \\mitW _ { \\mitalpha } - \\mitL _ { \\mitalpha } $",
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"$ \\mitW _ { \\mitalpha } $",
"$ \\mitW _ { \\mitalpha } - \\mittheta _ { \\mitbeta } \\mitsigma _ { \\mitalpha \\mitbeta } ^ { \\mitmu \\mitnu } \\mitB _ { \\mitmu \\mitnu } $",
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"$ \\mitN = 4 $",
"$ \\mitN = 4 $",
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"\\begin{equation*} \\mitL _ { \\mitalpha } = \\mittheta ^ { \\mitbeta } ( \\mitsigma _ { \\mitalpha \\mitbeta } ^ { \\mitmu \\mitnu } \\mitB _ { \\mitmu \\mitnu } + \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitphi ) + \\mittheta ^ { 2 } \\mitchi _ { \\mitalpha } , \\end{equation*}",
"\\begin{align*} \\mitV & \\mapsto & \\mitV + \\mitU \\\\ \\mitL _ { \\mitalpha } & \\mapsto & \\mitL _ { \\mitalpha } + \\bar { \\mitD } ^ { 2 } \\mitD _ { \\mitalpha } \\mitU . \\end{align*}",
"\\begin{equation*} \\frac { \\mitF ^ { 2 } } { \\vert \\mitF \\tilde { \\mitF } \\vert } \\end{equation*}",
"\\begin{equation*} \\mscrL _ { \\mitS \\mitB \\mitI } ( \\mitN = 2 ) = \\int \\mitd ^ { 4 } \\mittheta \\mitd ^ { 4 } \\bar { \\mittheta } \\; \\frac { \\mitW ^ { 2 } \\bar { \\mitW } ^ { 2 } } { \\mitD ^ { 4 } \\mitW ^ { 2 } - \\bar { \\mitD } ^ { 4 } \\bar { \\mitW } ^ { 2 } } ( \\frac { 1 } { \\mitD ^ { 4 } \\mitW ^ { 2 } } + \\frac { 1 } { \\bar { \\mitD } ^ { 4 } \\bar { \\mitW } ^ { 2 } } ) \\end{equation*}",
"\\begin{equation*} \\mscrL _ { \\mitS \\mitB \\mitI } ( \\mitN = 4 ) = \\int \\mitd ^ { 8 } \\mittheta \\mitd ^ { 8 } \\bar { \\mittheta } \\; \\frac { \\mitW ^ { 4 ( 0 , 4 , 0 ) } \\mitW ^ { 4 ( 0 , 4 , 0 ) } \\vert _ { \\mathrm { s i n g l e t } } } { \\mitF _ { + } ^ { 2 } - \\mitF _ { - } ^ { 2 } } ( \\frac { 1 } { \\mitF _ { + } ^ { 4 } \\mitF _ { - } ^ { 2 } } + \\frac { 1 } { \\mitF _ { - } ^ { 4 } \\mitF _ { + } ^ { 2 } } ) , \\end{equation*}",
"\\begin{align*} & & \\mitW ^ { \\miti \\mitj } = \\frac { 1 } { 2 } \\mitepsilon ^ { \\miti \\mitj \\mitk \\mitl } \\bar { \\mitW } ^ { \\mitk \\mitl } \\\\ & & \\bar { \\mitD } _ { \\miti \\dot { \\mitalpha } } \\mitW ^ { \\mitj \\mitk } = \\frac { 1 } { 3 } ( \\mitdelta _ { \\miti } ^ { \\mitj } \\mitW ^ { \\mitl \\mitk } - \\mitdelta _ { \\miti } ^ { \\mitk } \\bar { \\mitD } _ { \\mitl \\dot { \\mitalpha } } \\mitW ^ { \\mitl \\mitj } ) , \\\\ & & \\mitD _ { \\mitalpha } ^ { \\miti } \\mitW ^ { \\mitj \\mitk } + \\mitD _ { \\mitalpha } ^ { \\mitj } \\mitW ^ { \\miti \\mitk } = 0 , \\end{align*}"
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0002084_page20 | {
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0002106_page01 | {
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0002106_page03 | {
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0002106_page04 | {
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0002106_page05 | {
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"$ \\vert \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) \\vert \\approx \\sqrt { \\dot { \\mitx } ^ { 2 } ( \\mits ) } \\vert \\mits - \\mits ^ { \\prime } \\vert $",
"$ \\mits \\rightarrow \\mits ^ { \\prime } $",
"$ \\mits = \\mits ^ { \\prime } $",
"$ \\widehat { \\mitL } ( \\mits ) \\mitW $",
"$ \\mitdelta $",
"$ \\mitp $",
"$ \\mitdelta $",
"$ \\mitW [ \\mitC ] $",
"$ \\mupDelta _ { \\mitn } $",
"$ \\mitp ^ { \\mitk } ( \\mathrm { l o g } \\mitp + \\mathrm { c o n s t } ) $",
"$ \\nabla _ { \\mitalpha } \\mitF _ { \\mitbeta \\mitgamma } $",
"$ \\mitF _ { \\mitmu \\mitnu } $",
"$ \\dot { \\mitx } _ { \\mitlambda } ( \\mits ) \\dot { \\mitx } _ { \\mitsigma } ( \\mits ^ { \\prime } ) $",
"\\begin{equation*} \\int \\mitdelta ( \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) ) \\, \\mitf ( \\mits ^ { \\prime } ) \\, \\mitd \\mits ^ { \\prime } \\, = \\underset { \\mupDelta \\rightarrow 4 } { \\operatorname { l i m } } ( 4 - \\mupDelta ) \\int \\frac { \\mitf ( \\mits ^ { \\prime } ) \\, \\mitd \\mits ^ { \\prime } } { \\vert \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) \\vert ^ { \\mupDelta } } = 0 \\; , \\end{equation*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mitW } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } = ( \\widehat { \\mitL } ( \\mits ) \\mitW ) \\mitdelta ( \\mits - \\mits ^ { \\prime } ) + \\sum _ { \\mitn } \\frac { \\mitC _ { \\mitn } ( \\mits , \\{ \\mitx ( \\mits ) \\} ) } { \\vert \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) \\vert ^ { 4 - \\mupDelta _ { \\mitn } } } \\; , \\end{equation*}",
"\\begin{equation*} \\widehat { \\mitL } ( \\mits ) \\mitW = 0 \\; . \\end{equation*}",
"\\begin{equation*} \\underset { \\mitp \\rightarrow \\infty } { \\operatorname { l i m } } \\frac { \\mitdelta ^ { 2 } \\mitW } { \\mitdelta \\mitx _ { \\mitmu } \\left( \\frac { \\mitq } { 2 } + \\mitp \\right) \\mitdelta \\mitx _ { \\mitmu } \\left( \\frac { \\mitq } { 2 } - \\mitp \\right) } = ( \\widehat { \\mitL } _ { \\mitq } \\mitW ) \\mitp ^ { 0 } + \\sum _ { \\mitn } \\mitC _ { \\mitn } ( \\mitq ) \\vert \\mitp \\vert ^ { \\mitlambda _ { \\mitn } } \\; . \\end{equation*}",
"\\begin{equation*} \\mitF _ { \\mitmu \\mitlambda } ( \\mitx ( \\mits ) ) \\mitF _ { \\mitmu \\mitsigma } ( \\mitx ( \\mits ^ { \\prime } ) ) \\sim \\frac { ( \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) ) _ { \\mitmu } \\nabla _ { \\mitmu } \\mitF _ { \\mitlambda \\mitsigma } } { { \\left\\vert \\mitx ( \\mits ) - \\mitx ( \\mits ^ { \\prime } ) \\right\\vert } ^ { 2 } } \\; . \\end{equation*}"
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0002106_page06 | {
"latex": [
"\\( s\\Rightarrow \\alpha (s) \\)",
"\\( s\\Rightarrow \\alpha (s) \\)",
"\\( \\alpha '(s) \\)",
"\\( \\dot {x}^{2} \\)",
"\\( \\varpropto p^{n} \\)",
"\\( n \\)",
"\\( \\sigma \\protect \\)",
"\\( \\sigma \\)",
"\\begin {equation} \\label {2.8} W[C]=\\sum _{n}\\int _{s_{1}<s_{2}\\ldots <s_{n}}\\Gamma ^{(n)}_{\\mu _{1}\\ldots \\mu _{n}}(x(s_{1}),\\ldots x(s_{n}))\\, \\dot {x}_{\\mu _{1}}(s_{1})\\ldots \\dot {x}_{\\mu _{n}}(s_{n})\\, ds_{1}\\ldots ds_{n} \\end {equation}",
"\\[ I_{\\alpha \\beta }=\\int ds\\left ( \\frac {\\ddot {x}_{\\alpha }\\dot {x}_{\\beta }-\\ddot {x}_{\\beta }\\dot {x}_{\\alpha }}{\\dot {x}^{2}}\\right ) \\, .\\]",
"\\begin {eqnarray} S & = & \\frac {1}{2}\\int d^{2}\\xi \\sqrt {{g}}g^{ab}(\\xi )G_{MN}(z(\\xi ))\\partial _{a}z^{M}\\partial _{b}z^{N}+\\Phi (z(\\xi ))\\, ^{(2)}\\! R(g)\\sqrt {{g}}\\\\ && \\quad +\\, \\epsilon ^{ab}B_{MN}(z(\\xi ))\\partial _{a}z^{M}\\partial _{b}z^{N}+\\textrm {other background fields}+\\textrm {fermions}.\\end {eqnarray}"
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"$ n $",
"$ \\sigma $",
"$ \\sigma $",
"\\begin{equation*} W [ C ] = \\sum _ { n } \\int _ { s _ { 1 } < s _ { 2 } \\ldots < s _ { n } } \\Gamma _ { \\mu _ { 1 } \\ldots \\mu _ { n } } ^ { ( n ) } ( x ( s _ { 1 } ) , \\ldots x ( s _ { n } ) ) \\, \\dot { x } _ { \\mu _ { 1 } } ( s _ { 1 } ) \\ldots \\dot { x } _ { \\mu _ { n } } ( s _ { n } ) \\, d s _ { 1 } \\ldots d s _ { n } \\end{equation*}",
"\\begin{equation*} I _ { \\alpha \\beta } = \\int d s ( \\frac { \\ddot { x } _ { \\alpha } \\dot { x } _ { \\beta } - \\ddot { x } _ { \\beta } \\dot { x } _ { \\alpha } } { \\dot { x } ^ { 2 } } ) \\, . \\end{equation*}",
"\\begin{align*} S & = & \\frac { 1 } { 2 } \\int d ^ { 2 } \\xi \\sqrt { g } g ^ { a b } ( \\xi ) G _ { M N } ( z ( \\xi ) ) \\partial _ { a } z ^ { M } \\partial _ { b } z ^ { N } + \\Phi ( z ( \\xi ) ) \\, { } ^ { ( 2 ) } \\! R ( g ) \\sqrt { g } \\\\ & & \\quad + \\, \\epsilon ^ { a b } B _ { M N } ( z ( \\xi ) ) \\partial _ { a } z ^ { M } \\partial _ { b } z ^ { N } + o t h e r ~ b a c k g r o u n d ~ f i e l d s + f e r m i o n s . \\end{align*}"
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"$ \\mits \\Rightarrow \\mitalpha ( \\mits ) $",
"$ \\mitalpha ^ { \\prime } ( \\mits ) $",
"$ \\dot { \\mitx } ^ { 2 } $",
"$ \\propto \\mitp ^ { \\mitn } $",
"$ \\mitn $",
"$ \\mitsigma $",
"$ \\mitsigma $",
"\\begin{equation*} \\mitW [ \\mitC ] = \\sum _ { \\mitn } \\int _ { \\mits _ { 1 } < \\mits _ { 2 } \\ldots < \\mits _ { \\mitn } } \\mupGamma _ { \\mitmu _ { 1 } \\ldots \\mitmu _ { \\mitn } } ^ { ( \\mitn ) } ( \\mitx ( \\mits _ { 1 } ) , \\ldots \\mitx ( \\mits _ { \\mitn } ) ) \\, \\dot { \\mitx } _ { \\mitmu _ { 1 } } ( \\mits _ { 1 } ) \\ldots \\dot { \\mitx } _ { \\mitmu _ { \\mitn } } ( \\mits _ { \\mitn } ) \\, \\mitd \\mits _ { 1 } \\ldots \\mitd \\mits _ { \\mitn } \\end{equation*}",
"\\begin{equation*} \\mitI _ { \\mitalpha \\mitbeta } = \\int \\mitd \\mits \\left( \\frac { \\ddot { \\mitx } _ { \\mitalpha } \\dot { \\mitx } _ { \\mitbeta } - \\ddot { \\mitx } _ { \\mitbeta } \\dot { \\mitx } _ { \\mitalpha } } { \\dot { \\mitx } ^ { 2 } } \\right) \\, . \\end{equation*}",
"\\begin{align*} \\mitS & = & \\frac { 1 } { 2 } \\int \\mitd ^ { 2 } \\mitxi \\sqrt { \\mitg } \\mitg ^ { \\mita \\mitb } ( \\mitxi ) \\mitG _ { \\mitM \\mitN } ( \\mitz ( \\mitxi ) ) \\mitpartial _ { \\mita } \\mitz ^ { \\mitM } \\mitpartial _ { \\mitb } \\mitz ^ { \\mitN } + \\mupPhi ( \\mitz ( \\mitxi ) ) \\, { } ^ { ( 2 ) } \\! \\mitR ( \\mitg ) \\sqrt { \\mitg } \\\\ & & \\quad + \\, \\mitepsilon ^ { \\mita \\mitb } \\mitB _ { \\mitM \\mitN } ( \\mitz ( \\mitxi ) ) \\mitpartial _ { \\mita } \\mitz ^ { \\mitM } \\mitpartial _ { \\mitb } \\mitz ^ { \\mitN } + \\mathrm { o t h e r ~ b a c k g r o u n d ~ f i e l d s + f e r m i o n s } . \\end{align*}"
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0002106_page07 | {
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"\\( z^{M}=(x^{\\mu },y) \\)",
"\\( G_{MN} \\)",
"\\( \\beta \\)",
"\\( g^{ab} \\)",
"\\( \\rho (y) \\)",
"\\( V \\)",
"\\( \\textrm {Tr}F_{\\mu \\nu }^{2} \\)",
"\\( S \\)",
"\\begin {equation} \\label {3.2} ds^{2}=\\rho (y)(dy^{2}+d\\vec {x}\\, ^{2})\\; . \\end {equation}",
"\\[ V_{\\mu \\nu }=\\int d^{2}\\xi \\left ( \\vphantom {\\sum }\\psi _{\\vec {p}}(y(\\xi ))\\partial _{a}x^{\\mu }\\partial _{a}x^{\\nu }\\right ) e^{i\\vec {p}\\, \\vec {x}(\\xi )}\\]",
"\\begin {equation} \\label {3.4} \\begin {array}{c} \\textrm {Closed string}\\\\ \\textrm {states} \\end {array}\\Longleftrightarrow \\begin {array}{c} \\textrm {Gauge invariant}\\\\ \\textrm {operators}. \\end {array} \\end {equation}",
"\\[ \\textrm {Tr}P\\exp \\oint \\limits _{C}A\\, dx\\mathop {\\approx }_{C\\to 0}I+C_{\\mu \\nu \\lambda \\sigma }\\textrm {Tr}(F_{\\mu \\nu }F_{\\lambda \\sigma })+\\ldots \\]"
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"$ V $",
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"\\begin{equation*} d s ^ { 2 } = \\rho ( y ) ( d y ^ { 2 } + d \\vec { x } \\, { } ^ { 2 } ) \\; . \\end{equation*}",
"\\begin{equation*} V _ { \\mu \\nu } = \\int d ^ { 2 } \\xi ( \\psi _ { \\vec { p } } ( y ( \\xi ) ) \\partial _ { a } x ^ { \\mu } \\partial _ { a } x ^ { \\nu } ) e ^ { i \\vec { p } \\, \\vec { x } ( \\xi ) } \\end{equation*}",
"\\begin{align*} \\begin{array}{c} C l o s e d ~ s t r i n g \\\\ s t a t e s \\end{array} \\Longleftrightarrow \\begin{array}{c} G a u g e ~ i n v a r i a n t \\\\ o p e r a t o r s . \\end{array} \\end{align*}",
"\\begin{equation*} T r P \\operatorname { e x p } \\oint _ { C } A \\, d x \\underset { C \\rightarrow 0 } { \\approx } I + C _ { \\mu \\nu \\lambda \\sigma } T r ( F _ { \\mu \\nu } F _ { \\lambda \\sigma } ) + \\ldots \\end{equation*}"
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"$ \\mitrho ( \\mity ) $",
"$ \\mitV $",
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"\\begin{equation*} \\mitV _ { \\mitmu \\mitnu } = \\int \\mitd ^ { 2 } \\mitxi \\left( \\mitpsi _ { \\vec { \\mitp } } ( \\mity ( \\mitxi ) ) \\mitpartial _ { \\mita } \\mitx ^ { \\mitmu } \\mitpartial _ { \\mita } \\mitx ^ { \\mitnu } \\right) \\mite ^ { \\miti \\vec { \\mitp } \\, \\vec { \\mitx } ( \\mitxi ) } \\end{equation*}",
"\\begin{align*} \\begin{array}{c} \\mathrm { C l o s e d ~ s t r i n g } \\\\ \\mathrm { s t a t e s } \\end{array} \\Longleftrightarrow \\begin{array}{c} \\mathrm { G a u g e ~ i n v a r i a n t } \\\\ \\mathrm { o p e r a t o r s } . \\end{array} \\end{align*}",
"\\begin{equation*} \\mathrm { T r } \\mitP \\operatorname { e x p } \\oint \\limits _ { \\mitC } \\mitA \\, \\mitd \\mitx \\underset { \\mitC \\rightarrow 0 } { \\approx } \\mitI + \\mitC _ { \\mitmu \\mitnu \\mitlambda \\mitsigma } \\mathrm { T r } ( \\mitF _ { \\mitmu \\mitnu } \\mitF _ { \\mitlambda \\mitsigma } ) + \\ldots \\end{equation*}"
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0002106_page08 | {
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0002106_page09 | {
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0002106_page10 | {
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"\\begin {equation} \\label {3.75} W[C]\\varpropto e^{-\\sqrt {g_{YM}^{2}N}\\cdot A_{\\min }[C]}\\; . \\end {equation}",
"\\begin {equation} \\label {3.9} A_{\\min }[C]=\\min \\frac {1}{2}\\int \\frac {d^{2}\\xi }{y^{2}}\\left ( (\\partial _{a}\\vec {x})^{2}+(\\partial _{a}y)^{2}\\right ) \\end {equation}",
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"\\begin {equation} \\label {4.2} \\left \\{ \\begin {array}{l} \\dis \\left ( \\partial _{1}\\vec {x}\\right ) ^{2}+\\left ( \\partial _{1}y\\right ) ^{2}=\\left ( \\partial _{2}\\vec {x}\\right ) ^{2}+\\left ( \\partial _{2}y\\right ) ^{2}\\; ,\\\\ \\dis \\partial _{1}\\vec {x}\\, \\partial _{2}\\vec {x}+\\partial _{1}y\\, \\partial _{2}y=0\\; . \\end {array}\\right . \\end {equation}",
"\\[ \\left \\{ \\begin {array}{l} \\vec {x}(\\sigma ,0)=\\vec {c}(\\sigma )\\; ,\\\\ y(\\sigma ,0)=0\\; . \\end {array}\\right . \\]",
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"\\begin{equation*} W [ C ] \\propto e ^ { - \\sqrt { g _ { Y M } ^ { 2 } N } \\cdot A _ { \\operatorname { m i n } } [ C ] } \\; . \\end{equation*}",
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"\\begin{align*} \\{ \\begin{array}{l} { ( \\partial _ { 1 } \\vec { x } ) } ^ { 2 } + { ( \\partial _ { 1 } y ) } ^ { 2 } = { ( \\partial _ { 2 } \\vec { x } ) } ^ { 2 } + { ( \\partial _ { 2 } y ) } ^ { 2 } \\; , \\\\ \\partial _ { 1 } \\vec { x } \\, \\partial _ { 2 } \\vec { x } + \\partial _ { 1 } y \\, \\partial _ { 2 } y = 0 \\; . \\end{array} \\end{align*}",
"\\begin{equation*} \\{ \\begin{array}{l} \\vec { x } ( \\sigma , 0 ) = \\vec { c } ( \\sigma ) \\; , \\\\ y ( \\sigma , 0 ) = 0 \\; . \\end{array} \\end{equation*}",
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0002106_page11 | {
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"\\( y_{\\min }=a(\\sigma )\\tau _{\\min }=\\epsilon \\)",
"\\( L[C] \\)",
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"\\( \\mathcal {A}[\\vec {c}(\\sigma )] \\)",
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"\\begin {equation} \\label {4.5} \\left \\{ \\begin {array}{l} \\dis a^{2}(\\sigma )=\\left ( \\frac {d\\vec {c}}{d\\sigma }\\right ) ^{2}\\; ,\\\\ \\dis \\vec {f}(\\sigma )=\\left ( \\frac {d\\vec {c}}{d\\sigma }\\right ) ^{2}\\frac {d}{d\\sigma }\\left ( \\frac {\\partial _{\\sigma }\\vec {c}}{\\left ( \\partial _{\\sigma }\\vec {c}\\, \\right ) ^{2}}\\right ) \\; . \\end {array}\\right . \\end {equation}",
"\\[ \\theta _{\\bot \\parallel }=\\frac {1}{y^{2}}\\left ( \\partial _{\\tau }\\vec {x}\\, \\partial _{\\sigma }\\vec {x}+\\partial _{\\tau }y\\, \\partial _{\\sigma }y\\right ) =\\frac {1}{a^{2}\\tau }\\left [ \\vec {f}{\\vec {c}\\, }'+aa'\\right ] +\\frac {1}{a^{2}}\\left ( \\vec {g}{\\vec {c}\\, }'\\right ) +\\ldots \\]",
"\\[ \\theta _{\\bot \\parallel }=\\frac {1}{a^{2}}\\left ( \\vec {g}{\\vec {c}\\, }'\\right ) \\; .\\]",
"\\begin {eqnarray} &\\theta _{\\bot \\bot } & =\\dis \\frac {1}{2y^{2}}\\left [ \\left ( \\partial _{\\tau }\\vec {x}\\right ) ^{2}+\\left ( \\partial _{\\tau }y\\right ) ^{2}-\\left ( \\partial _{\\sigma }\\vec {x}\\right ) ^{2}-\\left ( \\partial _{\\sigma }y\\right ) ^{2}\\right ] \\\\ && =\\dis \\frac {1}{a^{2}}\\left [ {\\vec {f}\\, }^{2}+2ab-{\\vec {c}\\, }'\\vec {f}\\, '-{a'}^{2}\\right ] \\; .\\end {eqnarray}",
"\\begin {equation} \\label {4.9} A_{\\min }[C]=\\frac {L[C]}{\\epsilon }+\\mathcal {A}[\\vec {c}(\\sigma )]\\; , \\end {equation}",
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"\\begin{equation*} \\mittheta _ { \\bot \\parallel } = \\frac { 1 } { \\mita ^ { 2 } } \\left( \\vec { \\mitg } { \\vec { \\mitc } \\, } ^ { \\prime } \\right) \\; . \\end{equation*}",
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0002106_page12 | {
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"\\( \\tau \\)",
"\\( \\mathcal {A}[C] \\)",
"\\( \\mathcal {A}[C] \\)",
"\\( \\phi ^{i}(s) \\)",
"\\( \\mathcal {A}[\\vec {\\phi }(s)] \\)",
"\\( y(s)=y\\to 0. \\)",
"\\( \\frac {\\delta A}{\\delta y} \\)",
"\\( A(y) \\)",
"\\[ x^{1}(s)=s,\\quad x^{i}(s)=\\phi ^{i}(s),\\quad i=2,\\ldots D\\; ,\\]",
"\\[ G^{MN}(z)\\frac {\\delta A}{\\delta z^{M}(s)}\\frac {\\delta A}{\\delta z^{N}(s)}=G_{MN}\\frac {dz^{M}}{ds}\\frac {dz^{N}}{ds}\\; .\\]",
"\\begin {equation} \\label {5.3} \\left ( \\frac {\\delta A}{\\delta y(s)}\\right ) ^{2}+\\left ( \\frac {\\delta A}{\\delta \\vec {x}(s)}\\right ) ^{2}=\\frac {1}{y^{4}(s)}\\left \\{ \\left ( \\frac {dy}{ds}\\right ) ^{2}+\\left ( \\frac {d\\vec {x}}{ds}\\right ) ^{2}\\right \\} \\; . \\end {equation}",
"\\begin {equation} \\label {5.4} \\frac {\\partial A}{\\partial y}=\\int ds\\frac {\\delta A}{\\delta y(s)}\\Bigl |_{y(s)=y}\\; . \\end {equation}",
"\\begin {equation} \\label {5.5} \\frac {\\partial A}{\\partial y}=-\\frac {1}{y^{2}}\\int ds\\sqrt {{\\left ( \\frac {d\\vec {x}}{ds}\\right ) ^{2}-y^{4}\\left ( \\frac {\\delta A}{\\delta \\vec {x}(s)}\\right ) ^{2}}}\\; . \\end {equation}",
"\\[ A(y)\\mathop {\\approx }_{y\\to 0}\\frac {L[C]}{y}+O(1)\\; .\\]"
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"\\begin{equation*} x ^ { 1 } ( s ) = s , \\quad x ^ { i } ( s ) = \\phi ^ { i } ( s ) , \\quad i = 2 , \\ldots D \\; , \\end{equation*}",
"\\begin{equation*} G ^ { M N } ( z ) \\frac { \\delta A } { \\delta z ^ { M } ( s ) } \\frac { \\delta A } { \\delta z ^ { N } ( s ) } = G _ { M N } \\frac { d z ^ { M } } { d s } \\frac { d z ^ { N } } { d s } \\; . \\end{equation*}",
"\\begin{equation*} { ( \\frac { \\delta A } { \\delta y ( s ) } ) } ^ { 2 } + { ( \\frac { \\delta A } { \\delta \\vec { x } ( s ) } ) } ^ { 2 } = \\frac { 1 } { y ^ { 4 } ( s ) } \\{ { ( \\frac { d y } { d s } ) } ^ { 2 } + { ( \\frac { d \\vec { x } } { d s } ) } ^ { 2 } \\} \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\partial A } { \\partial y } = \\int d s \\frac { \\delta A } { \\delta y ( s ) } \\vert _ { y ( s ) = y } \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\partial A } { \\partial y } = - \\frac { 1 } { y ^ { 2 } } \\int d s \\sqrt { { ( \\frac { d \\vec { x } } { d s } ) } ^ { 2 } - y ^ { 4 } { ( \\frac { \\delta A } { \\delta \\vec { x } ( s ) } ) } ^ { 2 } } \\; . \\end{equation*}",
"\\begin{equation*} A ( y ) \\underset { y \\rightarrow 0 } { \\approx } \\frac { L [ C ] } { y } + O ( 1 ) \\; . \\end{equation*}"
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"\\begin{equation*} \\mitG ^ { \\mitM \\mitN } ( \\mitz ) \\frac { \\mitdelta \\mitA } { \\mitdelta \\mitz ^ { \\mitM } ( \\mits ) } \\frac { \\mitdelta \\mitA } { \\mitdelta \\mitz ^ { \\mitN } ( \\mits ) } = \\mitG _ { \\mitM \\mitN } \\frac { \\mitd \\mitz ^ { \\mitM } } { \\mitd \\mits } \\frac { \\mitd \\mitz ^ { \\mitN } } { \\mitd \\mits } \\; . \\end{equation*}",
"\\begin{equation*} { \\left( \\frac { \\mitdelta \\mitA } { \\mitdelta \\mity ( \\mits ) } \\right) } ^ { 2 } + { \\left( \\frac { \\mitdelta \\mitA } { \\mitdelta \\vec { \\mitx } ( \\mits ) } \\right) } ^ { 2 } = \\frac { 1 } { \\mity ^ { 4 } ( \\mits ) } \\left\\{ { \\left( \\frac { \\mitd \\mity } { \\mitd \\mits } \\right) } ^ { 2 } + { \\left( \\frac { \\mitd \\vec { \\mitx } } { \\mitd \\mits } \\right) } ^ { 2 } \\right\\} \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitpartial \\mitA } { \\mitpartial \\mity } = \\int \\mitd \\mits \\frac { \\mitdelta \\mitA } { \\mitdelta \\mity ( \\mits ) } \\Big| _ { \\mity ( \\mits ) = \\mity } \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitpartial \\mitA } { \\mitpartial \\mity } = - \\frac { 1 } { \\mity ^ { 2 } } \\int \\mitd \\mits \\sqrt { { \\left( \\frac { \\mitd \\vec { \\mitx } } { \\mitd \\mits } \\right) } ^ { 2 } - \\mity ^ { 4 } { \\left( \\frac { \\mitdelta \\mitA } { \\mitdelta \\vec { \\mitx } ( \\mits ) } \\right) } ^ { 2 } } \\; . \\end{equation*}",
"\\begin{equation*} \\mitA ( \\mity ) \\underset { \\mity \\rightarrow 0 } { \\approx } \\frac { \\mitL [ \\mitC ] } { \\mity } + \\mitO ( 1 ) \\; . \\end{equation*}"
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|
0002106_page13 | {
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"\\( A \\)",
"\\( \\Gamma \\)",
"\\( \\vec {\\pi }=\\delta A/\\delta \\vec {\\phi } \\)",
"\\( s \\)",
"\\( B_{4} \\)",
"\\( \\omega _{i}=\\Gamma _{2}(p_{i}) \\)",
"\\[ A=\\sum _{n}\\frac {1}{n!}\\int ds_{1}\\ldots ds_{n}\\Gamma _{i_{1}\\ldots i_{n}}(s_{1},\\ldots s_{n}|y)\\bigl (\\phi _{i_{1}}(s_{1})\\ldots \\phi _{i_{n}}(s_{n})\\bigl )\\; .\\]",
"\\begin {equation} \\label {5.75} \\frac {dx_{\\mu }}{ds}\\frac {\\delta A}{\\delta x_{\\mu }(s)}=\\frac {\\delta A}{\\delta x_{1}(s)}\\Bigl |_{x_{1}=s}+\\frac {d\\vec {\\phi }}{ds}\\frac {\\delta A}{\\delta \\vec {\\phi }}=0 \\end {equation}",
"\\begin {eqnarray} &\\dis \\frac {\\partial A}{\\partial y} & =-\\frac {L_{0}}{y^{2}}+\\frac {1}{2}\\int \\left ( y^{2}\\vec {\\pi }^{2}-\\frac {1}{y^{2}}{\\vec {\\phi }\\, '}^{2}\\right ) ds\\\\ && \\qquad \\qquad +\\frac {1}{8}\\int \\left ( \\frac {1}{y^{2}}\\left ( y^{4}\\vec {\\pi }^{2}-{\\vec {\\phi }\\, '}^{2}\\right ) ^{2}+4y^{2}\\left ( \\vec {\\phi }\\, '\\vec {\\pi }\\right ) ^{2}\\right ) ds+\\ldots \\end {eqnarray}",
"\\begin {equation} \\label {5.9} \\left \\{ \\begin {array}{l} \\dis \\frac {d\\Gamma _{2}}{dy}=y^{2}\\Gamma ^{2}_{2}-\\frac {p^{2}}{y^{2}}\\; ,\\\\ \\dis \\frac {d\\Gamma _{4}}{dy}=y^{2}\\left ( \\sum _{1}^{4}\\Gamma _{2}(p_{i})\\right ) \\Gamma _{4}(p_{1},\\ldots p_{4})-B_{4}(p_{1},\\ldots p_{4})\\; . \\end {array}\\right . \\end {equation}",
"\\begin {eqnarray} &A & =\\frac {L_{0}}{y}+\\frac {1}{2}\\int \\Gamma _{2}(p)\\left ( \\vec {\\phi }_{p}\\vec {\\phi }_{-p}\\right ) \\, dp-\\\\ && \\qquad -\\frac {1}{8}\\int \\Gamma _{4}(p_{1},\\ldots p_{4})\\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\left ( \\vec {\\phi }_{p_{3}}\\vec {\\phi }_{p_{4}}\\right ) \\delta \\left ( \\sum p_{i}\\right ) dp_{1}\\ldots dp_{4}+\\ldots \\end {eqnarray}",
"\\begin {eqnarray} (2\\pi )B_{4}(p_{1},\\ldots p_{4}) & = & C_{4}(p_{1},\\ldots p_{4})+D(p_{1},p_{2})+D(p_{3},p_{4})-\\\\ && \\qquad -D(p_{1},p_{3})-D(p_{1},p_{4})-D(p_{2},p_{3})-D(p_{2},p_{3})\\; .\\end {eqnarray}",
"\\begin {eqnarray} C_{4} & = & \\left ( p_{1}p_{2}p_{3}p_{4}\\right ) \\frac {1}{y^{2}}+y^{6}\\left ( \\omega _{1}\\omega _{2}\\omega _{3}\\omega _{4}\\right ) \\; ,\\\\ D(p_{1},p_{2}) & = & (p_{1}p_{2}\\omega _{3}\\omega _{4})y^{2}\\; ,\\end {eqnarray}"
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"\\begin{equation*} A = \\sum _ { n } \\frac { 1 } { n ! } \\int d s _ { 1 } \\ldots d s _ { n } \\Gamma _ { i _ { 1 } \\ldots i _ { n } } ( s _ { 1 } , \\ldots s _ { n } \\vert y ) ( \\phi _ { i _ { 1 } } ( s _ { 1 } ) \\ldots \\phi _ { i _ { n } } ( s _ { n } ) ) \\; . \\end{equation*}",
"\\begin{equation*} \\frac { d x _ { \\mu } } { d s } \\frac { \\delta A } { \\delta x _ { \\mu } ( s ) } = \\frac { \\delta A } { \\delta x _ { 1 } ( s ) } \\vert _ { x _ { 1 } = s } + \\frac { d \\vec { \\phi } } { d s } \\frac { \\delta A } { \\delta \\vec { \\phi } } = 0 \\end{equation*}",
"\\begin{align*} & \\frac { \\partial A } { \\partial y } & = - \\frac { L _ { 0 } } { y ^ { 2 } } + \\frac { 1 } { 2 } \\int ( y ^ { 2 } \\vec { \\pi } ^ { 2 } - \\frac { 1 } { y ^ { 2 } } { \\vec { \\phi } \\, { } ^ { \\prime } } ^ { 2 } ) d s \\\\ & & \\qquad \\qquad + \\frac { 1 } { 8 } \\int ( \\frac { 1 } { y ^ { 2 } } { ( y ^ { 4 } \\vec { \\pi } ^ { 2 } - { \\vec { \\phi } \\, { } ^ { \\prime } } ^ { 2 } ) } ^ { 2 } + 4 y ^ { 2 } { ( \\vec { \\phi } \\, { } ^ { \\prime } \\vec { \\pi } ) } ^ { 2 } ) d s + \\ldots \\end{align*}",
"\\begin{align*} \\{ \\begin{array}{l} \\frac { d \\Gamma _ { 2 } } { d y } = y ^ { 2 } \\Gamma _ { 2 } ^ { 2 } - \\frac { p ^ { 2 } } { y ^ { 2 } } \\; , \\\\ \\frac { d \\Gamma _ { 4 } } { d y } = y ^ { 2 } ( \\sum _ { 1 } ^ { 4 } \\Gamma _ { 2 } ( p _ { i } ) ) \\Gamma _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) - B _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) \\; . \\end{array} \\end{align*}",
"\\begin{align*} & A & = \\frac { L _ { 0 } } { y } + \\frac { 1 } { 2 } \\int \\Gamma _ { 2 } ( p ) ( \\vec { \\phi } _ { p } \\vec { \\phi } _ { - p } ) \\, d p - \\\\ & & \\qquad - \\frac { 1 } { 8 } \\int \\Gamma _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) ( \\vec { \\phi } _ { p _ { 3 } } \\vec { \\phi } _ { p _ { 4 } } ) \\delta ( \\sum p _ { i } ) d p _ { 1 } \\ldots d p _ { 4 } + \\ldots \\end{align*}",
"\\begin{align*} ( 2 \\pi ) B _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) & = & C _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) + D ( p _ { 1 } , p _ { 2 } ) + D ( p _ { 3 } , p _ { 4 } ) - \\\\ & & \\qquad - D ( p _ { 1 } , p _ { 3 } ) - D ( p _ { 1 } , p _ { 4 } ) - D ( p _ { 2 } , p _ { 3 } ) - D ( p _ { 2 } , p _ { 3 } ) \\; . \\end{align*}",
"\\begin{align*} C _ { 4 } & = & ( p _ { 1 } p _ { 2 } p _ { 3 } p _ { 4 } ) \\frac { 1 } { y ^ { 2 } } + y ^ { 6 } ( \\omega _ { 1 } \\omega _ { 2 } \\omega _ { 3 } \\omega _ { 4 } ) \\; , \\\\ D ( p _ { 1 } , p _ { 2 } ) & = & ( p _ { 1 } p _ { 2 } \\omega _ { 3 } \\omega _ { 4 } ) y ^ { 2 } \\; , \\end{align*}"
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"\\begin{equation*} \\frac { \\mitd \\mitx _ { \\mitmu } } { \\mitd \\mits } \\frac { \\mitdelta \\mitA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) } = \\frac { \\mitdelta \\mitA } { \\mitdelta \\mitx _ { 1 } ( \\mits ) } \\Big| _ { \\mitx _ { 1 } = \\mits } + \\frac { \\mitd \\vec { \\mitphi } } { \\mitd \\mits } \\frac { \\mitdelta \\mitA } { \\mitdelta \\vec { \\mitphi } } = 0 \\end{equation*}",
"\\begin{align*} & \\displaystyle \\frac { \\mitpartial \\mitA } { \\mitpartial \\mity } & = - \\frac { \\mitL _ { 0 } } { \\mity ^ { 2 } } + \\frac { 1 } { 2 } \\int \\left( \\mity ^ { 2 } \\vec { \\mitpi } ^ { 2 } - \\frac { 1 } { \\mity ^ { 2 } } { \\vec { \\mitphi } \\, { } ^ { \\prime } } ^ { 2 } \\right) \\mitd \\mits \\\\ & & \\qquad \\qquad + \\frac { 1 } { 8 } \\int \\left( \\frac { 1 } { \\mity ^ { 2 } } { \\left( \\mity ^ { 4 } \\vec { \\mitpi } ^ { 2 } - { \\vec { \\mitphi } \\, { } ^ { \\prime } } ^ { 2 } \\right) } ^ { 2 } + 4 \\mity ^ { 2 } { \\left( \\vec { \\mitphi } \\, { } ^ { \\prime } \\vec { \\mitpi } \\right) } ^ { 2 } \\right) \\mitd \\mits + \\ldots \\end{align*}",
"\\begin{align*} \\left\\{ \\begin{array}{l} \\displaystyle \\frac { \\mitd \\mupGamma _ { 2 } } { \\mitd \\mity } = \\mity ^ { 2 } \\mupGamma _ { 2 } ^ { 2 } - \\frac { \\mitp ^ { 2 } } { \\mity ^ { 2 } } \\; , \\\\ \\displaystyle \\frac { \\mitd \\mupGamma _ { 4 } } { \\mitd \\mity } = \\mity ^ { 2 } \\left( \\sum _ { 1 } ^ { 4 } \\mupGamma _ { 2 } ( \\mitp _ { \\miti } ) \\right) \\mupGamma _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) - \\mitB _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) \\; . \\end{array} \\right. \\end{align*}",
"\\begin{align*} & \\mitA & = \\frac { \\mitL _ { 0 } } { \\mity } + \\frac { 1 } { 2 } \\int \\mupGamma _ { 2 } ( \\mitp ) \\left( \\vec { \\mitphi } _ { \\mitp } \\vec { \\mitphi } _ { - \\mitp } \\right) \\, \\mitd \\mitp - \\\\ & & \\qquad - \\frac { 1 } { 8 } \\int \\mupGamma _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\left( \\vec { \\mitphi } _ { \\mitp _ { 3 } } \\vec { \\mitphi } _ { \\mitp _ { 4 } } \\right) \\mitdelta \\left( \\sum \\mitp _ { \\miti } \\right) \\mitd \\mitp _ { 1 } \\ldots \\mitd \\mitp _ { 4 } + \\ldots \\end{align*}",
"\\begin{align*} ( 2 \\mitpi ) \\mitB _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) & = & \\mitC _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) + \\mitD ( \\mitp _ { 1 } , \\mitp _ { 2 } ) + \\mitD ( \\mitp _ { 3 } , \\mitp _ { 4 } ) - \\\\ & & \\qquad - \\mitD ( \\mitp _ { 1 } , \\mitp _ { 3 } ) - \\mitD ( \\mitp _ { 1 } , \\mitp _ { 4 } ) - \\mitD ( \\mitp _ { 2 } , \\mitp _ { 3 } ) - \\mitD ( \\mitp _ { 2 } , \\mitp _ { 3 } ) \\; . \\end{align*}",
"\\begin{align*} \\mitC _ { 4 } & = & \\left( \\mitp _ { 1 } \\mitp _ { 2 } \\mitp _ { 3 } \\mitp _ { 4 } \\right) \\frac { 1 } { \\mity ^ { 2 } } + \\mity ^ { 6 } \\left( \\mitomega _ { 1 } \\mitomega _ { 2 } \\mitomega _ { 3 } \\mitomega _ { 4 } \\right) \\; , \\\\ \\mitD ( \\mitp _ { 1 } , \\mitp _ { 2 } ) & = & ( \\mitp _ { 1 } \\mitp _ { 2 } \\mitomega _ { 3 } \\mitomega _ { 4 } ) \\mity ^ { 2 } \\; , \\end{align*}"
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0002106_page14 | {
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"\\( y\\to \\infty \\)",
"\\( \\Gamma _{4} \\)",
"\\( y\\to 0 \\)",
"\\( \\Delta =\\sum |p_{i}| \\)",
"\\( \\epsilon _{i}=\\textrm {sgn }p_{i} \\)",
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"\\( \\vec {\\phi } \\)",
"\\begin {equation} \\label {5.13} \\Gamma _{2}(p,y)\\equiv \\omega (p,y)=\\frac {p^{2}}{y(1+|p|y)} \\end {equation}",
"\\begin {eqnarray} \\Gamma _{4} & = & \\int _{y}^{\\infty }dy\\, e^{-\\int _{0}^{y}dy_{1}y_{1}^{2}\\left ( \\sum \\omega _{i}\\right ) }B_{4}(p_{1},\\ldots p_{4}|y)=\\\\ &= & \\int _{y}^{\\infty }dy\\, \\prod _{i}(1+|p_{i}|y)e^{-\\sum |p_{i}|y}B_{4}(p_{1},\\ldots p_{4}|y).\\end {eqnarray}",
"\\begin {equation} \\label {5.15} (2\\pi )\\Gamma _{4}=F(p_{1},\\ldots p_{4})+\\Phi _{12}+\\Phi _{34}-\\Phi _{13}-\\Phi _{14}-\\Phi _{23}-\\Phi _{24}. \\end {equation}",
"\\begin {eqnarray} F & = & \\left ( 2\\frac {\\epsilon _{1}\\epsilon _{2}\\epsilon _{3}\\epsilon _{4}+1}{\\Delta ^{3}}+\\frac {\\epsilon _{1}\\epsilon _{2}\\epsilon _{3}\\epsilon _{4}}{\\Delta ^{2}}\\left ( \\sum \\frac {1}{|p_{i}|}\\right ) +\\frac {\\sum _{i<j}|p_{i}|\\cdot |p_{j}|}{\\Delta \\, p_{1}p_{2}p_{3}p_{4}}-\\frac {\\Delta }{p_{1}p_{2}p_{3}p_{4}}\\right ) p_{1}^{2}p_{2}^{2}p_{3}^{2}p_{4}^{2},\\\\ \\Phi _{12}\\! \\! \\! \\! & = & \\left ( \\frac {2\\epsilon _{1}\\epsilon _{2}}{\\Delta ^{3}}+\\frac {\\epsilon _{1}\\epsilon _{2}}{\\Delta ^{2}}\\left ( \\frac {1}{|p_{1}|}+\\frac {1}{|p_{2}|}\\right ) +\\frac {1}{\\Delta \\, p_{1}p_{2}}\\right ) p_{1}^{2}p_{2}^{2}p_{3}^{2}p_{4}^{2}.\\end {eqnarray}",
"\\begin {equation} \\label {5.18} A=\\int \\frac {d\\tau }{\\tau ^{2}}\\sqrt {{1+\\vec {\\phi }_{\\tau }^{\\, 2}+\\vec {\\phi }_{\\sigma }^{\\, 2}+\\vec {\\phi }_{\\tau }^{\\, 2}\\vec {\\phi }_{\\sigma }^{\\, 2}-\\left ( \\vec {\\phi }_{\\tau }\\vec {\\phi }_{\\sigma }\\right ) ^{2}}}. \\end {equation}",
"\\[ \\partial _{\\tau }\\left ( \\frac {1}{\\tau ^{2}}\\partial _{\\tau }\\vec {\\phi }\\right ) +\\frac {1}{\\tau ^{2}}\\partial _{\\sigma }^{2}\\vec {\\phi }=0\\]",
"\\[ \\vec {\\phi }_{\\textrm {cl}}(p,\\tau )=(|p|\\tau )^{3/2}K_{3/2}(|p|\\tau )\\vec {\\phi }(p)=(1+|p|\\tau )e^{-|p|\\tau }\\vec {\\phi }(p).\\]"
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"$ y \\rightarrow 0 $",
"$ \\Delta = \\sum \\vert p _ { i } \\vert $",
"$ \\epsilon _ { i } = s g n ~ p _ { i } $",
"$ \\Gamma _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } ) $",
"$ x _ { 1 } = \\sigma , \\, y = \\tau , \\, \\phi _ { i } = \\phi _ { i } ( \\sigma , \\tau ) , $",
"$ \\vec { \\phi } $",
"\\begin{equation*} \\Gamma _ { 2 } ( p , y ) \\equiv \\omega ( p , y ) = \\frac { p ^ { 2 } } { y ( 1 + \\vert p \\vert y ) } \\end{equation*}",
"\\begin{align*} \\Gamma _ { 4 } & = & \\int _ { y } ^ { \\infty } d y \\, e ^ { - \\int _ { 0 } ^ { y } d y _ { 1 } y _ { 1 } ^ { 2 } ( \\sum \\omega _ { i } ) } B _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } \\vert y ) = \\\\ & = & \\int _ { y } ^ { \\infty } d y \\, \\prod _ { i } ( 1 + \\vert p _ { i } \\vert y ) e ^ { - \\sum \\vert p _ { i } \\vert y } B _ { 4 } ( p _ { 1 } , \\ldots p _ { 4 } \\vert y ) . \\end{align*}",
"\\begin{equation*} ( 2 \\pi ) \\Gamma _ { 4 } = F ( p _ { 1 } , \\ldots p _ { 4 } ) + \\Phi _ { 1 2 } + \\Phi _ { 3 4 } - \\Phi _ { 1 3 } - \\Phi _ { 1 4 } - \\Phi _ { 2 3 } - \\Phi _ { 2 4 } . \\end{equation*}",
"\\begin{align*} F & = & ( 2 \\frac { \\epsilon _ { 1 } \\epsilon _ { 2 } \\epsilon _ { 3 } \\epsilon _ { 4 } + 1 } { \\Delta ^ { 3 } } + \\frac { \\epsilon _ { 1 } \\epsilon _ { 2 } \\epsilon _ { 3 } \\epsilon _ { 4 } } { \\Delta ^ { 2 } } ( \\sum \\frac { 1 } { \\vert p _ { i } \\vert } ) + \\frac { \\sum _ { i < j } \\vert p _ { i } \\vert \\cdot \\vert p _ { j } \\vert } { \\Delta \\, p _ { 1 } p _ { 2 } p _ { 3 } p _ { 4 } } - \\frac { \\Delta } { p _ { 1 } p _ { 2 } p _ { 3 } p _ { 4 } } ) p _ { 1 } ^ { 2 } p _ { 2 } ^ { 2 } p _ { 3 } ^ { 2 } p _ { 4 } ^ { 2 } , \\\\ \\Phi _ { 1 2 } \\! \\! \\! \\! & = & ( \\frac { 2 \\epsilon _ { 1 } \\epsilon _ { 2 } } { \\Delta ^ { 3 } } + \\frac { \\epsilon _ { 1 } \\epsilon _ { 2 } } { \\Delta ^ { 2 } } ( \\frac { 1 } { \\vert p _ { 1 } \\vert } + \\frac { 1 } { \\vert p _ { 2 } \\vert } ) + \\frac { 1 } { \\Delta \\, p _ { 1 } p _ { 2 } } ) p _ { 1 } ^ { 2 } p _ { 2 } ^ { 2 } p _ { 3 } ^ { 2 } p _ { 4 } ^ { 2 } . \\end{align*}",
"\\begin{equation*} A = \\int \\frac { d \\tau } { \\tau ^ { 2 } } \\sqrt { 1 + \\vec { \\phi } _ { \\tau } ^ { \\, 2 } + \\vec { \\phi } _ { \\sigma } ^ { \\, 2 } + \\vec { \\phi } _ { \\tau } ^ { \\, 2 } \\vec { \\phi } _ { \\sigma } ^ { \\, 2 } - { ( \\vec { \\phi } _ { \\tau } \\vec { \\phi } _ { \\sigma } ) } ^ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\partial _ { \\tau } ( \\frac { 1 } { \\tau ^ { 2 } } \\partial _ { \\tau } \\vec { \\phi } ) + \\frac { 1 } { \\tau ^ { 2 } } \\partial _ { \\sigma } ^ { 2 } \\vec { \\phi } = 0 \\end{equation*}",
"\\begin{equation*} \\vec { \\phi } _ { c l } ( p , \\tau ) = ( \\vert p \\vert \\tau ) ^ { 3 \\slash 2 } K _ { 3 \\slash 2 } ( \\vert p \\vert \\tau ) \\vec { \\phi } ( p ) = ( 1 + \\vert p \\vert \\tau ) e ^ { - \\vert p \\vert \\tau } \\vec { \\phi } ( p ) . \\end{equation*}"
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"\\begin{align*} \\mupGamma _ { 4 } & = & \\int _ { \\mity } ^ { \\infty } \\mitd \\mity \\, \\mite ^ { - \\int _ { 0 } ^ { \\mity } \\mitd \\mity _ { 1 } \\mity _ { 1 } ^ { 2 } \\left( \\sum \\mitomega _ { \\miti } \\right) } \\mitB _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } \\vert \\mity ) = \\\\ & = & \\int _ { \\mity } ^ { \\infty } \\mitd \\mity \\, \\prod _ { \\miti } ( 1 + \\vert \\mitp _ { \\miti } \\vert \\mity ) \\mite ^ { - \\sum \\vert \\mitp _ { \\miti } \\vert \\mity } \\mitB _ { 4 } ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } \\vert \\mity ) . \\end{align*}",
"\\begin{equation*} ( 2 \\mitpi ) \\mupGamma _ { 4 } = \\mitF ( \\mitp _ { 1 } , \\ldots \\mitp _ { 4 } ) + \\mupPhi _ { 1 2 } + \\mupPhi _ { 3 4 } - \\mupPhi _ { 1 3 } - \\mupPhi _ { 1 4 } - \\mupPhi _ { 2 3 } - \\mupPhi _ { 2 4 } . \\end{equation*}",
"\\begin{align*} \\mitF & = & \\left( 2 \\frac { \\mitepsilon _ { 1 } \\mitepsilon _ { 2 } \\mitepsilon _ { 3 } \\mitepsilon _ { 4 } + 1 } { \\mupDelta ^ { 3 } } + \\frac { \\mitepsilon _ { 1 } \\mitepsilon _ { 2 } \\mitepsilon _ { 3 } \\mitepsilon _ { 4 } } { \\mupDelta ^ { 2 } } \\left( \\sum \\frac { 1 } { \\vert \\mitp _ { \\miti } \\vert } \\right) + \\frac { \\sum _ { \\miti < \\mitj } \\vert \\mitp _ { \\miti } \\vert \\cdot \\vert \\mitp _ { \\mitj } \\vert } { \\mupDelta \\, \\mitp _ { 1 } \\mitp _ { 2 } \\mitp _ { 3 } \\mitp _ { 4 } } - \\frac { \\mupDelta } { \\mitp _ { 1 } \\mitp _ { 2 } \\mitp _ { 3 } \\mitp _ { 4 } } \\right) \\mitp _ { 1 } ^ { 2 } \\mitp _ { 2 } ^ { 2 } \\mitp _ { 3 } ^ { 2 } \\mitp _ { 4 } ^ { 2 } , \\\\ \\mupPhi _ { 1 2 } \\! \\! \\! \\! & = & \\left( \\frac { 2 \\mitepsilon _ { 1 } \\mitepsilon _ { 2 } } { \\mupDelta ^ { 3 } } + \\frac { \\mitepsilon _ { 1 } \\mitepsilon _ { 2 } } { \\mupDelta ^ { 2 } } \\left( \\frac { 1 } { \\vert \\mitp _ { 1 } \\vert } + \\frac { 1 } { \\vert \\mitp _ { 2 } \\vert } \\right) + \\frac { 1 } { \\mupDelta \\, \\mitp _ { 1 } \\mitp _ { 2 } } \\right) \\mitp _ { 1 } ^ { 2 } \\mitp _ { 2 } ^ { 2 } \\mitp _ { 3 } ^ { 2 } \\mitp _ { 4 } ^ { 2 } . \\end{align*}",
"\\begin{equation*} \\mitA = \\int \\frac { \\mitd \\mittau } { \\mittau ^ { 2 } } \\sqrt { 1 + \\vec { \\mitphi } _ { \\mittau } ^ { \\, 2 } + \\vec { \\mitphi } _ { \\mitsigma } ^ { \\, 2 } + \\vec { \\mitphi } _ { \\mittau } ^ { \\, 2 } \\vec { \\mitphi } _ { \\mitsigma } ^ { \\, 2 } - { \\left( \\vec { \\mitphi } _ { \\mittau } \\vec { \\mitphi } _ { \\mitsigma } \\right) } ^ { 2 } } . \\end{equation*}",
"\\begin{equation*} \\mitpartial _ { \\mittau } \\left( \\frac { 1 } { \\mittau ^ { 2 } } \\mitpartial _ { \\mittau } \\vec { \\mitphi } \\right) + \\frac { 1 } { \\mittau ^ { 2 } } \\mitpartial _ { \\mitsigma } ^ { 2 } \\vec { \\mitphi } = 0 \\end{equation*}",
"\\begin{equation*} \\vec { \\mitphi } _ { \\mathrm { c l } } ( \\mitp , \\mittau ) = ( \\vert \\mitp \\vert \\mittau ) ^ { 3 \\slash 2 } \\mitK _ { 3 \\slash 2 } ( \\vert \\mitp \\vert \\mittau ) \\vec { \\mitphi } ( \\mitp ) = ( 1 + \\vert \\mitp \\vert \\mittau ) \\mite ^ { - \\vert \\mitp \\vert \\mittau } \\vec { \\mitphi } ( \\mitp ) . \\end{equation*}"
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0002106_page15 | {
"latex": [
"\\( s\\to s' \\)",
"\\( \\mathcal {A} \\)",
"\\( \\mathcal {A} \\)",
"\\( \\phi \\)",
"\\( \\Gamma _{2}(p)=-|p|^{3} \\)",
"\\( \\Gamma _{2}(p,y) \\)",
"\\( k=-k' \\)",
"\\( q=0 \\)",
"\\( H \\)",
"\\( \\Phi (k,p) \\)",
"\\( k\\to -k \\)",
"\\( k\\to \\infty . \\)",
"\\( x=k/p \\)",
"\\( p_{1}=-p_{2}=p \\)",
"\\begin {equation} \\label {7.05} W[C]=e^{-\\sqrt {{g^{2}_{YM}N}}\\mathcal {A}[C]}\\: . \\end {equation}",
"\\[ \\frac {\\delta ^{2}W}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}=\\left ( g^{2}_{YM}N\\frac {\\delta \\mathcal {A}}{\\delta x_{\\mu }(s)}\\frac {\\delta \\mathcal {A}}{\\delta x_{\\mu }(s')}-\\sqrt {{g^{2}_{YM}N}}\\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}\\right ) W\\; .\\]",
"\\begin {equation} \\label {7.1} \\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}=\\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{1}(s)\\delta x_{1}(s')}+\\frac {\\delta ^{2}\\mathcal {A}}{\\delta \\vec {\\phi }(s)\\delta \\vec {\\phi }(s')}\\; . \\end {equation}",
"\\[ \\lim _{p\\to \\infty }\\frac {\\delta ^{2}\\mathcal {A}^{(2)}}{\\delta \\vec {\\phi }\\left ( \\frac {q}{2}+p\\right ) \\delta \\vec {\\phi }\\left ( \\frac {q}{2}-p\\right ) }=(D-1)\\delta (q)\\Gamma _{2}(p)=(1-D)\\delta (q)|p|^{3}\\; ,\\]",
"\\begin {eqnarray} && (2\\pi )\\frac {\\delta ^{2}\\mathcal {A}^{(4)}}{\\delta \\vec {\\phi }(k)\\delta \\vec {\\phi }(k')}=-\\frac {1}{2}\\int H(k,k',p_{1},p_{2})\\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\, \\delta (k+k'+p_{1}+p_{2})\\, dp_{1}dp_{2}\\; ,\\\\ && H=(D+1)F(k,k',p_{1},p_{2})+(D-3)\\bigl [\\Phi (k,k')+\\Phi (p_{1},p_{2})\\bigr ]\\\\ && \\qquad \\qquad \\qquad \\qquad \\qquad -(D-1)\\bigr [\\Phi (k,p_{1})+\\Phi (k,p_{2})+\\Phi (k',p_{1})+\\Phi (k',p_{2})\\bigl ]\\; .\\end {eqnarray}"
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"latex_norm": [
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"$ A $",
"$ A $",
"$ \\phi $",
"$ \\Gamma _ { 2 } ( p ) = - \\vert p \\vert ^ { 3 } $",
"$ \\Gamma _ { 2 } ( p , y ) $",
"$ k = - k ^ { \\prime } $",
"$ q = 0 $",
"$ H $",
"$ \\Phi ( k , p ) $",
"$ k \\rightarrow - k $",
"$ k \\rightarrow \\infty . $",
"$ x = k \\slash p $",
"$ p _ { 1 } = - p _ { 2 } = p $",
"\\begin{equation*} W [ C ] = e ^ { - \\sqrt { g _ { Y M } ^ { 2 } N } A [ C ] } \\> . \\end{equation*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } W } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } = ( g _ { Y M } ^ { 2 } N \\frac { \\delta A } { \\delta x _ { \\mu } ( s ) } \\frac { \\delta A } { \\delta x _ { \\mu } ( s ^ { \\prime } ) } - \\sqrt { g _ { Y M } ^ { 2 } N } \\frac { \\delta ^ { 2 } A } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } ) W \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } A } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } = \\frac { \\delta ^ { 2 } A } { \\delta x _ { 1 } ( s ) \\delta x _ { 1 } ( s ^ { \\prime } ) } + \\frac { \\delta ^ { 2 } A } { \\delta \\vec { \\phi } ( s ) \\delta \\vec { \\phi } ( s ^ { \\prime } ) } \\; . \\end{equation*}",
"\\begin{equation*} \\underset { p \\rightarrow \\infty } { \\operatorname { l i m } } \\frac { \\delta ^ { 2 } A ^ { ( 2 ) } } { \\delta \\vec { \\phi } ( \\frac { q } { 2 } + p ) \\delta \\vec { \\phi } ( \\frac { q } { 2 } - p ) } = ( D - 1 ) \\delta ( q ) \\Gamma _ { 2 } ( p ) = ( 1 - D ) \\delta ( q ) \\vert p \\vert ^ { 3 } \\; , \\end{equation*}",
"\\begin{align*} & & ( 2 \\pi ) \\frac { \\delta ^ { 2 } A ^ { ( 4 ) } } { \\delta \\vec { \\phi } ( k ) \\delta \\vec { \\phi } ( k ^ { \\prime } ) } = - \\frac { 1 } { 2 } \\int H ( k , k ^ { \\prime } , p _ { 1 } , p _ { 2 } ) ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) \\, \\delta ( k + k ^ { \\prime } + p _ { 1 } + p _ { 2 } ) \\, d p _ { 1 } d p _ { 2 } \\; , \\\\ & & H = ( D + 1 ) F ( k , k ^ { \\prime } , p _ { 1 } , p _ { 2 } ) + ( D - 3 ) [ \\Phi ( k , k ^ { \\prime } ) + \\Phi ( p _ { 1 } , p _ { 2 } ) ] \\\\ & & \\qquad \\qquad \\qquad \\qquad \\qquad - ( D - 1 ) [ \\Phi ( k , p _ { 1 } ) + \\Phi ( k , p _ { 2 } ) + \\Phi ( k ^ { \\prime } , p _ { 1 } ) + \\Phi ( k ^ { \\prime } , p _ { 2 } ) ] \\; . \\end{align*}"
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"$ \\mscrA $",
"$ \\mscrA $",
"$ \\mitphi $",
"$ \\mupGamma _ { 2 } ( \\mitp ) = - \\vert \\mitp \\vert ^ { 3 } $",
"$ \\mupGamma _ { 2 } ( \\mitp , \\mity ) $",
"$ \\mitk = - \\mitk ^ { \\prime } $",
"$ \\mitq = 0 $",
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"$ \\mitk \\rightarrow \\infty . $",
"$ \\mitx = \\mitk \\slash \\mitp $",
"$ \\mitp _ { 1 } = - \\mitp _ { 2 } = \\mitp $",
"\\begin{equation*} \\mitW [ \\mitC ] = \\mite ^ { - \\sqrt { \\mitg _ { \\mitY \\mitM } ^ { 2 } \\mitN } \\mscrA [ \\mitC ] } \\> . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mitW } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } = \\left( \\mitg _ { \\mitY \\mitM } ^ { 2 } \\mitN \\frac { \\mitdelta \\mscrA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) } \\frac { \\mitdelta \\mscrA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } - \\sqrt { \\mitg _ { \\mitY \\mitM } ^ { 2 } \\mitN } \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } \\right) \\mitW \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } = \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { 1 } ( \\mits ) \\mitdelta \\mitx _ { 1 } ( \\mits ^ { \\prime } ) } + \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\vec { \\mitphi } ( \\mits ) \\mitdelta \\vec { \\mitphi } ( \\mits ^ { \\prime } ) } \\; . \\end{equation*}",
"\\begin{equation*} \\underset { \\mitp \\rightarrow \\infty } { \\operatorname { l i m } } \\frac { \\mitdelta ^ { 2 } \\mscrA ^ { ( 2 ) } } { \\mitdelta \\vec { \\mitphi } \\left( \\frac { \\mitq } { 2 } + \\mitp \\right) \\mitdelta \\vec { \\mitphi } \\left( \\frac { \\mitq } { 2 } - \\mitp \\right) } = ( \\mitD - 1 ) \\mitdelta ( \\mitq ) \\mupGamma _ { 2 } ( \\mitp ) = ( 1 - \\mitD ) \\mitdelta ( \\mitq ) \\vert \\mitp \\vert ^ { 3 } \\; , \\end{equation*}",
"\\begin{align*} & & ( 2 \\mitpi ) \\frac { \\mitdelta ^ { 2 } \\mscrA ^ { ( 4 ) } } { \\mitdelta \\vec { \\mitphi } ( \\mitk ) \\mitdelta \\vec { \\mitphi } ( \\mitk ^ { \\prime } ) } = - \\frac { 1 } { 2 } \\int \\mitH ( \\mitk , \\mitk ^ { \\prime } , \\mitp _ { 1 } , \\mitp _ { 2 } ) \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\, \\mitdelta ( \\mitk + \\mitk ^ { \\prime } + \\mitp _ { 1 } + \\mitp _ { 2 } ) \\, \\mitd \\mitp _ { 1 } \\mitd \\mitp _ { 2 } \\; , \\\\ & & \\mitH = ( \\mitD + 1 ) \\mitF ( \\mitk , \\mitk ^ { \\prime } , \\mitp _ { 1 } , \\mitp _ { 2 } ) + ( \\mitD - 3 ) \\big [ \\mupPhi ( \\mitk , \\mitk ^ { \\prime } ) + \\mupPhi ( \\mitp _ { 1 } , \\mitp _ { 2 } ) \\big ] \\\\ & & \\qquad \\qquad \\qquad \\qquad \\qquad - ( \\mitD - 1 ) \\big [ \\mupPhi ( \\mitk , \\mitp _ { 1 } ) + \\mupPhi ( \\mitk , \\mitp _ { 2 } ) + \\mupPhi ( \\mitk ^ { \\prime } , \\mitp _ { 1 } ) + \\mupPhi ( \\mitk ^ { \\prime } , \\mitp _ { 2 } ) \\big ] \\; . \\end{align*}"
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0002106_page16 | {
"latex": [
"\\( p=1 \\)",
"\\( k\\to \\infty \\)",
"\\( q=0 \\)",
"\\begin {eqnarray} F(k,-k,p,-p) & = & |p|^{5}\\Bigl \\{\\frac {x^{4}}{2(1+x)^{3}}+\\frac {x^{3}}{2(1+x)}+\\frac {x^{2}(1+4x+x^{2})}{2(1+x)}-2x^{2}(1+x)\\Bigr \\}\\\\ &= & |p|^{5}\\Bigl \\{-\\frac {3}{2}x^{3}-x+O\\left ( \\frac {1}{x}\\right ) \\Bigl \\}\\; ,\\\\ \\Phi (k,-k)+\\Phi (p,-p) & = & |p|^{5}\\Bigl \\{-\\frac {x^{2}(1+x^{2})}{2(1+x)}-\\frac {x^{3}}{2(1+x)}-\\frac {x^{4}}{2(1+x)^{3}}\\Bigr \\}\\\\ &= & |p|^{5}\\Bigr \\{-\\frac {1}{2}x^{3}-x+2+O\\left ( \\frac {1}{x}\\right ) \\Bigl \\}\\; .\\end {eqnarray}",
"\\begin {eqnarray} \\frac {\\delta \\mathcal {A}^{(4)}}{\\delta \\vec {\\phi }(k)\\delta \\vec {\\phi }(-k)} & = & \\frac {1}{2\\pi }\\int \\Bigl \\{Dp^{2}|k|^{3}+(D-1)p^{4}|k|\\\\ && \\qquad \\qquad +(3-D)|p|^{5}\\Bigl \\}\\left ( \\vec {\\phi }_{p}\\vec {\\phi }_{-p}\\right ) \\, dp+O\\left ( \\frac {1}{k}\\right ) \\; .\\end {eqnarray}",
"\\begin {eqnarray} && \\frac {\\delta ^{2}\\mathcal {A}^{(4)}}{\\delta \\vec {\\phi }\\left ( \\frac {q}{2}+k\\right ) \\delta \\vec {\\phi }\\left ( \\frac {q}{2}-k\\right ) }=\\frac {1}{2\\pi }\\int \\Bigl \\{-Dp_{1}p_{2}|k|^{3}\\\\ && \\qquad \\qquad +\\left ( \\frac {D-4}{2}p_{1}p_{2}^{3}+\\frac {3D-6}{2}p_{1}^{2}p_{2}^{2}\\right ) |k|+\\left ( (4-D)p_{1}^{2}|p_{2}|^{3}+p_{1}p_{2}|p_{2}|^{3}\\right ) \\Bigl \\}\\\\ && \\qquad \\qquad \\qquad \\qquad \\qquad \\times \\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\, \\delta (p_{1}+p_{2}+q)\\, dp_{1}dp_{2}+O\\left ( \\frac {1}{k}\\right ) \\; .\\end {eqnarray}",
"\\[ \\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{1}(s)\\delta x_{1}(s')}=\\dot {\\phi }_{i}(s)\\dot {\\phi }_{k}(s')\\frac {\\delta ^{2}\\mathcal {A}}{\\delta \\phi _{i}(s)\\delta \\phi _{k}(s')}-\\delta (s-s')\\dvp (s)\\frac {d}{ds}\\left ( \\frac {\\delta A}{\\delta \\vec {\\phi }(s)}\\right ) \\; .\\]",
"\\begin {eqnarray} (2\\pi )\\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{1}\\left ( k\\right ) \\delta x_{1}\\left ( k'\\right ) }=-\\int dp\\, dp'\\, p\\, p'\\phi _{i}(p)\\phi _{k}(p')\\frac {\\delta ^{2}\\mathcal {A}}{\\delta \\phi _{i}(p+k)\\delta \\phi _{k}(p'+k')} & & \\\\ -\\int dp\\, p(p+k+k')\\vec {\\phi }(p)\\frac {\\delta \\mathcal {A}}{\\delta \\vec {\\phi }(p+k+k')}\\; . & & \\end {eqnarray}"
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"$ p = 1 $",
"$ k \\rightarrow \\infty $",
"$ q = 0 $",
"\\begin{align*} F ( k , - k , p , - p ) & = & \\vert p \\vert ^ { 5 } \\{ \\frac { x ^ { 4 } } { 2 ( 1 + x ) ^ { 3 } } + \\frac { x ^ { 3 } } { 2 ( 1 + x ) } + \\frac { x ^ { 2 } ( 1 + 4 x + x ^ { 2 } ) } { 2 ( 1 + x ) } - 2 x ^ { 2 } ( 1 + x ) \\} \\\\ & = & \\vert p \\vert ^ { 5 } \\{ - \\frac { 3 } { 2 } x ^ { 3 } - x + O ( \\frac { 1 } { x } ) \\} \\; , \\\\ \\Phi ( k , - k ) + \\Phi ( p , - p ) & = & \\vert p \\vert ^ { 5 } \\{ - \\frac { x ^ { 2 } ( 1 + x ^ { 2 } ) } { 2 ( 1 + x ) } - \\frac { x ^ { 3 } } { 2 ( 1 + x ) } - \\frac { x ^ { 4 } } { 2 ( 1 + x ) ^ { 3 } } \\} \\\\ & = & \\vert p \\vert ^ { 5 } \\{ - \\frac { 1 } { 2 } x ^ { 3 } - x + 2 + O ( \\frac { 1 } { x } ) \\} \\; . \\end{align*}",
"\\begin{align*} \\frac { \\delta A ^ { ( 4 ) } } { \\delta \\vec { \\phi } ( k ) \\delta \\vec { \\phi } ( - k ) } & = & \\frac { 1 } { 2 \\pi } \\int \\{ D p ^ { 2 } \\vert k \\vert ^ { 3 } + ( D - 1 ) p ^ { 4 } \\vert k \\vert \\\\ & & \\qquad \\qquad + ( 3 - D ) \\vert p \\vert ^ { 5 } \\} ( \\vec { \\phi } _ { p } \\vec { \\phi } _ { - p } ) \\, d p + O ( \\frac { 1 } { k } ) \\; . \\end{align*}",
"\\begin{align*} & & \\frac { \\delta ^ { 2 } A ^ { ( 4 ) } } { \\delta \\vec { \\phi } ( \\frac { q } { 2 } + k ) \\delta \\vec { \\phi } ( \\frac { q } { 2 } - k ) } = \\frac { 1 } { 2 \\pi } \\int \\{ - D p _ { 1 } p _ { 2 } \\vert k \\vert ^ { 3 } \\\\ & & \\qquad \\qquad + ( \\frac { D - 4 } { 2 } p _ { 1 } p _ { 2 } ^ { 3 } + \\frac { 3 D - 6 } { 2 } p _ { 1 } ^ { 2 } p _ { 2 } ^ { 2 } ) \\vert k \\vert + ( ( 4 - D ) p _ { 1 } ^ { 2 } \\vert p _ { 2 } \\vert ^ { 3 } + p _ { 1 } p _ { 2 } \\vert p _ { 2 } \\vert ^ { 3 } ) \\} \\\\ & & \\qquad \\qquad \\qquad \\qquad \\qquad \\times ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) \\, \\delta ( p _ { 1 } + p _ { 2 } + q ) \\, d p _ { 1 } d p _ { 2 } + O ( \\frac { 1 } { k } ) \\; . \\end{align*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } A } { \\delta x _ { 1 } ( s ) \\delta x _ { 1 } ( s ^ { \\prime } ) } = \\dot { \\phi } _ { i } ( s ) \\dot { \\phi } _ { k } ( s ^ { \\prime } ) \\frac { \\delta ^ { 2 } A } { \\delta \\phi _ { i } ( s ) \\delta \\phi _ { k } ( s ^ { \\prime } ) } - \\delta ( s - s ^ { \\prime } ) \\dot { \\vec { \\phi } \\; } \\! ( s ) \\frac { d } { d s } ( \\frac { \\delta A } { \\delta \\vec { \\phi } ( s ) } ) \\; . \\end{equation*}",
"\\begin{align*} ( 2 \\pi ) \\frac { \\delta ^ { 2 } A } { \\delta x _ { 1 } ( k ) \\delta x _ { 1 } ( k ^ { \\prime } ) } = - \\int d p \\, d p ^ { \\prime } \\, p \\, p ^ { \\prime } \\phi _ { i } ( p ) \\phi _ { k } ( p ^ { \\prime } ) \\frac { \\delta ^ { 2 } A } { \\delta \\phi _ { i } ( p + k ) \\delta \\phi _ { k } ( p ^ { \\prime } + k ^ { \\prime } ) } \\\\ - \\int d p \\, p ( p + k + k ^ { \\prime } ) \\vec { \\phi } ( p ) \\frac { \\delta A } { \\delta \\vec { \\phi } ( p + k + k ^ { \\prime } ) } \\; . \\end{align*}"
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"$ \\mitp = 1 $",
"$ \\mitk \\rightarrow \\infty $",
"$ \\mitq = 0 $",
"\\begin{align*} \\mitF ( \\mitk , - \\mitk , \\mitp , - \\mitp ) & = & \\vert \\mitp \\vert ^ { 5 } \\Big \\{ \\frac { \\mitx ^ { 4 } } { 2 ( 1 + \\mitx ) ^ { 3 } } + \\frac { \\mitx ^ { 3 } } { 2 ( 1 + \\mitx ) } + \\frac { \\mitx ^ { 2 } ( 1 + 4 \\mitx + \\mitx ^ { 2 } ) } { 2 ( 1 + \\mitx ) } - 2 \\mitx ^ { 2 } ( 1 + \\mitx ) \\Big \\} \\\\ & = & \\vert \\mitp \\vert ^ { 5 } \\Big \\{ - \\frac { 3 } { 2 } \\mitx ^ { 3 } - \\mitx + \\mitO \\left( \\frac { 1 } { \\mitx } \\right) \\Big \\} \\; , \\\\ \\mupPhi ( \\mitk , - \\mitk ) + \\mupPhi ( \\mitp , - \\mitp ) & = & \\vert \\mitp \\vert ^ { 5 } \\Big \\{ - \\frac { \\mitx ^ { 2 } ( 1 + \\mitx ^ { 2 } ) } { 2 ( 1 + \\mitx ) } - \\frac { \\mitx ^ { 3 } } { 2 ( 1 + \\mitx ) } - \\frac { \\mitx ^ { 4 } } { 2 ( 1 + \\mitx ) ^ { 3 } } \\Big \\} \\\\ & = & \\vert \\mitp \\vert ^ { 5 } \\Big \\{ - \\frac { 1 } { 2 } \\mitx ^ { 3 } - \\mitx + 2 + \\mitO \\left( \\frac { 1 } { \\mitx } \\right) \\Big \\} \\; . \\end{align*}",
"\\begin{align*} \\frac { \\mitdelta \\mscrA ^ { ( 4 ) } } { \\mitdelta \\vec { \\mitphi } ( \\mitk ) \\mitdelta \\vec { \\mitphi } ( - \\mitk ) } & = & \\frac { 1 } { 2 \\mitpi } \\int \\Big \\{ \\mitD \\mitp ^ { 2 } \\vert \\mitk \\vert ^ { 3 } + ( \\mitD - 1 ) \\mitp ^ { 4 } \\vert \\mitk \\vert \\\\ & & \\qquad \\qquad + ( 3 - \\mitD ) \\vert \\mitp \\vert ^ { 5 } \\Big \\} \\left( \\vec { \\mitphi } _ { \\mitp } \\vec { \\mitphi } _ { - \\mitp } \\right) \\, \\mitd \\mitp + \\mitO \\left( \\frac { 1 } { \\mitk } \\right) \\; . \\end{align*}",
"\\begin{align*} & & \\frac { \\mitdelta ^ { 2 } \\mscrA ^ { ( 4 ) } } { \\mitdelta \\vec { \\mitphi } \\left( \\frac { \\mitq } { 2 } + \\mitk \\right) \\mitdelta \\vec { \\mitphi } \\left( \\frac { \\mitq } { 2 } - \\mitk \\right) } = \\frac { 1 } { 2 \\mitpi } \\int \\Big \\{ - \\mitD \\mitp _ { 1 } \\mitp _ { 2 } \\vert \\mitk \\vert ^ { 3 } \\\\ & & \\qquad \\qquad + \\left( \\frac { \\mitD - 4 } { 2 } \\mitp _ { 1 } \\mitp _ { 2 } ^ { 3 } + \\frac { 3 \\mitD - 6 } { 2 } \\mitp _ { 1 } ^ { 2 } \\mitp _ { 2 } ^ { 2 } \\right) \\vert \\mitk \\vert + \\left( ( 4 - \\mitD ) \\mitp _ { 1 } ^ { 2 } \\vert \\mitp _ { 2 } \\vert ^ { 3 } + \\mitp _ { 1 } \\mitp _ { 2 } \\vert \\mitp _ { 2 } \\vert ^ { 3 } \\right) \\Big \\} \\\\ & & \\qquad \\qquad \\qquad \\qquad \\qquad \\times \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\, \\mitdelta ( \\mitp _ { 1 } + \\mitp _ { 2 } + \\mitq ) \\, \\mitd \\mitp _ { 1 } \\mitd \\mitp _ { 2 } + \\mitO \\left( \\frac { 1 } { \\mitk } \\right) \\; . \\end{align*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { 1 } ( \\mits ) \\mitdelta \\mitx _ { 1 } ( \\mits ^ { \\prime } ) } = \\dot { \\mitphi } _ { \\miti } ( \\mits ) \\dot { \\mitphi } _ { \\mitk } ( \\mits ^ { \\prime } ) \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitphi _ { \\miti } ( \\mits ) \\mitdelta \\mitphi _ { \\mitk } ( \\mits ^ { \\prime } ) } - \\mitdelta ( \\mits - \\mits ^ { \\prime } ) \\dot { \\vec { \\mitphi } \\; } \\! ( \\mits ) \\frac { \\mitd } { \\mitd \\mits } \\left( \\frac { \\mitdelta \\mitA } { \\mitdelta \\vec { \\mitphi } ( \\mits ) } \\right) \\; . \\end{equation*}",
"\\begin{align*} ( 2 \\mitpi ) \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { 1 } \\left( \\mitk \\right) \\mitdelta \\mitx _ { 1 } \\left( \\mitk ^ { \\prime } \\right) } = - \\int \\mitd \\mitp \\, \\mitd \\mitp ^ { \\prime } \\, \\mitp \\, \\mitp ^ { \\prime } \\mitphi _ { \\miti } ( \\mitp ) \\mitphi _ { \\mitk } ( \\mitp ^ { \\prime } ) \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitphi _ { \\miti } ( \\mitp + \\mitk ) \\mitdelta \\mitphi _ { \\mitk } ( \\mitp ^ { \\prime } + \\mitk ^ { \\prime } ) } \\\\ - \\int \\mitd \\mitp \\, \\mitp ( \\mitp + \\mitk + \\mitk ^ { \\prime } ) \\vec { \\mitphi } ( \\mitp ) \\frac { \\mitdelta \\mscrA } { \\mitdelta \\vec { \\mitphi } ( \\mitp + \\mitk + \\mitk ^ { \\prime } ) } \\; . \\end{align*}"
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0002106_page17 | {
"latex": [
"\\( \\mathcal {A}^{(2)} \\)",
"\\( \\propto \\delta ''(s-s') \\)",
"\\( \\propto k^{2} \\)",
"\\( D \\)",
"\\( \\widehat {{L}}_{q} \\)",
"\\( k^{0} \\)",
"\\( D=4 \\)",
"\\begin {eqnarray} \\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{1}\\left ( \\frac {q}{2}+k\\right ) \\delta x_{1}\\left ( \\frac {q}{2}-k\\right ) } & = & \\frac {1}{2\\pi }\\int \\Bigl \\{\\Bigl |\\frac {p_{1}-p_{2}}{2}+k\\Bigr |^{3}-|p_{2}|^{3}\\Bigl \\}\\\\ && \\qquad \\qquad \\times p_{1}p_{2}\\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\delta (p_{1}+p_{2}+q)\\, dp_{1}dp_{2}\\\\ &\\mathop {=}_{k\\to \\infty } & \\frac {1}{2\\pi }\\int \\Bigl \\{|k|^{3}+\\frac {3}{4}(p_{1}-p_{2})^{2}|k|-p_{1}p_{2}|p_{2}|^{3}\\Bigl \\}\\times \\ldots \\end {eqnarray}",
"\\begin {eqnarray} && \\frac {\\delta ^{2}\\mathcal {A}^{(4)}}{\\delta x_{\\mu }\\left ( \\frac {q}{2}+k\\right ) \\delta x_{\\mu }\\left ( \\frac {q}{2}-k\\right ) }=(1-D)\\delta (q)|k|^{3}+\\frac {1}{2\\pi }\\int \\Bigl \\{(1-D)p_{1}p_{2}|k|^{3}\\\\ && \\qquad \\qquad +\\left ( \\frac {D-1}{2}p_{1}p_{2}^{3}+\\frac {3D-9}{2}p_{1}^{2}p_{2}^{2}\\right ) |k|+(4-D)p_{1}^{2}|p_{2}|^{3}\\Bigl \\}\\\\ && \\qquad \\qquad \\qquad \\qquad \\times \\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\, \\delta (p_{1}+p_{2}+q)\\, dp_{1}dp_{2}+O\\left ( \\frac {1}{k}\\right ) +O(\\phi ^{4})\\; .\\end {eqnarray}",
"\\begin {equation} \\label {8.1} \\widehat {{L}}_{q}\\mathcal {A}=\\frac {4-D}{2\\pi }\\int p_{1}^{2}|p_{2}|^{3}\\left ( \\vec {\\phi }_{p_{1}}\\vec {\\phi }_{p_{2}}\\right ) \\, \\delta (p_{1}+p_{2}+q)\\, dp_{1}dp_{2}\\; . \\end {equation}"
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"\\begin{align*} \\frac { \\delta ^ { 2 } A } { \\delta x _ { 1 } ( \\frac { q } { 2 } + k ) \\delta x _ { 1 } ( \\frac { q } { 2 } - k ) } & = & \\frac { 1 } { 2 \\pi } \\int \\{ \\vert \\frac { p _ { 1 } - p _ { 2 } } { 2 } + k \\vert ^ { 3 } - \\vert p _ { 2 } \\vert ^ { 3 } \\} \\\\ & & \\qquad \\qquad \\times p _ { 1 } p _ { 2 } ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) \\delta ( p _ { 1 } + p _ { 2 } + q ) \\, d p _ { 1 } d p _ { 2 } \\\\ & \\underset { k \\rightarrow \\infty } { = } & \\frac { 1 } { 2 \\pi } \\int \\{ \\vert k \\vert ^ { 3 } + \\frac { 3 } { 4 } ( p _ { 1 } - p _ { 2 } ) ^ { 2 } \\vert k \\vert - p _ { 1 } p _ { 2 } \\vert p _ { 2 } \\vert ^ { 3 } \\} \\times \\ldots \\end{align*}",
"\\begin{align*} & & \\frac { \\delta ^ { 2 } A ^ { ( 4 ) } } { \\delta x _ { \\mu } ( \\frac { q } { 2 } + k ) \\delta x _ { \\mu } ( \\frac { q } { 2 } - k ) } = ( 1 - D ) \\delta ( q ) \\vert k \\vert ^ { 3 } + \\frac { 1 } { 2 \\pi } \\int \\{ ( 1 - D ) p _ { 1 } p _ { 2 } \\vert k \\vert ^ { 3 } \\\\ & & \\qquad \\qquad + ( \\frac { D - 1 } { 2 } p _ { 1 } p _ { 2 } ^ { 3 } + \\frac { 3 D - 9 } { 2 } p _ { 1 } ^ { 2 } p _ { 2 } ^ { 2 } ) \\vert k \\vert + ( 4 - D ) p _ { 1 } ^ { 2 } \\vert p _ { 2 } \\vert ^ { 3 } \\} \\\\ & & \\qquad \\qquad \\qquad \\qquad \\times ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) \\, \\delta ( p _ { 1 } + p _ { 2 } + q ) \\, d p _ { 1 } d p _ { 2 } + O ( \\frac { 1 } { k } ) + O ( \\phi ^ { 4 } ) \\; . \\end{align*}",
"\\begin{equation*} \\hat { L } _ { q } A = \\frac { 4 - D } { 2 \\pi } \\int p _ { 1 } ^ { 2 } \\vert p _ { 2 } \\vert ^ { 3 } ( \\vec { \\phi } _ { p _ { 1 } } \\vec { \\phi } _ { p _ { 2 } } ) \\, \\delta ( p _ { 1 } + p _ { 2 } + q ) \\, d p _ { 1 } d p _ { 2 } \\; . \\end{equation*}"
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"\\begin{align*} \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { 1 } \\left( \\frac { \\mitq } { 2 } + \\mitk \\right) \\mitdelta \\mitx _ { 1 } \\left( \\frac { \\mitq } { 2 } - \\mitk \\right) } & = & \\frac { 1 } { 2 \\mitpi } \\int \\Big \\{ \\Big| \\frac { \\mitp _ { 1 } - \\mitp _ { 2 } } { 2 } + \\mitk \\Big| ^ { 3 } - \\vert \\mitp _ { 2 } \\vert ^ { 3 } \\Big \\} \\\\ & & \\qquad \\qquad \\times \\mitp _ { 1 } \\mitp _ { 2 } \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\mitdelta ( \\mitp _ { 1 } + \\mitp _ { 2 } + \\mitq ) \\, \\mitd \\mitp _ { 1 } \\mitd \\mitp _ { 2 } \\\\ & \\underset { \\mitk \\rightarrow \\infty } { = } & \\frac { 1 } { 2 \\mitpi } \\int \\Big \\{ \\vert \\mitk \\vert ^ { 3 } + \\frac { 3 } { 4 } ( \\mitp _ { 1 } - \\mitp _ { 2 } ) ^ { 2 } \\vert \\mitk \\vert - \\mitp _ { 1 } \\mitp _ { 2 } \\vert \\mitp _ { 2 } \\vert ^ { 3 } \\Big \\} \\times \\ldots \\end{align*}",
"\\begin{align*} & & \\frac { \\mitdelta ^ { 2 } \\mscrA ^ { ( 4 ) } } { \\mitdelta \\mitx _ { \\mitmu } \\left( \\frac { \\mitq } { 2 } + \\mitk \\right) \\mitdelta \\mitx _ { \\mitmu } \\left( \\frac { \\mitq } { 2 } - \\mitk \\right) } = ( 1 - \\mitD ) \\mitdelta ( \\mitq ) \\vert \\mitk \\vert ^ { 3 } + \\frac { 1 } { 2 \\mitpi } \\int \\Big \\{ ( 1 - \\mitD ) \\mitp _ { 1 } \\mitp _ { 2 } \\vert \\mitk \\vert ^ { 3 } \\\\ & & \\qquad \\qquad + \\left( \\frac { \\mitD - 1 } { 2 } \\mitp _ { 1 } \\mitp _ { 2 } ^ { 3 } + \\frac { 3 \\mitD - 9 } { 2 } \\mitp _ { 1 } ^ { 2 } \\mitp _ { 2 } ^ { 2 } \\right) \\vert \\mitk \\vert + ( 4 - \\mitD ) \\mitp _ { 1 } ^ { 2 } \\vert \\mitp _ { 2 } \\vert ^ { 3 } \\Big \\} \\\\ & & \\qquad \\qquad \\qquad \\qquad \\times \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\, \\mitdelta ( \\mitp _ { 1 } + \\mitp _ { 2 } + \\mitq ) \\, \\mitd \\mitp _ { 1 } \\mitd \\mitp _ { 2 } + \\mitO \\left( \\frac { 1 } { \\mitk } \\right) + \\mitO ( \\mitphi ^ { 4 } ) \\; . \\end{align*}",
"\\begin{equation*} \\widehat { \\mitL } _ { \\mitq } \\mscrA = \\frac { 4 - \\mitD } { 2 \\mitpi } \\int \\mitp _ { 1 } ^ { 2 } \\vert \\mitp _ { 2 } \\vert ^ { 3 } \\left( \\vec { \\mitphi } _ { \\mitp _ { 1 } } \\vec { \\mitphi } _ { \\mitp _ { 2 } } \\right) \\, \\mitdelta ( \\mitp _ { 1 } + \\mitp _ { 2 } + \\mitq ) \\, \\mitd \\mitp _ { 1 } \\mitd \\mitp _ { 2 } \\; . \\end{equation*}"
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0002106_page18 | {
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"\\( x=\\frac {x_{1}+x_{2}}{2} \\)",
"\\( \\bar {s}=\\frac {s+s'}{2}. \\)",
"\\( \\propto |k|^{3} \\)",
"\\( |k| \\)",
"\\( s \\)",
"\\( C_{1}=D-3 \\)",
"\\( C_{2}=2 \\)",
"\\( C_{1} \\)",
"\\( C_{2} \\)",
"\\( F_{\\mu \\nu } \\)",
"\\begin {eqnarray} F_{\\mu \\lambda }(x_{1})F_{\\mu \\sigma }(x_{2}) & \\sim & C_{1}\\frac {\\delta _{\\lambda \\sigma }}{|x_{1}-x_{2}|^{4}}+C_{2}\\frac {(x_{1}-x_{2})_{\\lambda }(x_{1}-x_{2})_{\\sigma }}{|x_{1}-x_{2}|^{6}}\\\\ && +\\, C_{3}\\frac {(x_{1}-x_{2})_{(\\lambda }(x_{1}-x_{2})_{\\mu }F_{\\mu \\sigma )}(x)}{|x_{1}-x_{2}|^{4}}+C_{4}\\frac {(x_{1}-x_{2})_{\\mu }\\nabla _{\\mu }F_{\\lambda \\sigma }(x)}{\\left | x_{1}-x_{2}\\right | ^{2}}\\; .\\end {eqnarray}",
"\\[ \\frac {\\delta ^{2}W}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}\\mathop {=}_{s\\to s'}\\frac {1}{|s-s'|^{4}}\\frac {C_{1}+C_{2}}{\\dot {x}^{2}}+\\frac {1}{|s-s'|^{2}}\\left ( \\frac {(C_{1}+C_{2})(\\dot {x}\\! \\stackrel {{\\, ...}}{x})}{12\\dot {x}^{4}}+\\frac {C_{1}\\ddot {x}^{2}}{4\\dot {x}^{4}}+\\frac {C_{2}(\\dot {x}\\ddot {x})^{2}}{4\\dot {x}^{4}}\\right ) \\; .\\]",
"\\begin {equation} \\label {8.35} \\frac {\\delta ^{2}W}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}\\mathop {=}_{s\\to s'}\\frac {1}{|s-s'|^{4}}(C_{1}+C_{2})(1-\\dot {\\phi }^{2})+\\frac {1}{|s-s'|^{2}}\\left ( \\frac {(C_{1}+C_{2})}{12}\\dot {\\phi }\\stackrel {{\\, ...}}{\\phi }+\\frac {C_{1}}{4}\\ddot {\\phi }^{2}\\right ) +O(\\phi ^{4})\\; . \\end {equation}",
"\\[ |k|^{3}\\leftrightarrow \\frac {1}{|s-s'|^{4}},\\qquad |k|\\leftrightarrow -\\frac {1}{6|s-s'|^{2}}\\]",
"\\begin {equation} \\label {8.4} \\frac {\\delta ^{2}\\mathcal {A}}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}\\propto \\frac {1}{|s-s'|^{4}}(1-D)(1-\\dot {\\phi }^{2})+\\frac {1}{|s-s'|^{2}}\\left ( \\frac {1-D}{12}\\dot {\\phi }\\stackrel {{\\, ...}}{\\phi }+\\frac {3-D}{4}\\ddot {\\phi }^{2}\\right ) . \\end {equation}"
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"$ \\bar { s } = \\frac { s + s ^ { \\prime } } { 2 } . $",
"$ \\propto \\vert k \\vert ^ { 3 } $",
"$ \\vert k \\vert $",
"$ s $",
"$ C _ { 1 } = D - 3 $",
"$ C _ { 2 } = 2 $",
"$ C _ { 1 } $",
"$ C _ { 2 } $",
"$ F _ { \\mu \\nu } $",
"\\begin{align*} F _ { \\mu \\lambda } ( x _ { 1 } ) F _ { \\mu \\sigma } ( x _ { 2 } ) & \\sim & C _ { 1 } \\frac { \\delta _ { \\lambda \\sigma } } { \\vert x _ { 1 } - x _ { 2 } \\vert ^ { 4 } } + C _ { 2 } \\frac { ( x _ { 1 } - x _ { 2 } ) _ { \\lambda } ( x _ { 1 } - x _ { 2 } ) _ { \\sigma } } { \\vert x _ { 1 } - x _ { 2 } \\vert ^ { 6 } } \\\\ & & + \\, C _ { 3 } \\frac { ( x _ { 1 } - x _ { 2 } ) _ { ( \\lambda } ( x _ { 1 } - x _ { 2 } ) _ { \\mu } F _ { \\mu \\sigma ) } ( x ) } { \\vert x _ { 1 } - x _ { 2 } \\vert ^ { 4 } } + C _ { 4 } \\frac { ( x _ { 1 } - x _ { 2 } ) _ { \\mu } \\nabla _ { \\mu } F _ { \\lambda \\sigma } ( x ) } { { \\vert x _ { 1 } - x _ { 2 } \\vert } ^ { 2 } } \\; . \\end{align*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } W } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } \\underset { s \\rightarrow s ^ { \\prime } } { = } \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 4 } } \\frac { C _ { 1 } + C _ { 2 } } { \\dot { x } ^ { 2 } } + \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 2 } } ( \\frac { ( C _ { 1 } + C _ { 2 } ) ( \\dot { x } \\! \\overset { \\, . . . } { x } ) } { 1 2 \\dot { x } ^ { 4 } } + \\frac { C _ { 1 } \\ddot { x } ^ { 2 } } { 4 \\dot { x } ^ { 4 } } + \\frac { C _ { 2 } ( \\dot { x } \\ddot { x } ) ^ { 2 } } { 4 \\dot { x } ^ { 4 } } ) \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } W } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } \\underset { s \\rightarrow s ^ { \\prime } } { = } \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 4 } } ( C _ { 1 } + C _ { 2 } ) ( 1 - \\dot { \\phi } ^ { 2 } ) + \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 2 } } ( \\frac { ( C _ { 1 } + C _ { 2 } ) } { 1 2 } \\dot { \\phi } \\overset { \\, . . . } { \\phi } + \\frac { C _ { 1 } } { 4 } \\ddot { \\phi } ^ { 2 } ) + O ( \\phi ^ { 4 } ) \\; . \\end{equation*}",
"\\begin{equation*} \\vert k \\vert ^ { 3 } \\leftrightarrow \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 4 } } , \\qquad \\vert k \\vert \\leftrightarrow - \\frac { 1 } { 6 \\vert s - s ^ { \\prime } \\vert ^ { 2 } } \\end{equation*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } A } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } \\propto \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 4 } } ( 1 - D ) ( 1 - \\dot { \\phi } ^ { 2 } ) + \\frac { 1 } { \\vert s - s ^ { \\prime } \\vert ^ { 2 } } ( \\frac { 1 - D } { 1 2 } \\dot { \\phi } \\overset { \\, . . . } { \\phi } + \\frac { 3 - D } { 4 } \\ddot { \\phi } ^ { 2 } ) . \\end{equation*}"
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"$ \\vert \\mitk \\vert $",
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"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mitW } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } \\underset { \\mits \\rightarrow \\mits ^ { \\prime } } { = } \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 4 } } \\frac { \\mitC _ { 1 } + \\mitC _ { 2 } } { \\dot { \\mitx } ^ { 2 } } + \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 2 } } \\left( \\frac { ( \\mitC _ { 1 } + \\mitC _ { 2 } ) ( \\dot { \\mitx } \\! \\overset { \\, . . . } { \\mitx } ) } { 1 2 \\dot { \\mitx } ^ { 4 } } + \\frac { \\mitC _ { 1 } \\ddot { \\mitx } ^ { 2 } } { 4 \\dot { \\mitx } ^ { 4 } } + \\frac { \\mitC _ { 2 } ( \\dot { \\mitx } \\ddot { \\mitx } ) ^ { 2 } } { 4 \\dot { \\mitx } ^ { 4 } } \\right) \\; . \\end{equation*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mitW } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } \\underset { \\mits \\rightarrow \\mits ^ { \\prime } } { = } \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 4 } } ( \\mitC _ { 1 } + \\mitC _ { 2 } ) ( 1 - \\dot { \\mitphi } ^ { 2 } ) + \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 2 } } \\left( \\frac { ( \\mitC _ { 1 } + \\mitC _ { 2 } ) } { 1 2 } \\dot { \\mitphi } \\overset { \\, . . . } { \\mitphi } + \\frac { \\mitC _ { 1 } } { 4 } \\ddot { \\mitphi } ^ { 2 } \\right) + \\mitO ( \\mitphi ^ { 4 } ) \\; . \\end{equation*}",
"\\begin{equation*} \\vert \\mitk \\vert ^ { 3 } \\leftrightarrow \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 4 } } , \\qquad \\vert \\mitk \\vert \\leftrightarrow - \\frac { 1 } { 6 \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 2 } } \\end{equation*}",
"\\begin{equation*} \\frac { \\mitdelta ^ { 2 } \\mscrA } { \\mitdelta \\mitx _ { \\mitmu } ( \\mits ) \\mitdelta \\mitx _ { \\mitmu } ( \\mits ^ { \\prime } ) } \\propto \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 4 } } ( 1 - \\mitD ) ( 1 - \\dot { \\mitphi } ^ { 2 } ) + \\frac { 1 } { \\vert \\mits - \\mits ^ { \\prime } \\vert ^ { 2 } } \\left( \\frac { 1 - \\mitD } { 1 2 } \\dot { \\mitphi } \\overset { \\, . . . } { \\mitphi } + \\frac { 3 - \\mitD } { 4 } \\ddot { \\mitphi } ^ { 2 } \\right) . \\end{equation*}"
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0002106_page19 | {
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"\\[ \\nabla _{\\mu }F_{\\mu \\nu }=J_{\\nu }\\ne 0\\]"
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0002106_page20 | {
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"\\( \\widehat {L}_{q}W\\propto \\epsilon \\)",
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"\\begin {equation} \\label {9.5} \\mathcal {A}[C]=\\mathcal {A}[f(C)]\\; \\end {equation}",
"\\begin {equation} \\label {9.2} (x_{\\mu },y)\\stackrel {{F}}{\\rightarrow }\\left ( \\frac {x_{\\mu }}{x^{2}+y^{2}},\\frac {y}{x^{2}+y^{2}}\\right ) \\; . \\end {equation}",
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0002106_page21 | {
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"\\( \\frac {dx_{\\lambda }}{ds}\\frac {\\delta U}{\\delta x_{\\lambda }(s)}=0 \\)",
"\\[ \\frac {L[C]}{\\epsilon }+\\mathcal {A}[C]\\mathop {=}_{\\epsilon ,\\epsilon '\\to 0}\\frac {L[f(C)]}{\\epsilon '}+\\mathcal {A}[f(C)]+\\oint _{C}\\left | \\frac {df(x(s))}{ds}\\right | ds\\int _{\\epsilon /x(s)^{2}}^{\\epsilon '}\\frac {dy}{y^{2}}\\; .\\]",
"\\begin {equation} \\label {9.6} \\widehat {{L}}(s)U_{f}[C]=\\rho \\left ( \\widehat {{L}}(s)U\\right ) [f(C)] \\end {equation}",
"\\[ \\frac {\\delta ^{2}U_{f}[x(s)]}{\\delta x_{\\mu }(s)\\delta x_{\\mu }(s')}=\\partial _{\\mu }f_{\\lambda }(x(s))\\partial _{\\mu }f_{\\sigma }(x(s'))\\frac {\\delta ^{2}U}{\\delta f_{\\lambda }(s)\\delta f_{\\sigma }(s')}+\\partial ^{2}f_{\\lambda }(x(s))\\frac {\\delta U}{\\delta f_{\\lambda }(s)}\\delta (s-s')\\; .\\]",
"\\[ \\partial _{\\mu }f_{\\lambda }\\partial _{\\mu }f_{\\sigma }=\\rho (f)\\delta _{\\lambda \\sigma }\\; .\\]",
"\\begin {eqnarray*} && \\widehat {{L}}(s)U_{f}[C]\\delta (s-s')=\\rho (f)\\left ( \\widehat {{L}}(s)U\\right ) [f(C)]\\delta (s-s')\\\\ && \\qquad \\qquad +\\, \\partial ^{2}f_{\\lambda }\\frac {\\delta U}{\\delta f_{\\lambda }}\\delta (s-s')+\\partial _{\\mu }f_{\\lambda }(x(s))\\partial _{\\mu }f_{\\sigma }(x(s'))N_{[\\lambda \\sigma ]}\\delta '(s-s')\\; ,\\end {eqnarray*}",
"\\[ \\frac {\\delta ^{2}U}{\\delta f_{\\lambda }(s)\\delta f_{\\sigma }(s')}=N_{[\\lambda \\sigma ]}\\Bigl (\\frac {s+s'}{2}\\Bigr )\\delta '(s-s')+\\ldots \\]",
"\\[ \\frac {\\delta U}{\\delta x_{\\lambda }(s)}=N_{[\\lambda \\sigma ]}(s)\\dot {x}_{\\sigma }(s)\\; .\\]"
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"\\begin{equation*} \\hat { L } ( s ) U _ { f } [ C ] = \\rho ( \\hat { L } ( s ) U ) [ f ( C ) ] \\end{equation*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } U _ { f } [ x ( s ) ] } { \\delta x _ { \\mu } ( s ) \\delta x _ { \\mu } ( s ^ { \\prime } ) } = \\partial _ { \\mu } f _ { \\lambda } ( x ( s ) ) \\partial _ { \\mu } f _ { \\sigma } ( x ( s ^ { \\prime } ) ) \\frac { \\delta ^ { 2 } U } { \\delta f _ { \\lambda } ( s ) \\delta f _ { \\sigma } ( s ^ { \\prime } ) } + \\partial ^ { 2 } f _ { \\lambda } ( x ( s ) ) \\frac { \\delta U } { \\delta f _ { \\lambda } ( s ) } \\delta ( s - s ^ { \\prime } ) \\; . \\end{equation*}",
"\\begin{equation*} \\partial _ { \\mu } f _ { \\lambda } \\partial _ { \\mu } f _ { \\sigma } = \\rho ( f ) \\delta _ { \\lambda \\sigma } \\; . \\end{equation*}",
"\\begin{align*} & & \\hat { L } ( s ) U _ { f } [ C ] \\delta ( s - s ^ { \\prime } ) = \\rho ( f ) ( \\hat { L } ( s ) U ) [ f ( C ) ] \\delta ( s - s ^ { \\prime } ) \\\\ & & \\qquad \\qquad + \\, \\partial ^ { 2 } f _ { \\lambda } \\frac { \\delta U } { \\delta f _ { \\lambda } } \\delta ( s - s ^ { \\prime } ) + \\partial _ { \\mu } f _ { \\lambda } ( x ( s ) ) \\partial _ { \\mu } f _ { \\sigma } ( x ( s ^ { \\prime } ) ) N _ { [ \\lambda \\sigma ] } \\delta ^ { \\prime } ( s - s ^ { \\prime } ) \\; , \\end{align*}",
"\\begin{equation*} \\frac { \\delta ^ { 2 } U } { \\delta f _ { \\lambda } ( s ) \\delta f _ { \\sigma } ( s ^ { \\prime } ) } = N _ { [ \\lambda \\sigma ] } ( \\frac { s + s ^ { \\prime } } { 2 } ) \\delta ^ { \\prime } ( s - s ^ { \\prime } ) + \\ldots \\end{equation*}",
"\\begin{equation*} \\frac { \\delta U } { \\delta x _ { \\lambda } ( s ) } = N _ { [ \\lambda \\sigma ] } ( s ) \\dot { x } _ { \\sigma } ( s ) \\; . \\end{equation*}"
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0002106_page22 | {
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0002133_page01 | {
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0002133_page02 | {
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"\\begin{equation*} D \\equiv \\sum _ { p = 0 } ^ { 4 } D _ { p } , \\quad F \\equiv \\sum _ { p = 1 } ^ { 5 } F _ { p } . \\end{equation*}",
"\\begin{align*} \\begin{array}{lcl} F _ { 1 } \\, = \\, d D _ { 0 } , & \\quad & F _ { 2 } \\, = \\, d D _ { 1 } , \\\\ F _ { 3 } \\, = \\, d D _ { 2 } - B _ { 2 } \\wedge d D _ { 0 } , & & F _ { 4 } \\, = \\, d D _ { 3 } - B _ { 2 } \\wedge d D _ { 1 } , \\\\ F _ { 5 } \\, = \\, d D _ { 4 } - B _ { 2 } \\wedge d D _ { 2 } + \\frac { 1 } { 2 } B _ { 2 } \\wedge B _ { 2 } \\wedge d D _ { 0 } . & & ~ \\end{array} \\end{align*}",
"\\begin{equation*} \\int d ^ { d } x \\sqrt { - g } \\vert F _ { p } \\vert ^ { 2 } = \\int d ^ { d } x \\frac { \\sqrt { - g } } { p ! } g ^ { \\mu _ { 1 } \\nu _ { 1 } } \\cdots g ^ { \\mu _ { p } \\nu _ { p } } F _ { \\mu _ { 1 } \\mu _ { p } } F _ { \\nu _ { 1 } \\nu _ { p } } . \\end{equation*}"
],
"latex_expand": [
"$ \\mitd = 9 $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitS \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitd = 9 $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ \\mitd $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ \\mitS \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitS \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitS \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitS \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ \\mitC _ { \\mitp + 1 } = ( 1 \\slash ( \\mitp + 1 ) ! ) \\, \\mitC _ { \\mitmu _ { 1 } . . . \\mitmu _ { \\mitp + 1 } } \\, \\mitd \\mitx ^ { \\mitmu _ { 1 } } \\wedge \\cdots \\wedge \\mitd \\mitx ^ { \\mitmu _ { \\mitp + 1 } } $",
"$ \\mitF = \\mite ^ { - \\mitB } \\wedge \\mitd \\mitD $",
"\\begin{align*} \\begin{array}{lcl} \\mitD _ { 0 } \\equiv \\mitC _ { 0 } , & \\quad & \\mitD _ { 1 } \\equiv \\mitC _ { 1 } , \\\\ \\mitD _ { 2 } \\equiv \\mitC _ { 2 } + \\mitB _ { 2 } \\wedge \\mitC _ { 0 } , & & \\mitD _ { 3 } \\equiv \\mitC _ { 3 } + \\mitB _ { 2 } \\wedge \\mitC _ { 1 } , \\\\ \\mitD _ { 4 } \\equiv \\mitC _ { 4 } + \\frac { 1 } { 2 } \\mitB _ { 2 } \\wedge \\mitC _ { 2 } + \\frac { 1 } { 2 } \\mitB _ { 2 } \\wedge \\mitB _ { 2 } \\wedge \\mitC _ { 0 } . & & \\end{array} \\end{align*}",
"\\begin{equation*} \\mitD \\equiv \\sum _ { \\mitp = 0 } ^ { 4 } \\mitD _ { \\mitp } , \\quad \\mitF \\equiv \\sum _ { \\mitp = 1 } ^ { 5 } \\mitF _ { \\mitp } . \\end{equation*}",
"\\begin{align*} \\begin{array}{lcl} \\mitF _ { 1 } \\, = \\, \\mitd \\mitD _ { 0 } , & \\quad & \\mitF _ { 2 } \\, = \\, \\mitd \\mitD _ { 1 } , \\\\ \\mitF _ { 3 } \\, = \\, \\mitd \\mitD _ { 2 } - \\mitB _ { 2 } \\wedge \\mitd \\mitD _ { 0 } , & & \\mitF _ { 4 } \\, = \\, \\mitd \\mitD _ { 3 } - \\mitB _ { 2 } \\wedge \\mitd \\mitD _ { 1 } , \\\\ \\mitF _ { 5 } \\, = \\, \\mitd \\mitD _ { 4 } - \\mitB _ { 2 } \\wedge \\mitd \\mitD _ { 2 } + \\frac { 1 } { 2 } \\mitB _ { 2 } \\wedge \\mitB _ { 2 } \\wedge \\mitd \\mitD _ { 0 } . & & ~ \\end{array} \\end{align*}",
"\\begin{equation*} \\int \\mitd ^ { \\mitd } \\mitx \\sqrt { - \\mitg } \\vert \\mitF _ { \\mitp } \\vert ^ { 2 } = \\int \\mitd ^ { \\mitd } \\mitx \\frac { \\sqrt { - \\mitg } } { \\mitp ! } \\mitg ^ { \\mitmu _ { 1 } \\mitnu _ { 1 } } \\cdots \\mitg ^ { \\mitmu _ { \\mitp } \\mitnu _ { \\mitp } } \\mitF _ { \\mitmu _ { 1 } \\mitmu _ { \\mitp } } \\mitF _ { \\mitnu _ { 1 } \\mitnu _ { \\mitp } } . \\end{equation*}"
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0002133_page03 | {
"latex": [
"$d=10$",
"$d=9$",
"$H_3=dB_2$",
"$F_2=dC_1=dD_1$",
"$F_4=dC_3+H_3\\wedge C_1=dD_3-B_2\\wedge d D_1$",
"$x$",
"$H^{(1)}_{\\mu \\nu \\rho }=e^a_\\mu e^b_\\nu e^c_\\rho E^M_a E^N_b E^P_c H_{MNP}$",
"\\begin {equation} \\label {iia} \\begin {split} S^{IIA}_{10}=&{1\\over 2\\kappa ^2_{10}}\\int d^{10}x \\sqrt {-G} e^{-2\\Phi }\\left [R(G)+4 G^{MN}\\partial _M \\Phi \\partial _N \\Phi -{1\\over 2} |H_3|^2\\right ]\\\\ &-{1\\over 4\\kappa ^2_{10}}\\int d^{10}x \\sqrt {-G} \\Bigl (|F_2|^2+|F_4|^2\\,\\Bigr )+{1\\over 4\\kappa ^2_{10}}\\int d^{10}x B_2\\wedge dC_3\\wedge dC_3, \\end {split} \\end {equation}",
"\\begin {equation} E^A_M=\\begin {pmatrix} e^a_\\mu & e A^{(1)}_\\mu \\\\ 0 & e \\end {pmatrix}, \\quad E^M_A=\\begin {pmatrix} e_a^\\mu & -e^\\nu _aA^{(1)}_\\nu \\\\ 0 & e^{-1} \\end {pmatrix}. \\end {equation}",
"\\begin {equation} \\begin {split} \\label {9sugr} S_9=&{1\\over 2\\kappa ^2_9}\\int d^9x \\sqrt {-g} e^{-2\\phi }\\biggl [R(g)+4g^{\\mu \\nu }\\partial _\\mu \\phi \\partial _\\nu \\phi -e^{-2}g^{\\mu \\nu }\\partial _\\mu e\\partial _\\nu e\\\\ &-{1\\over 2}e^2|F^{(1)}_2|^2 -{1\\over 2}e^{-2}|F^{(2)}_2|^2 -{1\\over 2}|H_3^{(1)}|^2\\biggr ]\\\\ &-{1\\over 4\\kappa ^2_9}\\int d^9 x \\sqrt {-g} \\Bigl (e|F_2|^2+e^{-1}g^{\\mu \\nu }\\partial _\\mu D_x\\partial _\\nu D_x +e^{-1}|H^{(2)}_3|^2+e|F_4|^2\\Bigr ), \\end {split} \\end {equation}",
"\\begin {gather} e^2=G_{xx},\\qquad g_{\\mu \\nu }=G_{\\mu \\nu }- G_{xx}A^{(1)}_\\mu A^{(1)}_\\nu ,\\\\ A^{(1)}_\\mu ={G_{\\mu x}\\over G_{xx}}, \\qquad A^{(2)}_\\mu =B_{\\mu x}\\\\ A_\\mu =D_\\mu -A^{(1)}_\\mu D_x,\\quad F^{i}_{\\mu \\nu }=\\partial _\\mu A^{(i)}_\\nu -\\partial _\\nu A^{(i)}_\\mu ,\\\\ B^{(1)}_{\\mu \\nu }=B_{\\mu \\nu }+ {1\\over 2}A^{(1)}_\\mu A^{(2)}_\\nu -{1\\over 2}A^{(1)}_\\nu A^{(2)}_\\mu ,\\quad B^{(2)}_{\\mu \\nu }=D_{\\mu \\nu x},\\\\ \\phi =\\Phi -\\ln \\,G_{xx}/4, \\qquad {\\cal D}_{\\mu \\nu \\rho }=D_{\\mu \\nu \\rho },\\\\ H_3^{(1)}=d B_2^{(1)}-{1\\over 2}(A_1^{(1)}\\wedge F_2^{(2)}+A_1^{(2)}\\wedge F_2^{(1)}),\\\\ H_3^{(2)}=d B_2^{(2)}-B_2^{(1)}\\wedge d D_x +{1\\over 2}A_1^{(2)} \\wedge A_1^{(1)}\\wedge d D_x -A_1^{(2)} \\wedge (F_2+ F_2^{(1)} D_x),\\\\ F_4=d {\\cal D}_3-B_2^{(1)}\\wedge d D_1 +{1\\over 2}A_1^{(1)}\\wedge A_1^{(2)} \\wedge d D_1 +H_3^{(2)} \\wedge A_1^{(1)}, \\end {gather}",
"\\begin {equation} {\\mathst E}^A_M=\\begin {pmatrix} e^a_\\mu & e^{-1} A^{(2)}_\\mu \\\\ 0 & e^{-1} \\end {pmatrix}, \\quad {\\mathst E}^M_A=\\begin {pmatrix} e_a^\\mu & -e^\\nu _a A^{(2)}_\\nu \\\\ 0 & e \\end {pmatrix}, \\end {equation}"
],
"latex_norm": [
"$ d = 1 0 $",
"$ d = 9 $",
"$ H _ { 3 } = d B _ { 2 } $",
"$ F _ { 2 } = d C _ { 1 } = d D _ { 1 } $",
"$ F _ { 4 } = d C _ { 3 } + H _ { 3 } \\wedge C _ { 1 } = d D _ { 3 } - B _ { 2 } \\wedge d D _ { 1 } $",
"$ x $",
"$ H _ { \\mu \\nu \\rho } ^ { ( 1 ) } = e _ { \\mu } ^ { a } e _ { \\nu } ^ { b } e _ { \\rho } ^ { c } E _ { a } ^ { M } E _ { b } ^ { N } E _ { c } ^ { P } H _ { M N P } $",
"\\begin{align*} \\begin{array}{rl} S _ { 1 0 } ^ { I I A } = & \\frac { 1 } { 2 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x \\sqrt { - G } e ^ { - 2 \\Phi } [ R ( G ) + 4 G ^ { M N } \\partial _ { M } \\Phi \\partial _ { N } \\Phi - \\frac { 1 } { 2 } \\vert H _ { 3 } \\vert ^ { 2 } ] \\\\ & - \\frac { 1 } { 4 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x \\sqrt { - G } ( \\vert F _ { 2 } \\vert ^ { 2 } + \\vert F _ { 4 } \\vert ^ { 2 } \\, ) + \\frac { 1 } { 4 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x B _ { 2 } \\wedge d C _ { 3 } \\wedge d C _ { 3 } , \\end{array} \\end{align*}",
"\\begin{align*} E _ { M } ^ { A } = ( \\begin{array}{cc} e _ { \\mu } ^ { a } & e A _ { \\mu } ^ { ( 1 ) } \\\\ 0 & e \\end{array} ) , \\quad E _ { A } ^ { M } = ( \\begin{array}{cc} e _ { a } ^ { \\mu } & - e _ { a } ^ { \\nu } A _ { \\nu } ^ { ( 1 ) } \\\\ 0 & e ^ { - 1 } \\end{array} ) . \\end{align*}",
"\\begin{align*} \\begin{array}{rl} S _ { 9 } = & \\frac { 1 } { 2 \\kappa _ { 9 } ^ { 2 } } \\int d ^ { 9 } x \\sqrt { - g } e ^ { - 2 \\phi } [ R ( g ) + 4 g ^ { \\mu \\nu } \\partial _ { \\mu } \\phi \\partial _ { \\nu } \\phi - e ^ { - 2 } g ^ { \\mu \\nu } \\partial _ { \\mu } e \\partial _ { \\nu } e \\\\ & - \\frac { 1 } { 2 } e ^ { 2 } \\vert F _ { 2 } ^ { ( 1 ) } \\vert ^ { 2 } - \\frac { 1 } { 2 } e ^ { - 2 } \\vert F _ { 2 } ^ { ( 2 ) } \\vert ^ { 2 } - \\frac { 1 } { 2 } \\vert H _ { 3 } ^ { ( 1 ) } \\vert ^ { 2 } ] \\\\ & - \\frac { 1 } { 4 \\kappa _ { 9 } ^ { 2 } } \\int d ^ { 9 } x \\sqrt { - g } ( e \\vert F _ { 2 } \\vert ^ { 2 } + e ^ { - 1 } g ^ { \\mu \\nu } \\partial _ { \\mu } D _ { x } \\partial _ { \\nu } D _ { x } + e ^ { - 1 } \\vert H _ { 3 } ^ { ( 2 ) } \\vert ^ { 2 } + e \\vert F _ { 4 } \\vert ^ { 2 } ) , \\end{array} \\end{align*}",
"\\begin{align*} e ^ { 2 } = G _ { x x } , \\qquad g _ { \\mu \\nu } = G _ { \\mu \\nu } - G _ { x x } A _ { \\mu } ^ { ( 1 ) } A _ { \\nu } ^ { ( 1 ) } , \\\\ A _ { \\mu } ^ { ( 1 ) } = \\frac { G _ { \\mu x } } { G _ { x x } } , \\qquad A _ { \\mu } ^ { ( 2 ) } = B _ { \\mu x } \\\\ A _ { \\mu } = D _ { \\mu } - A _ { \\mu } ^ { ( 1 ) } D _ { x } , \\quad F _ { \\mu \\nu } ^ { i } = \\partial _ { \\mu } A _ { \\nu } ^ { ( i ) } - \\partial _ { \\nu } A _ { \\mu } ^ { ( i ) } , \\\\ B _ { \\mu \\nu } ^ { ( 1 ) } = B _ { \\mu \\nu } + \\frac { 1 } { 2 } A _ { \\mu } ^ { ( 1 ) } A _ { \\nu } ^ { ( 2 ) } - \\frac { 1 } { 2 } A _ { \\nu } ^ { ( 1 ) } A _ { \\mu } ^ { ( 2 ) } , \\quad B _ { \\mu \\nu } ^ { ( 2 ) } = D _ { \\mu \\nu x } , \\\\ \\phi = \\Phi - \\operatorname { l n } \\, G _ { x x } \\slash 4 , \\qquad D _ { \\mu \\nu \\rho } = D _ { \\mu \\nu \\rho } , \\\\ H _ { 3 } ^ { ( 1 ) } = d B _ { 2 } ^ { ( 1 ) } - \\frac { 1 } { 2 } ( A _ { 1 } ^ { ( 1 ) } \\wedge F _ { 2 } ^ { ( 2 ) } + A _ { 1 } ^ { ( 2 ) } \\wedge F _ { 2 } ^ { ( 1 ) } ) , \\\\ H _ { 3 } ^ { ( 2 ) } = d B _ { 2 } ^ { ( 2 ) } - B _ { 2 } ^ { ( 1 ) } \\wedge d D _ { x } + \\frac { 1 } { 2 } A _ { 1 } ^ { ( 2 ) } \\wedge A _ { 1 } ^ { ( 1 ) } \\wedge d D _ { x } - A _ { 1 } ^ { ( 2 ) } \\wedge ( F _ { 2 } + F _ { 2 } ^ { ( 1 ) } D _ { x } ) , \\\\ F _ { 4 } = d D _ { 3 } - B _ { 2 } ^ { ( 1 ) } \\wedge d D _ { 1 } + \\frac { 1 } { 2 } A _ { 1 } ^ { ( 1 ) } \\wedge A _ { 1 } ^ { ( 2 ) } \\wedge d D _ { 1 } + H _ { 3 } ^ { ( 2 ) } \\wedge A _ { 1 } ^ { ( 1 ) } , \\end{align*}",
"\\begin{align*} E _ { M } ^ { A } = ( \\begin{array}{cc} e _ { \\mu } ^ { a } & e ^ { - 1 } A _ { \\mu } ^ { ( 2 ) } \\\\ 0 & e ^ { - 1 } \\end{array} ) , \\quad E _ { A } ^ { M } = ( \\begin{array}{cc} e _ { a } ^ { \\mu } & - e _ { a } ^ { \\nu } A _ { \\nu } ^ { ( 2 ) } \\\\ 0 & e \\end{array} ) , \\end{align*}"
],
"latex_expand": [
"$ \\mitd = 1 0 $",
"$ \\mitd = 9 $",
"$ \\mitH _ { 3 } = \\mitd \\mitB _ { 2 } $",
"$ \\mitF _ { 2 } = \\mitd \\mitC _ { 1 } = \\mitd \\mitD _ { 1 } $",
"$ \\mitF _ { 4 } = \\mitd \\mitC _ { 3 } + \\mitH _ { 3 } \\wedge \\mitC _ { 1 } = \\mitd \\mitD _ { 3 } - \\mitB _ { 2 } \\wedge \\mitd \\mitD _ { 1 } $",
"$ \\mitx $",
"$ \\mitH _ { \\mitmu \\mitnu \\mitrho } ^ { ( 1 ) } = \\mite _ { \\mitmu } ^ { \\mita } \\mite _ { \\mitnu } ^ { \\mitb } \\mite _ { \\mitrho } ^ { \\mitc } \\mitE _ { \\mita } ^ { \\mitM } \\mitE _ { \\mitb } ^ { \\mitN } \\mitE _ { \\mitc } ^ { \\mitP } \\mitH _ { \\mitM \\mitN \\mitP } $",
"\\begin{align*} \\begin{array}{rl} \\displaystyle \\mitS _ { 1 0 } ^ { \\mitI \\mitI \\mitA } = & \\displaystyle \\frac { 1 } { 2 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\sqrt { - \\mitG } \\mite ^ { - 2 \\mupPhi } \\left[ \\mitR ( \\mitG ) + 4 \\mitG ^ { \\mitM \\mitN } \\mitpartial _ { \\mitM } \\mupPhi \\mitpartial _ { \\mitN } \\mupPhi - \\frac { 1 } { 2 } \\vert \\mitH _ { 3 } \\vert ^ { 2 } \\right] \\\\ \\displaystyle & \\displaystyle - \\frac { 1 } { 4 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\sqrt { - \\mitG } \\Big ( \\vert \\mitF _ { 2 } \\vert ^ { 2 } + \\vert \\mitF _ { 4 } \\vert ^ { 2 } \\, \\Big ) + \\frac { 1 } { 4 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\mitB _ { 2 } \\wedge \\mitd \\mitC _ { 3 } \\wedge \\mitd \\mitC _ { 3 } , \\end{array} \\end{align*}",
"\\begin{align*} \\mitE _ { \\mitM } ^ { \\mitA } = \\left( \\begin{array}{cc} \\mite _ { \\mitmu } ^ { \\mita } & \\mite \\mitA _ { \\mitmu } ^ { ( 1 ) } \\\\ 0 & \\mite \\end{array} \\right) , \\quad \\mitE _ { \\mitA } ^ { \\mitM } = \\left( \\begin{array}{cc} \\mite _ { \\mita } ^ { \\mitmu } & - \\mite _ { \\mita } ^ { \\mitnu } \\mitA _ { \\mitnu } ^ { ( 1 ) } \\\\ 0 & \\mite ^ { - 1 } \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\begin{array}{rl} \\displaystyle \\mitS _ { 9 } = & \\displaystyle \\frac { 1 } { 2 \\mitkappa _ { 9 } ^ { 2 } } \\int \\mitd ^ { 9 } \\mitx \\sqrt { - \\mitg } \\mite ^ { - 2 \\mitphi } \\bigg [ \\mitR ( \\mitg ) + 4 \\mitg ^ { \\mitmu \\mitnu } \\mitpartial _ { \\mitmu } \\mitphi \\mitpartial _ { \\mitnu } \\mitphi - \\mite ^ { - 2 } \\mitg ^ { \\mitmu \\mitnu } \\mitpartial _ { \\mitmu } \\mite \\mitpartial _ { \\mitnu } \\mite \\\\ \\displaystyle & \\displaystyle - \\frac { 1 } { 2 } \\mite ^ { 2 } \\vert \\mitF _ { 2 } ^ { ( 1 ) } \\vert ^ { 2 } - \\frac { 1 } { 2 } \\mite ^ { - 2 } \\vert \\mitF _ { 2 } ^ { ( 2 ) } \\vert ^ { 2 } - \\frac { 1 } { 2 } \\vert \\mitH _ { 3 } ^ { ( 1 ) } \\vert ^ { 2 } \\bigg ] \\\\ \\displaystyle & \\displaystyle - \\frac { 1 } { 4 \\mitkappa _ { 9 } ^ { 2 } } \\int \\mitd ^ { 9 } \\mitx \\sqrt { - \\mitg } \\Big ( \\mite \\vert \\mitF _ { 2 } \\vert ^ { 2 } + \\mite ^ { - 1 } \\mitg ^ { \\mitmu \\mitnu } \\mitpartial _ { \\mitmu } \\mitD _ { \\mitx } \\mitpartial _ { \\mitnu } \\mitD _ { \\mitx } + \\mite ^ { - 1 } \\vert \\mitH _ { 3 } ^ { ( 2 ) } \\vert ^ { 2 } + \\mite \\vert \\mitF _ { 4 } \\vert ^ { 2 } \\Big ) , \\end{array} \\end{align*}",
"\\begin{align*} \\mite ^ { 2 } = \\mitG _ { \\mitx \\mitx } , \\qquad \\mitg _ { \\mitmu \\mitnu } = \\mitG _ { \\mitmu \\mitnu } - \\mitG _ { \\mitx \\mitx } \\mitA _ { \\mitmu } ^ { ( 1 ) } \\mitA _ { \\mitnu } ^ { ( 1 ) } , \\\\ \\mitA _ { \\mitmu } ^ { ( 1 ) } = \\frac { \\mitG _ { \\mitmu \\mitx } } { \\mitG _ { \\mitx \\mitx } } , \\qquad \\mitA _ { \\mitmu } ^ { ( 2 ) } = \\mitB _ { \\mitmu \\mitx } \\\\ \\mitA _ { \\mitmu } = \\mitD _ { \\mitmu } - \\mitA _ { \\mitmu } ^ { ( 1 ) } \\mitD _ { \\mitx } , \\quad \\mitF _ { \\mitmu \\mitnu } ^ { \\miti } = \\mitpartial _ { \\mitmu } \\mitA _ { \\mitnu } ^ { ( \\miti ) } - \\mitpartial _ { \\mitnu } \\mitA _ { \\mitmu } ^ { ( \\miti ) } , \\\\ \\mitB _ { \\mitmu \\mitnu } ^ { ( 1 ) } = \\mitB _ { \\mitmu \\mitnu } + \\frac { 1 } { 2 } \\mitA _ { \\mitmu } ^ { ( 1 ) } \\mitA _ { \\mitnu } ^ { ( 2 ) } - \\frac { 1 } { 2 } \\mitA _ { \\mitnu } ^ { ( 1 ) } \\mitA _ { \\mitmu } ^ { ( 2 ) } , \\quad \\mitB _ { \\mitmu \\mitnu } ^ { ( 2 ) } = \\mitD _ { \\mitmu \\mitnu \\mitx } , \\\\ \\mitphi = \\mupPhi - \\operatorname { l n } \\, \\mitG _ { \\mitx \\mitx } \\slash 4 , \\qquad \\mitD _ { \\mitmu \\mitnu \\mitrho } = \\mitD _ { \\mitmu \\mitnu \\mitrho } , \\\\ \\mitH _ { 3 } ^ { ( 1 ) } = \\mitd \\mitB _ { 2 } ^ { ( 1 ) } - \\frac { 1 } { 2 } ( \\mitA _ { 1 } ^ { ( 1 ) } \\wedge \\mitF _ { 2 } ^ { ( 2 ) } + \\mitA _ { 1 } ^ { ( 2 ) } \\wedge \\mitF _ { 2 } ^ { ( 1 ) } ) , \\\\ \\mitH _ { 3 } ^ { ( 2 ) } = \\mitd \\mitB _ { 2 } ^ { ( 2 ) } - \\mitB _ { 2 } ^ { ( 1 ) } \\wedge \\mitd \\mitD _ { \\mitx } + \\frac { 1 } { 2 } \\mitA _ { 1 } ^ { ( 2 ) } \\wedge \\mitA _ { 1 } ^ { ( 1 ) } \\wedge \\mitd \\mitD _ { \\mitx } - \\mitA _ { 1 } ^ { ( 2 ) } \\wedge ( \\mitF _ { 2 } + \\mitF _ { 2 } ^ { ( 1 ) } \\mitD _ { \\mitx } ) , \\\\ \\mitF _ { 4 } = \\mitd \\mitD _ { 3 } - \\mitB _ { 2 } ^ { ( 1 ) } \\wedge \\mitd \\mitD _ { 1 } + \\frac { 1 } { 2 } \\mitA _ { 1 } ^ { ( 1 ) } \\wedge \\mitA _ { 1 } ^ { ( 2 ) } \\wedge \\mitd \\mitD _ { 1 } + \\mitH _ { 3 } ^ { ( 2 ) } \\wedge \\mitA _ { 1 } ^ { ( 1 ) } , \\end{align*}",
"\\begin{align*} \\mscrE _ { \\mitM } ^ { \\mitA } = \\left( \\begin{array}{cc} \\mite _ { \\mitmu } ^ { \\mita } & \\mite ^ { - 1 } \\mitA _ { \\mitmu } ^ { ( 2 ) } \\\\ 0 & \\mite ^ { - 1 } \\end{array} \\right) , \\quad \\mscrE _ { \\mitA } ^ { \\mitM } = \\left( \\begin{array}{cc} \\mite _ { \\mita } ^ { \\mitmu } & - \\mite _ { \\mita } ^ { \\mitnu } \\mitA _ { \\mitnu } ^ { ( 2 ) } \\\\ 0 & \\mite \\end{array} \\right) , \\end{align*}"
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0002133_page04 | {
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"$F_5$",
"$O(d,d,R)$",
"$\\Omega $",
"$O(d,d,R)$",
"$2d$",
"$2d$",
"\\begin {gather} e^{-2}={\\mathst G}_{xx},\\qquad g_{\\mu \\nu } ={\\mathst G}_{\\mu \\nu }- {\\mathst G}_{xx}A^{(2)}_\\mu A^{(2)}_\\nu ,\\\\ A^{(1)}_\\mu ={\\mathst B}_{\\mu x}, \\qquad A^{(2)}_\\mu ={{\\mathst G}_{\\mu x}\\over {\\mathst G}_{xx}},\\qquad D=D_x,\\\\ A_\\mu =D_{\\mu x}-{\\mathst B}_{\\mu x} D =D_{\\mu x}-A^{(1)}_\\mu D,\\\\ B^{(1)}_{\\mu \\nu }={\\mathst B}_{\\mu \\nu }- {1\\over 2}A^{(1)}_\\mu A^{(2)}_\\nu +{1\\over 2}A^{(1)}_\\nu A^{(2)}_\\mu ,\\qquad B^{(2)}_{\\mu \\nu }=D_{\\mu \\nu },\\\\ \\phi ={\\hat \\Phi }-\\ln \\,{\\mathst G}_{xx}/4, \\qquad {\\cal D}_{\\mu \\nu \\rho }=D_{\\mu \\nu \\rho x}. \\end {gather}",
"\\begin {equation} \\label {iib} \\begin {split} S^{IIB}_{10}=&{1\\over 2\\kappa ^2_{10}}\\int d^{10}x \\sqrt {-{\\mathst G}} e^{-2\\hat \\Phi }\\left [R({\\mathst G})+ 4 {\\mathst G}^{MN}\\partial _M \\hat \\Phi \\partial _N \\hat \\Phi -{1\\over 2} |{\\mathst H}_3|^2\\right ]\\\\ &-{1\\over 4\\kappa ^2_{10}}\\int d^{10}x \\sqrt {-{\\mathst G}} \\left (|F_1|^2+|F_3|^2+{1\\over 2}|F_5|^2\\,\\right ) +{1\\over 4\\kappa ^2_{10}}\\int d^{10}x {\\mathst B}_2\\wedge dC_4\\wedge dC_2, \\end {split} \\end {equation}",
"\\begin {gather} {\\tilde g}_{xx}={1\\over g_{xx}},\\quad {\\tilde g}_{\\mu x}={B_{\\mu x}\\over g_{xx}},\\quad {\\tilde g}_{\\mu \\nu }=g_{\\mu \\nu }-{g_{\\mu x}g_{\\nu x}-B_{\\mu x}B_{\\nu x} \\over g_{xx}},\\\\ {\\tilde B}_{\\mu x}={g_{\\mu x}\\over g_{xx}},\\quad {\\tilde B}_{\\mu \\nu }=B_{\\mu \\nu }-{B_{\\mu x}g_{\\nu x} -B_{\\nu x}g_{\\mu x} \\over g_{xx}},\\\\ {\\tilde \\phi }=\\phi -{1\\over 2}\\ln g_{xx},\\\\ {\\tilde D}_x=D,\\quad {\\tilde D}_\\mu =D_{\\mu x},\\quad {\\tilde D}_{\\mu \\nu x}=D_{\\mu \\nu },\\quad {\\tilde D}_{\\mu \\nu \\rho }=D_{\\mu \\nu \\rho x}. \\end {gather}",
"\\begin {gather} {\\tilde C}_x=C,\\quad {\\tilde C}_\\mu =C_{\\mu x}+B_{\\mu x}C,\\quad {\\tilde C}_{\\mu \\nu x}=C_{\\mu \\nu }+{g_{\\mu x}C_{\\nu x} -g_{\\nu x}C_{\\mu x}\\over g_{xx}},\\\\ {\\tilde C}_{\\mu \\nu \\rho }=C_{\\mu \\nu \\rho x}-{3\\over 2} B_{[\\mu \\nu }C_{\\rho ] x}-{3\\over 2}B_{x[\\mu }C_{\\nu \\rho ]} -{6g_{x[\\mu }B_{\\nu |x|}C_{\\rho ]x}\\over g_{xx}}. \\end {gather}",
"\\begin {equation} \\label {4dL} \\Omega ^T J \\Omega = J ,\\qquad J =\\begin {pmatrix} 0 & 1\\!\\!1_d\\\\ 1\\!\\!1_d & 0 \\end {pmatrix}. \\end {equation}",
"\\begin {equation} \\label {modulthree} M=\\begin {pmatrix} G^{-1} & -G^{-1}B \\\\ B G^{-1} & G -B G^{-1} B \\end {pmatrix}=\\begin {pmatrix} 1\\!\\!1 &0\\\\ B&1\\!\\!1 \\end {pmatrix} \\begin {pmatrix} G^{-1}&0\\\\ 0&G \\end {pmatrix} \\begin {pmatrix} 1\\!\\!1 &-B\\\\ 0&1\\!\\!1 \\end {pmatrix}, \\end {equation}"
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"$ O ( d , d , R ) $",
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"\\begin{align*} e ^ { - 2 } = G _ { x x } , \\qquad g _ { \\mu \\nu } = G _ { \\mu \\nu } - G _ { x x } A _ { \\mu } ^ { ( 2 ) } A _ { \\nu } ^ { ( 2 ) } , \\\\ A _ { \\mu } ^ { ( 1 ) } = B _ { \\mu x } , \\qquad A _ { \\mu } ^ { ( 2 ) } = \\frac { G _ { \\mu x } } { G _ { x x } } , \\qquad D = D _ { x } , \\\\ A _ { \\mu } = D _ { \\mu x } - B _ { \\mu x } D = D _ { \\mu x } - A _ { \\mu } ^ { ( 1 ) } D , \\\\ B _ { \\mu \\nu } ^ { ( 1 ) } = B _ { \\mu \\nu } - \\frac { 1 } { 2 } A _ { \\mu } ^ { ( 1 ) } A _ { \\nu } ^ { ( 2 ) } + \\frac { 1 } { 2 } A _ { \\nu } ^ { ( 1 ) } A _ { \\mu } ^ { ( 2 ) } , \\qquad B _ { \\mu \\nu } ^ { ( 2 ) } = D _ { \\mu \\nu } , \\\\ \\phi = \\hat { \\Phi } - \\operatorname { l n } \\, G _ { x x } \\slash 4 , \\qquad D _ { \\mu \\nu \\rho } = D _ { \\mu \\nu \\rho x } . \\end{align*}",
"\\begin{align*} \\begin{array}{rl} S _ { 1 0 } ^ { I I B } = & \\frac { 1 } { 2 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x \\sqrt { - G } e ^ { - 2 \\hat { \\Phi } } [ R ( G ) + 4 G ^ { M N } \\partial _ { M } \\hat { \\Phi } \\partial _ { N } \\hat { \\Phi } - \\frac { 1 } { 2 } \\vert H _ { 3 } \\vert ^ { 2 } ] \\\\ & - \\frac { 1 } { 4 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x \\sqrt { - G } ( \\vert F _ { 1 } \\vert ^ { 2 } + \\vert F _ { 3 } \\vert ^ { 2 } + \\frac { 1 } { 2 } \\vert F _ { 5 } \\vert ^ { 2 } \\, ) + \\frac { 1 } { 4 \\kappa _ { 1 0 } ^ { 2 } } \\int d ^ { 1 0 } x B _ { 2 } \\wedge d C _ { 4 } \\wedge d C _ { 2 } , \\end{array} \\end{align*}",
"\\begin{align*} \\widetilde { g } _ { x x } = \\frac { 1 } { g _ { x x } } , \\quad \\widetilde { g } _ { \\mu x } = \\frac { B _ { \\mu x } } { g _ { x x } } , \\quad \\widetilde { g } _ { \\mu \\nu } = g _ { \\mu \\nu } - \\frac { g _ { \\mu x } g _ { \\nu x } - B _ { \\mu x } B _ { \\nu x } } { g _ { x x } } , \\\\ \\widetilde { B } _ { \\mu x } = \\frac { g _ { \\mu x } } { g _ { x x } } , \\quad \\widetilde { B } _ { \\mu \\nu } = B _ { \\mu \\nu } - \\frac { B _ { \\mu x } g _ { \\nu x } - B _ { \\nu x } g _ { \\mu x } } { g _ { x x } } , \\\\ \\widetilde { \\phi } = \\phi - \\frac { 1 } { 2 } \\operatorname { l n } g _ { x x } , \\\\ \\widetilde { D } _ { x } = D , \\quad \\widetilde { D } _ { \\mu } = D _ { \\mu x } , \\quad \\widetilde { D } _ { \\mu \\nu x } = D _ { \\mu \\nu } , \\quad \\widetilde { D } _ { \\mu \\nu \\rho } = D _ { \\mu \\nu \\rho x } . \\end{align*}",
"\\begin{align*} \\widetilde { C } _ { x } = C , \\quad \\widetilde { C } _ { \\mu } = C _ { \\mu x } + B _ { \\mu x } C , \\quad \\widetilde { C } _ { \\mu \\nu x } = C _ { \\mu \\nu } + \\frac { g _ { \\mu x } C _ { \\nu x } - g _ { \\nu x } C _ { \\mu x } } { g _ { x x } } , \\\\ \\widetilde { C } _ { \\mu \\nu \\rho } = C _ { \\mu \\nu \\rho x } - \\frac { 3 } { 2 } B _ { [ \\mu \\nu } C _ { \\rho ] x } - \\frac { 3 } { 2 } B _ { x [ \\mu } C _ { \\nu \\rho ] } - \\frac { 6 g _ { x [ \\mu } B _ { \\nu \\vert x \\vert } C _ { \\rho ] x } } { g _ { x x } } . \\end{align*}",
"\\begin{align*} \\Omega ^ { T } J \\Omega = J , \\qquad J = ( \\begin{array}{cc} 0 & 1 \\! \\! 1 _ { d } \\\\ 1 \\! \\! 1 _ { d } & 0 \\end{array} ) . \\end{align*}",
"\\begin{align*} M = ( \\begin{array}{cc} G ^ { - 1 } & - G ^ { - 1 } B \\\\ B G ^ { - 1 } & G - B G ^ { - 1 } B \\end{array} ) = ( \\begin{array}{cc} 1 \\! \\! 1 & 0 \\\\ B & 1 \\! \\! 1 \\end{array} ) ( \\begin{array}{cc} G ^ { - 1 } & 0 \\\\ 0 & G \\end{array} ) ( \\begin{array}{cc} 1 \\! \\! 1 & - B \\\\ 0 & 1 \\! \\! 1 \\end{array} ) , \\end{align*}"
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"$ \\mitF _ { 5 } $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ \\mupOmega $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ 2 \\mitd $",
"$ 2 \\mitd $",
"\\begin{align*} \\mite ^ { - 2 } = \\mscrG _ { \\mitx \\mitx } , \\qquad \\mitg _ { \\mitmu \\mitnu } = \\mscrG _ { \\mitmu \\mitnu } - \\mscrG _ { \\mitx \\mitx } \\mitA _ { \\mitmu } ^ { ( 2 ) } \\mitA _ { \\mitnu } ^ { ( 2 ) } , \\\\ \\mitA _ { \\mitmu } ^ { ( 1 ) } = \\mscrB _ { \\mitmu \\mitx } , \\qquad \\mitA _ { \\mitmu } ^ { ( 2 ) } = \\frac { \\mscrG _ { \\mitmu \\mitx } } { \\mscrG _ { \\mitx \\mitx } } , \\qquad \\mitD = \\mitD _ { \\mitx } , \\\\ \\mitA _ { \\mitmu } = \\mitD _ { \\mitmu \\mitx } - \\mscrB _ { \\mitmu \\mitx } \\mitD = \\mitD _ { \\mitmu \\mitx } - \\mitA _ { \\mitmu } ^ { ( 1 ) } \\mitD , \\\\ \\mitB _ { \\mitmu \\mitnu } ^ { ( 1 ) } = \\mscrB _ { \\mitmu \\mitnu } - \\frac { 1 } { 2 } \\mitA _ { \\mitmu } ^ { ( 1 ) } \\mitA _ { \\mitnu } ^ { ( 2 ) } + \\frac { 1 } { 2 } \\mitA _ { \\mitnu } ^ { ( 1 ) } \\mitA _ { \\mitmu } ^ { ( 2 ) } , \\qquad \\mitB _ { \\mitmu \\mitnu } ^ { ( 2 ) } = \\mitD _ { \\mitmu \\mitnu } , \\\\ \\mitphi = \\hat { \\mupPhi } - \\operatorname { l n } \\, \\mscrG _ { \\mitx \\mitx } \\slash 4 , \\qquad \\mitD _ { \\mitmu \\mitnu \\mitrho } = \\mitD _ { \\mitmu \\mitnu \\mitrho \\mitx } . \\end{align*}",
"\\begin{align*} \\begin{array}{rl} \\displaystyle \\mitS _ { 1 0 } ^ { \\mitI \\mitI \\mitB } = & \\displaystyle \\frac { 1 } { 2 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\sqrt { - \\mscrG } \\mite ^ { - 2 \\hat { \\mupPhi } } \\left[ \\mitR ( \\mscrG ) + 4 \\mscrG ^ { \\mitM \\mitN } \\mitpartial _ { \\mitM } \\hat { \\mupPhi } \\mitpartial _ { \\mitN } \\hat { \\mupPhi } - \\frac { 1 } { 2 } \\vert \\mscrH _ { 3 } \\vert ^ { 2 } \\right] \\\\ \\displaystyle & \\displaystyle - \\frac { 1 } { 4 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\sqrt { - \\mscrG } \\left( \\vert \\mitF _ { 1 } \\vert ^ { 2 } + \\vert \\mitF _ { 3 } \\vert ^ { 2 } + \\frac { 1 } { 2 } \\vert \\mitF _ { 5 } \\vert ^ { 2 } \\, \\right) + \\frac { 1 } { 4 \\mitkappa _ { 1 0 } ^ { 2 } } \\int \\mitd ^ { 1 0 } \\mitx \\mscrB _ { 2 } \\wedge \\mitd \\mitC _ { 4 } \\wedge \\mitd \\mitC _ { 2 } , \\end{array} \\end{align*}",
"\\begin{align*} \\tilde { \\mitg } _ { \\mitx \\mitx } = \\frac { 1 } { \\mitg _ { \\mitx \\mitx } } , \\quad \\tilde { \\mitg } _ { \\mitmu \\mitx } = \\frac { \\mitB _ { \\mitmu \\mitx } } { \\mitg _ { \\mitx \\mitx } } , \\quad \\tilde { \\mitg } _ { \\mitmu \\mitnu } = \\mitg _ { \\mitmu \\mitnu } - \\frac { \\mitg _ { \\mitmu \\mitx } \\mitg _ { \\mitnu \\mitx } - \\mitB _ { \\mitmu \\mitx } \\mitB _ { \\mitnu \\mitx } } { \\mitg _ { \\mitx \\mitx } } , \\\\ \\tilde { \\mitB } _ { \\mitmu \\mitx } = \\frac { \\mitg _ { \\mitmu \\mitx } } { \\mitg _ { \\mitx \\mitx } } , \\quad \\tilde { \\mitB } _ { \\mitmu \\mitnu } = \\mitB _ { \\mitmu \\mitnu } - \\frac { \\mitB _ { \\mitmu \\mitx } \\mitg _ { \\mitnu \\mitx } - \\mitB _ { \\mitnu \\mitx } \\mitg _ { \\mitmu \\mitx } } { \\mitg _ { \\mitx \\mitx } } , \\\\ \\tilde { \\mitphi } = \\mitphi - \\frac { 1 } { 2 } \\operatorname { l n } \\mitg _ { \\mitx \\mitx } , \\\\ \\tilde { \\mitD } _ { \\mitx } = \\mitD , \\quad \\tilde { \\mitD } _ { \\mitmu } = \\mitD _ { \\mitmu \\mitx } , \\quad \\tilde { \\mitD } _ { \\mitmu \\mitnu \\mitx } = \\mitD _ { \\mitmu \\mitnu } , \\quad \\tilde { \\mitD } _ { \\mitmu \\mitnu \\mitrho } = \\mitD _ { \\mitmu \\mitnu \\mitrho \\mitx } . \\end{align*}",
"\\begin{align*} \\tilde { \\mitC } _ { \\mitx } = \\mitC , \\quad \\tilde { \\mitC } _ { \\mitmu } = \\mitC _ { \\mitmu \\mitx } + \\mitB _ { \\mitmu \\mitx } \\mitC , \\quad \\tilde { \\mitC } _ { \\mitmu \\mitnu \\mitx } = \\mitC _ { \\mitmu \\mitnu } + \\frac { \\mitg _ { \\mitmu \\mitx } \\mitC _ { \\mitnu \\mitx } - \\mitg _ { \\mitnu \\mitx } \\mitC _ { \\mitmu \\mitx } } { \\mitg _ { \\mitx \\mitx } } , \\\\ \\tilde { \\mitC } _ { \\mitmu \\mitnu \\mitrho } = \\mitC _ { \\mitmu \\mitnu \\mitrho \\mitx } - \\frac { 3 } { 2 } \\mitB _ { [ \\mitmu \\mitnu } \\mitC _ { \\mitrho ] \\mitx } - \\frac { 3 } { 2 } \\mitB _ { \\mitx [ \\mitmu } \\mitC _ { \\mitnu \\mitrho ] } - \\frac { 6 \\mitg _ { \\mitx [ \\mitmu } \\mitB _ { \\mitnu \\vert \\mitx \\vert } \\mitC _ { \\mitrho ] \\mitx } } { \\mitg _ { \\mitx \\mitx } } . \\end{align*}",
"\\begin{align*} \\mupOmega ^ { \\mitT } \\mitJ \\mupOmega = \\mitJ , \\qquad \\mitJ = \\left( \\begin{array}{cc} 0 & 1 \\! \\! 1 _ { \\mitd } \\\\ 1 \\! \\! 1 _ { \\mitd } & 0 \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mitM = \\left( \\begin{array}{cc} \\mitG ^ { - 1 } & - \\mitG ^ { - 1 } \\mitB \\\\ \\mitB \\mitG ^ { - 1 } & \\mitG - \\mitB \\mitG ^ { - 1 } \\mitB \\end{array} \\right) = \\left( \\begin{array}{cc} 1 \\! \\! 1 & 0 \\\\ \\mitB & 1 \\! \\! 1 \\end{array} \\right) \\left( \\begin{array}{cc} \\mitG ^ { - 1 } & 0 \\\\ 0 & \\mitG \\end{array} \\right) \\left( \\begin{array}{cc} 1 \\! \\! 1 & - \\mitB \\\\ 0 & 1 \\! \\! 1 \\end{array} \\right) , \\end{align*}"
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0002133_page05 | {
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"${\\mathst A}{\\mathst B}^T+{\\mathst B}{\\mathst A}^T= {\\mathst C}{\\mathst D}^T+{\\mathst D}{\\mathst C}^T=0$",
"${\\mathst A}{\\mathst D}^T+{\\mathst B}{\\mathst C}^T= {\\mathst C}{\\mathst B}^T+{\\mathst D}{\\mathst A}^T=1$",
"${\\mathst A}$",
"${\\mathst B}$",
"${\\mathst C}$",
"${\\mathst D}$",
"$d\\times d$",
"${\\mathst D}={\\mathst C}{\\mathst A}^{-1} {\\mathst B}+({\\mathst A}^{-1})^T$",
"$O(d,d,R)$",
"$C^T=-C$",
"$R\\in GL(d,R)$",
"$i$",
"$j$",
"$k=1$",
"$\\dots $",
"$d$",
"$\\Lambda _C$",
"$C$",
"$\\Lambda _R$",
"$G\\rightarrow RGR^T$",
"$B\\rightarrow RBR^T$",
"$O(d,d,Z)$",
"$\\{\\Gamma _r,\\ \\Gamma _s\\}=2J_{rs}$",
"$r$",
"$s=1$",
"$\\dots $",
"$2d$",
"$\\{a_i$",
"$a_j^\\dag \\}=\\delta _{ij} 1\\!\\!1$",
"$\\{a_i$",
"$a_j\\}=\\{a_i^\\dag $",
"$a_j^\\dag \\}=0$",
"$a_i|0\\rangle =0$",
"$O(d,d)$",
"\\begin {gather} A_{\\mu m}^{(1)}=G_{\\mu m},\\quad A^{(1)m}_\\mu =G^{mn} A_{\\mu n}^{(1)},\\\\ A^{(2)}_{\\mu m} = B_{\\mu m} + B_{m n} A^{(1) n}_{\\mu },\\quad {\\cal A}^i_{\\mu } =\\begin {pmatrix} A^{(1)m}_{\\mu }\\\\ A^{(2)}_{\\mu \\,m} \\end {pmatrix},\\\\ g_{\\mu \\nu }=G_{\\mu \\nu }-G_{mn} A^{(1)m}_\\mu A^{(1)n}_{\\nu },\\\\ \\phi =\\Phi -{1\\over 4}\\ln \\, {\\rm det}(G_{mn}),\\\\ B_{\\mu \\nu } = \\hat B_{\\mu \\nu } + {1 \\over 2} A^{(1) m}_{\\mu } A^{(2)}_{\\nu m} - {1 \\over 2} A^{(1) m}_{\\nu } A^{(2)}_{\\mu m} - A^{(1) m}_{\\mu } B_{m n} A^{(1) n}_\\nu , \\end {gather}",
"\\begin {equation} M \\to \\Omega M \\Omega ^T ,\\ \\ \\ {\\cal A}^i_{\\mu } \\to \\Omega _{ij} {\\cal A}^j_{\\mu }, \\ \\ \\ g_{\\mu \\nu } \\to g_{\\mu \\nu }, \\ \\ \\^^M\\phi \\to \\phi , \\ \\ \\ B_{\\mu \\nu } \\to B_{\\mu \\nu }. \\label {tdual} \\end {equation}",
"\\begin {equation} \\Omega =\\begin {pmatrix} {\\mathst A} & {\\mathst B}\\\\ {\\mathst C} & {\\mathst D} \\end {pmatrix}, \\end {equation}",
"\\begin {equation} \\Lambda _C=\\begin {pmatrix} 1\\!\\!1& 0\\\\ C & 1\\!\\!1 \\end {pmatrix},\\quad \\Lambda _R=\\begin {pmatrix} (R^T)^{-1} & 0\\\\ 0 & R \\end {pmatrix},\\quad \\Lambda _i=\\begin {pmatrix} -1\\!\\!1 + e_i & e_i\\\\ e_i & -1\\!\\!1 +e_i \\end {pmatrix},\\quad (e_i)_{jk}=\\delta _{ij}\\delta _{jk}, \\end {equation}",
"\\begin {equation} \\label {crean} a_i={\\Gamma _{d+i}\\over \\sqrt {2}},\\quad a_i^\\dag ={\\Gamma _i\\over \\sqrt {2}},\\quad i=1,\\ \\dots ,\\ d. \\end {equation}",
"\\begin {equation} \\label {state} |\\alpha \\rangle =(a_1^\\dag )^{i_1}\\cdots (a_d^\\dag )^{i_d} |0\\rangle , \\quad i_1,\\ \\dots ,\\ i_d=0\\ {\\rm or}\\ 1. \\end {equation}",
"\\begin {equation} \\label {spinrep} S(\\Omega )\\Gamma _s S(\\Omega )^{-1}=\\sum _r \\Gamma _r\\Omega ^r\\,_s. \\end {equation}"
],
"latex_norm": [
"$ G = [ G _ { i j } ] $",
"$ B = [ B _ { i j } ] $",
"$ d \\times d $",
"$ i $",
"$ j $",
"$ d $",
"$ \\Phi $",
"$ G _ { \\mu m } $",
"$ G _ { \\mu \\nu } $",
"$ G _ { m n } $",
"$ \\hat { B } _ { \\mu \\nu } $",
"$ B _ { \\mu m } $",
"$ B _ { m n } $",
"$ O ( d , d ) $",
"$ \\Omega $",
"$ O ( d , d , R ) $",
"$ A B ^ { T } + B A ^ { T } = C D ^ { T } + D C ^ { T } = 0 $",
"$ A D ^ { T } + B C ^ { T } = C B ^ { T } + D A ^ { T } = 1 $",
"$ A $",
"$ B $",
"$ C $",
"$ D $",
"$ d \\times d $",
"$ D = C A ^ { - 1 } B + ( A ^ { - 1 } ) ^ { T } $",
"$ O ( d , d , R ) $",
"$ C ^ { T } = - C $",
"$ R \\in G L ( d , R ) $",
"$ i $",
"$ j $",
"$ k = 1 $",
"$ \\dots $",
"$ d $",
"$ \\Lambda _ { C } $",
"$ C $",
"$ \\Lambda _ { R } $",
"$ G \\rightarrow R G R ^ { T } $",
"$ B \\rightarrow R B R ^ { T } $",
"$ O ( d , d , Z ) $",
"$ \\{ \\Gamma _ { r } , ~ \\Gamma _ { s } \\} = 2 J _ { r s } $",
"$ r $",
"$ s = 1 $",
"$ \\ldots $",
"$ 2 d $",
"$ \\{ a _ { i } $",
"$ a _ { j } ^ { \\dagger } \\} = \\delta _ { i j } 1 \\! \\! 1 $",
"$ \\{ a _ { i } $",
"$ a _ { j } \\} = \\{ a _ { i } ^ { \\dagger } $",
"$ a _ { j } ^ { \\dagger } \\} = 0 $",
"$ a _ { i } \\vert 0 \\rangle = 0 $",
"$ O ( d , d ) $",
"\\begin{align*} A _ { \\mu m } ^ { ( 1 ) } = G _ { \\mu m } , \\quad A _ { \\mu } ^ { ( 1 ) m } = G ^ { m n } A _ { \\mu n } ^ { ( 1 ) } , \\\\ \\\\ \\\\ g _ { \\mu \\nu } = G _ { \\mu \\nu } - G _ { m n } A _ { \\mu } ^ { ( 1 ) m } A _ { \\nu } ^ { ( 1 ) n } , \\\\ \\phi = \\Phi - \\frac { 1 } { 4 } \\operatorname { l n } \\, d e t ( G _ { m n } ) , \\\\ B _ { \\mu \\nu } = \\hat { B } _ { \\mu \\nu } + \\frac { 1 } { 2 } A _ { \\mu } ^ { ( 1 ) m } A _ { \\nu m } ^ { ( 2 ) } - \\frac { 1 } { 2 } A _ { \\nu } ^ { ( 1 ) m } A _ { \\mu m } ^ { ( 2 ) } - A _ { \\mu } ^ { ( 1 ) m } B _ { m n } A _ { \\nu } ^ { ( 1 ) n } , \\end{align*}",
"\\begin{equation*} M \\rightarrow \\Omega M \\Omega ^ { T } , ~ ~ ~ A _ { \\mu } ^ { i } \\rightarrow \\Omega _ { i j } A _ { \\mu } ^ { j } , ~ ~ ~ g _ { \\mu \\nu } \\rightarrow g _ { \\mu \\nu } , ~ ~ ~ \\phi \\rightarrow \\phi , ~ ~ ~ B _ { \\mu \\nu } \\rightarrow B _ { \\mu \\nu } . \\end{equation*}",
"\\begin{align*} \\Omega = ( \\begin{array}{cc} A & B \\\\ C & D \\end{array} ) , \\end{align*}",
"\\begin{align*} \\Lambda _ { C } = ( \\begin{array}{cc} 1 \\! \\! 1 & 0 \\\\ C & 1 \\! \\! 1 \\end{array} ) , \\quad \\Lambda _ { R } = ( \\begin{array}{cc} ( R ^ { T } ) ^ { - 1 } & 0 \\\\ 0 & R \\end{array} ) , \\quad \\Lambda _ { i } = ( \\begin{array}{cc} - 1 \\! \\! 1 + e _ { i } & e _ { i } \\\\ e _ { i } & - 1 \\! \\! 1 + e _ { i } \\end{array} ) , \\quad ( e _ { i } ) _ { j k } = \\delta _ { i j } \\delta _ { j k } , \\end{align*}",
"\\begin{equation*} a _ { i } = \\frac { \\Gamma _ { d + i } } { \\sqrt { 2 } } , \\quad a _ { i } ^ { \\dagger } = \\frac { \\Gamma _ { i } } { \\sqrt { 2 } } , \\quad i = 1 , ~ \\ldots , ~ d . \\end{equation*}",
"\\begin{equation*} \\vert \\alpha \\rangle = ( a _ { 1 } ^ { \\dagger } ) ^ { i _ { 1 } } \\cdots ( a _ { d } ^ { \\dagger } ) ^ { i _ { d } } \\vert 0 \\rangle , \\quad i _ { 1 } , ~ \\ldots , ~ i _ { d } = 0 ~ o r ~ 1 . \\end{equation*}",
"\\begin{equation*} S ( \\Omega ) \\Gamma _ { s } S ( \\Omega ) ^ { - 1 } = \\sum _ { r } \\Gamma _ { r } \\Omega ^ { r } \\, { } _ { s } . \\end{equation*}"
],
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"$ \\mitd \\times \\mitd $",
"$ \\miti $",
"$ \\mitj $",
"$ \\mitd $",
"$ \\mupPhi $",
"$ \\mitG _ { \\mitmu \\mitm } $",
"$ \\mitG _ { \\mitmu \\mitnu } $",
"$ \\mitG _ { \\mitm \\mitn } $",
"$ \\hat { \\mitB } _ { \\mitmu \\mitnu } $",
"$ \\mitB _ { \\mitmu \\mitm } $",
"$ \\mitB _ { \\mitm \\mitn } $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mupOmega $",
"$ \\mitO ( \\mitd , \\mitd , \\mitR ) $",
"$ \\mscrA \\mscrB ^ { \\mitT } + \\mscrB \\mscrA ^ { \\mitT } = \\mscrC \\mscrD ^ { \\mitT } + \\mscrD \\mscrC ^ { \\mitT } = 0 $",
"$ \\mscrA \\mscrD ^ { \\mitT } + \\mscrB \\mscrC ^ { \\mitT } = \\mscrC \\mscrB ^ { \\mitT } + \\mscrD \\mscrA ^ { \\mitT } = 1 $",
"$ \\mscrA $",
"$ \\mscrB $",
"$ \\mscrC $",
"$ \\mscrD $",
"$ \\mitd \\times \\mitd $",
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"$ \\miti $",
"$ \\mitj $",
"$ \\mitk = 1 $",
"$ \\dots $",
"$ \\mitd $",
"$ \\mupLambda _ { \\mitC } $",
"$ \\mitC $",
"$ \\mupLambda _ { \\mitR } $",
"$ \\mitG \\rightarrow \\mitR \\mitG \\mitR ^ { \\mitT } $",
"$ \\mitB \\rightarrow \\mitR \\mitB \\mitR ^ { \\mitT } $",
"$ \\mitO ( \\mitd , \\mitd , \\mitZ ) $",
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"$ \\mitr $",
"$ \\mits = 1 $",
"$ \\ldots $",
"$ 2 \\mitd $",
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"$ \\{ \\mita _ { \\miti } $",
"$ \\mita _ { \\mitj } \\} = \\{ \\mita _ { \\miti } ^ { \\dagger } $",
"$ \\mita _ { \\mitj } ^ { \\dagger } \\} = 0 $",
"$ \\mita _ { \\miti } \\vert 0 \\rangle = 0 $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"\\begin{align*} \\mitA _ { \\mitmu \\mitm } ^ { ( 1 ) } = \\mitG _ { \\mitmu \\mitm } , \\quad \\mitA _ { \\mitmu } ^ { ( 1 ) \\mitm } = \\mitG ^ { \\mitm \\mitn } \\mitA _ { \\mitmu \\mitn } ^ { ( 1 ) } , \\\\ \\\\ \\\\ \\mitg _ { \\mitmu \\mitnu } = \\mitG _ { \\mitmu \\mitnu } - \\mitG _ { \\mitm \\mitn } \\mitA _ { \\mitmu } ^ { ( 1 ) \\mitm } \\mitA _ { \\mitnu } ^ { ( 1 ) \\mitn } , \\\\ \\mitphi = \\mupPhi - \\frac { 1 } { 4 } \\operatorname { l n } \\, \\mathrm { d e t } ( \\mitG _ { \\mitm \\mitn } ) , \\\\ \\mitB _ { \\mitmu \\mitnu } = \\hat { \\mitB } _ { \\mitmu \\mitnu } + \\frac { 1 } { 2 } \\mitA _ { \\mitmu } ^ { ( 1 ) \\mitm } \\mitA _ { \\mitnu \\mitm } ^ { ( 2 ) } - \\frac { 1 } { 2 } \\mitA _ { \\mitnu } ^ { ( 1 ) \\mitm } \\mitA _ { \\mitmu \\mitm } ^ { ( 2 ) } - \\mitA _ { \\mitmu } ^ { ( 1 ) \\mitm } \\mitB _ { \\mitm \\mitn } \\mitA _ { \\mitnu } ^ { ( 1 ) \\mitn } , \\end{align*}",
"\\begin{equation*} \\mitM \\rightarrow \\mupOmega \\mitM \\mupOmega ^ { \\mitT } , ~ ~ ~ \\mitA _ { \\mitmu } ^ { \\miti } \\rightarrow \\mupOmega _ { \\miti \\mitj } \\mitA _ { \\mitmu } ^ { \\mitj } , ~ ~ ~ \\mitg _ { \\mitmu \\mitnu } \\rightarrow \\mitg _ { \\mitmu \\mitnu } , ~ ~ ~ \\mitphi \\rightarrow \\mitphi , ~ ~ ~ \\mitB _ { \\mitmu \\mitnu } \\rightarrow \\mitB _ { \\mitmu \\mitnu } . \\end{equation*}",
"\\begin{align*} \\mupOmega = \\left( \\begin{array}{cc} \\mscrA & \\mscrB \\\\ \\mscrC & \\mscrD \\end{array} \\right) , \\end{align*}",
"\\begin{align*} \\mupLambda _ { \\mitC } = \\left( \\begin{array}{cc} 1 \\! \\! 1 & 0 \\\\ \\mitC & 1 \\! \\! 1 \\end{array} \\right) , \\quad \\mupLambda _ { \\mitR } = \\left( \\begin{array}{cc} ( \\mitR ^ { \\mitT } ) ^ { - 1 } & 0 \\\\ 0 & \\mitR \\end{array} \\right) , \\quad \\mupLambda _ { \\miti } = \\left( \\begin{array}{cc} - 1 \\! \\! 1 + \\mite _ { \\miti } & \\mite _ { \\miti } \\\\ \\mite _ { \\miti } & - 1 \\! \\! 1 + \\mite _ { \\miti } \\end{array} \\right) , \\quad ( \\mite _ { \\miti } ) _ { \\mitj \\mitk } = \\mitdelta _ { \\miti \\mitj } \\mitdelta _ { \\mitj \\mitk } , \\end{align*}",
"\\begin{equation*} \\mita _ { \\miti } = \\frac { \\mupGamma _ { \\mitd + \\miti } } { \\sqrt { 2 } } , \\quad \\mita _ { \\miti } ^ { \\dagger } = \\frac { \\mupGamma _ { \\miti } } { \\sqrt { 2 } } , \\quad \\miti = 1 , ~ \\ldots , ~ \\mitd . \\end{equation*}",
"\\begin{equation*} \\vert \\mitalpha \\rangle = ( \\mita _ { 1 } ^ { \\dagger } ) ^ { \\miti _ { 1 } } \\cdots ( \\mita _ { \\mitd } ^ { \\dagger } ) ^ { \\miti _ { \\mitd } } \\vert 0 \\rangle , \\quad \\miti _ { 1 } , ~ \\ldots , ~ \\miti _ { \\mitd } = 0 ~ \\mathrm { o r } ~ 1 . \\end{equation*}",
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0002133_page06 | {
"latex": [
"$\\Omega $",
"$+$",
"${\\mathbold \\Lambda }_i$",
"$D$",
"$d=1$",
"$|\\alpha \\rangle =(|0\\rangle ,\\ a^\\dag |0\\rangle )$",
"$d=2$",
"$|\\alpha \\rangle =(|0\\rangle ,\\ a_x^\\dag |0\\rangle ,\\^^Ma_y^\\dag |0\\rangle ,\\ a_x^\\dag a_y^\\dag |0\\rangle )$",
"$\\chi $",
"$O(1,1)$",
"$\\Lambda _i$",
"$SO(1,1)$",
"$\\Lambda _i\\Lambda _j$",
"$O(d)\\otimes O(d)$",
"$O(d)$",
"$R$",
"$S$",
"$O(d,d)$",
"$\\Omega $",
"$J$",
"$O(d,d)$",
"$\\eta $",
"\\begin {equation} \\label {operator} {\\mathbold \\Omega }\\Gamma _s = \\sum _r\\Gamma _r\\Omega ^r\\,_s {\\mathbold \\Omega },\\quad {\\mathbold \\Omega }|\\beta \\rangle =\\sum _\\alpha |\\alpha \\rangle S_{\\alpha \\beta }(\\Omega ). \\end {equation}",
"\\begin {gather} {\\mathbold \\Lambda }_C=\\exp \\left ({1\\over 2} C_{ij}a_i a_j\\right ),\\quad {\\mathbold \\Lambda }_i=\\pm (a_i+a_i^\\dag ),\\\\ {\\mathbold \\Lambda }_R=({\\rm det}R)^{-1/2}\\,\\exp \\left (a_i A_i\\,^j a_j^\\dag \\right ),\\quad R=R_i\\,^j=\\exp (A_i\\,^j), \\end {gather}",
"\\begin {gather*} \\chi _\\alpha =(D,\\ D_x),\\quad \\chi _{\\mu \\alpha }=(D_\\mu ,\\ D_{\\mu x}),\\\\ \\chi _{\\mu \\nu \\alpha }=(D_{\\mu \\nu },\\ D_{\\mu \\nu x}),\\quad \\chi _{\\mu \\nu \\rho \\alpha }=(D_{\\mu \\nu \\rho },\\ D_{\\mu \\nu \\rho x}),\\\\ \\cdots , \\end {gather*}",
"\\begin {gather*} \\chi _\\alpha =(D,\\ D_x,\\ D_y,\\ D_{yx}),\\\\ \\chi _{\\mu \\alpha }=(D_\\mu ,\\ D_{\\mu x},\\ D_{\\mu y}, \\ D_{\\mu yx}),\\\\ \\chi _{\\mu \\nu \\alpha }=(D_{\\mu \\nu },\\ D_{\\mu \\nu x}, \\ D_{\\mu \\nu y},\\ D_{\\mu \\nu yx}),\\\\ \\cdots , \\end {gather*}",
"\\begin {equation} \\label {spintrans} |{\\tilde \\chi }_{\\mu _1\\dots \\mu _p\\alpha }\\rangle =\\sum _\\beta S^{-1}(\\Omega ^T)_{\\alpha \\beta } |{\\tilde \\chi }_{\\mu _1\\dots \\mu _p\\beta }\\rangle . \\end {equation}",
"\\begin {equation} \\label {rdualex} S\\left ((\\Lambda ^T)^{-1}\\right )=S(\\Lambda ) =\\Lambda =\\begin {pmatrix} 0 & 1\\\\ 1 & 0 \\end {pmatrix}. \\end {equation}",
"\\begin {equation} \\label {rdualex1} S(\\Lambda ^2)=\\Lambda ^2=\\begin {pmatrix} 1 & 0\\\\ 0 & 1 \\end {pmatrix}. \\end {equation}",
"$$J={\\mathst R}\\eta {\\mathst R},\\quad \\eta =\\begin {pmatrix} -1\\!\\!1 & 0\\\\ 0 & 1\\!\\!1 \\end {pmatrix},\\quad {\\mathst R}={\\sqrt {2}\\over 2} \\begin {pmatrix} -1\\!\\!1 & 1\\!\\!1\\\\ 1\\!\\!1 & 1\\!\\!1 \\end {pmatrix},$$"
],
"latex_norm": [
"$ \\Omega $",
"$ + $",
"$ \\Lambda i $",
"$ D $",
"$ d = 1 $",
"$ \\vert \\alpha \\rangle = ( \\vert 0 \\rangle , ~ a ^ { \\dagger } \\vert 0 \\rangle ) $",
"$ d = 2 $",
"$ \\vert \\alpha \\rangle = ( \\vert 0 \\rangle , ~ a _ { x } ^ { \\dagger } \\vert 0 \\rangle , ~ a _ { y } ^ { \\dagger } \\vert 0 \\rangle , ~ a _ { x } ^ { \\dagger } a _ { y } ^ { \\dagger } \\vert 0 \\rangle ) $",
"$ \\chi $",
"$ O ( 1 , 1 ) $",
"$ \\Lambda _ { i } $",
"$ S O ( 1 , 1 ) $",
"$ \\Lambda _ { i } \\Lambda _ { j } $",
"$ O ( d ) \\otimes O ( d ) $",
"$ O ( d ) $",
"$ R $",
"$ S $",
"$ O ( d , d ) $",
"$ \\Omega $",
"$ J $",
"$ O ( d , d ) $",
"$ \\eta $",
"\\begin{equation*} \\Omega \\Gamma _ { s } = \\sum _ { r } \\Gamma _ { r } \\Omega ^ { r } \\, { } _ { s } \\Omega , \\quad \\Omega \\vert \\beta \\rangle = \\sum _ { \\alpha } \\vert \\alpha \\rangle S _ { \\alpha \\beta } ( \\Omega ) . \\end{equation*}",
"\\begin{align*} \\Lambda _ { C } = \\operatorname { e x p } ( \\frac { 1 } { 2 } C _ { i j } a _ { i } a _ { j } ) , \\quad \\Lambda _ { i } = \\pm ( a _ { i } + a _ { i } ^ { \\dagger } ) , \\\\ \\Lambda _ { R } = ( d e t R ) ^ { - 1 \\slash 2 } \\, \\operatorname { e x p } ( a _ { i } A _ { i } \\, { } ^ { j } a _ { j } ^ { \\dagger } ) , \\quad R = R _ { i } \\, { } ^ { j } = \\operatorname { e x p } ( A _ { i } \\, { } ^ { j } ) , \\end{align*}",
"\\begin{align*} \\chi _ { \\alpha } = ( D , ~ D _ { x } ) , \\quad \\chi _ { \\mu \\alpha } = ( D _ { \\mu } , ~ D _ { \\mu x } ) , \\\\ \\chi _ { \\mu \\nu \\alpha } = ( D _ { \\mu \\nu } , ~ D _ { \\mu \\nu x } ) , \\quad \\chi _ { \\mu \\nu \\rho \\alpha } = ( D _ { \\mu \\nu \\rho } , ~ D _ { \\mu \\nu \\rho x } ) , \\\\ \\cdots , \\end{align*}",
"\\begin{align*} \\chi _ { \\alpha } = ( D , ~ D _ { x } , ~ D _ { y } , ~ D _ { y x } ) , \\\\ \\chi _ { \\mu \\alpha } = ( D _ { \\mu } , ~ D _ { \\mu x } , ~ D _ { \\mu y } , ~ D _ { \\mu y x } ) , \\\\ \\chi _ { \\mu \\nu \\alpha } = ( D _ { \\mu \\nu } , ~ D _ { \\mu \\nu x } , ~ D _ { \\mu \\nu y } , ~ D _ { \\mu \\nu y x } ) , \\\\ \\cdots , \\end{align*}",
"\\begin{equation*} \\vert \\widetilde { \\chi } _ { \\mu _ { 1 } \\ldots \\mu _ { p } \\alpha } \\rangle = \\sum _ { \\beta } S ^ { - 1 } ( \\Omega ^ { T } ) _ { \\alpha \\beta } \\vert \\widetilde { \\chi } _ { \\mu _ { 1 } \\ldots \\mu _ { p } \\beta } \\rangle . \\end{equation*}",
"\\begin{align*} S ( ( \\Lambda ^ { T } ) ^ { - 1 } ) = S ( \\Lambda ) = \\Lambda = ( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} ) . \\end{align*}",
"\\begin{align*} S ( \\Lambda ^ { 2 } ) = \\Lambda ^ { 2 } = ( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} ) . \\end{align*}",
"\\begin{align*} J = R \\eta R , \\quad \\eta = ( \\begin{array}{cc} - 1 \\! \\! 1 & 0 \\\\ 0 & 1 \\! \\! 1 \\end{array} ) , \\quad R = \\frac { \\sqrt { 2 } } { 2 } ( \\begin{array}{cc} - 1 \\! \\! 1 & 1 \\! \\! 1 \\\\ 1 \\! \\! 1 & 1 \\! \\! 1 \\end{array} ) , \\end{align*}"
],
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"$ \\mupOmega $",
"$ + $",
"$ \\mupLambda \\miti $",
"$ \\mitD $",
"$ \\mitd = 1 $",
"$ \\vert \\mitalpha \\rangle = ( \\vert 0 \\rangle , ~ \\mita ^ { \\dagger } \\vert 0 \\rangle ) $",
"$ \\mitd = 2 $",
"$ \\vert \\mitalpha \\rangle = ( \\vert 0 \\rangle , ~ \\mita _ { \\mitx } ^ { \\dagger } \\vert 0 \\rangle , ~ \\mita _ { \\mity } ^ { \\dagger } \\vert 0 \\rangle , ~ \\mita _ { \\mitx } ^ { \\dagger } \\mita _ { \\mity } ^ { \\dagger } \\vert 0 \\rangle ) $",
"$ \\mitchi $",
"$ \\mitO ( 1 , 1 ) $",
"$ \\mupLambda _ { \\miti } $",
"$ \\mitS \\mitO ( 1 , 1 ) $",
"$ \\mupLambda _ { \\miti } \\mupLambda _ { \\mitj } $",
"$ \\mitO ( \\mitd ) \\otimes \\mitO ( \\mitd ) $",
"$ \\mitO ( \\mitd ) $",
"$ \\mitR $",
"$ \\mitS $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\mupOmega $",
"$ \\mitJ $",
"$ \\mitO ( \\mitd , \\mitd ) $",
"$ \\miteta $",
"\\begin{equation*} \\mbfOmega \\mupGamma _ { \\mits } = \\sum _ { \\mitr } \\mupGamma _ { \\mitr } \\mupOmega ^ { \\mitr } \\, { } _ { \\mits } \\mbfOmega , \\quad \\mbfOmega \\vert \\mitbeta \\rangle = \\sum _ { \\mitalpha } \\vert \\mitalpha \\rangle \\mitS _ { \\mitalpha \\mitbeta } ( \\mupOmega ) . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mbfLambda _ { \\mitC } = \\operatorname { e x p } \\left( \\frac { 1 } { 2 } \\mitC _ { \\miti \\mitj } \\mita _ { \\miti } \\mita _ { \\mitj } \\right) , \\quad \\mbfLambda _ { \\miti } = \\pm ( \\mita _ { \\miti } + \\mita _ { \\miti } ^ { \\dagger } ) , \\\\ \\displaystyle \\mbfLambda _ { \\mitR } = ( \\mathrm { d e t } \\mitR ) ^ { - 1 \\slash 2 } \\, \\operatorname { e x p } \\left( \\mita _ { \\miti } \\mitA _ { \\miti } \\, { } ^ { \\mitj } \\mita _ { \\mitj } ^ { \\dagger } \\right) , \\quad \\mitR = \\mitR _ { \\miti } \\, { } ^ { \\mitj } = \\operatorname { e x p } ( \\mitA _ { \\miti } \\, { } ^ { \\mitj } ) , \\end{align*}",
"\\begin{align*} \\mitchi _ { \\mitalpha } = ( \\mitD , ~ \\mitD _ { \\mitx } ) , \\quad \\mitchi _ { \\mitmu \\mitalpha } = ( \\mitD _ { \\mitmu } , ~ \\mitD _ { \\mitmu \\mitx } ) , \\\\ \\mitchi _ { \\mitmu \\mitnu \\mitalpha } = ( \\mitD _ { \\mitmu \\mitnu } , ~ \\mitD _ { \\mitmu \\mitnu \\mitx } ) , \\quad \\mitchi _ { \\mitmu \\mitnu \\mitrho \\mitalpha } = ( \\mitD _ { \\mitmu \\mitnu \\mitrho } , ~ \\mitD _ { \\mitmu \\mitnu \\mitrho \\mitx } ) , \\\\ \\cdots , \\end{align*}",
"\\begin{align*} \\mitchi _ { \\mitalpha } = ( \\mitD , ~ \\mitD _ { \\mitx } , ~ \\mitD _ { \\mity } , ~ \\mitD _ { \\mity \\mitx } ) , \\\\ \\mitchi _ { \\mitmu \\mitalpha } = ( \\mitD _ { \\mitmu } , ~ \\mitD _ { \\mitmu \\mitx } , ~ \\mitD _ { \\mitmu \\mity } , ~ \\mitD _ { \\mitmu \\mity \\mitx } ) , \\\\ \\mitchi _ { \\mitmu \\mitnu \\mitalpha } = ( \\mitD _ { \\mitmu \\mitnu } , ~ \\mitD _ { \\mitmu \\mitnu \\mitx } , ~ \\mitD _ { \\mitmu \\mitnu \\mity } , ~ \\mitD _ { \\mitmu \\mitnu \\mity \\mitx } ) , \\\\ \\cdots , \\end{align*}",
"\\begin{equation*} \\vert \\tilde { \\mitchi } _ { \\mitmu _ { 1 } \\ldots \\mitmu _ { \\mitp } \\mitalpha } \\rangle = \\sum _ { \\mitbeta } \\mitS ^ { - 1 } ( \\mupOmega ^ { \\mitT } ) _ { \\mitalpha \\mitbeta } \\vert \\tilde { \\mitchi } _ { \\mitmu _ { 1 } \\ldots \\mitmu _ { \\mitp } \\mitbeta } \\rangle . \\end{equation*}",
"\\begin{align*} \\mitS \\left( ( \\mupLambda ^ { \\mitT } ) ^ { - 1 } \\right) = \\mitS ( \\mupLambda ) = \\mupLambda = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mitS ( \\mupLambda ^ { 2 } ) = \\mupLambda ^ { 2 } = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mitJ = \\mscrR \\miteta \\mscrR , \\quad \\miteta = \\left( \\begin{array}{cc} - 1 \\! \\! 1 & 0 \\\\ 0 & 1 \\! \\! 1 \\end{array} \\right) , \\quad \\mscrR = \\frac { \\sqrt { 2 } } { 2 } \\left( \\begin{array}{cc} - 1 \\! \\! 1 & 1 \\! \\! 1 \\\\ 1 \\! \\! 1 & 1 \\! \\! 1 \\end{array} \\right) , \\end{align*}"
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0002133_page07 | {
"latex": [
"$\\Omega $",
"$O(2d)$",
"$(\\Omega ^T)^{-1}=\\Omega $",
"$R=-1\\!\\!1 +2 e_i$",
"$S=-1\\!\\!1$",
"$\\Lambda _i$",
"$B$",
"$D$",
"$\\eta $",
"$S$",
"$R$",
"$O(d-1,1)$",
"$S\\eta S^T=\\eta $",
"$R\\eta R^T=\\eta $",
"$t$",
"$x$",
"$B$",
"$$\\Omega ={\\mathst R}^{-1}\\begin {pmatrix} S & 0\\\\ 0 & R \\end {pmatrix}{\\mathst R}={1\\over 2}\\begin {pmatrix} R+S & R-S\\\\ R-S & R+S \\end {pmatrix}.$$",
"$$S=1\\!\\!1,\\qquad R=\\begin {pmatrix} \\cos \\theta & \\sin \\theta \\\\ -\\sin \\theta & \\cos \\theta \\end {pmatrix},$$",
"\\begin {equation} \\label {soluex1b} S(\\Omega )=\\begin {pmatrix} \\cos {\\theta \\over 2} & 0 & 0 &-\\sin {\\theta \\over 2}\\\\ 0 & \\cos {\\theta \\over 2} & \\sin {\\theta \\over 2}& 0 \\\\ 0 & -\\sin {\\theta \\over 2} & \\cos {\\theta \\over 2} & 0 \\\\ \\sin {\\theta \\over 2} & 0 & 0 & \\cos {\\theta \\over 2} \\end {pmatrix}. \\end {equation}",
"\\begin {equation} \\label {gentec2} \\Omega ={1\\over 2}\\begin {pmatrix} \\eta (S+R)\\eta & \\eta (R-S)\\\\ (R-S)\\eta & S+R \\end {pmatrix},\\quad {\\mathst R}={1\\over \\sqrt {2}}\\begin {pmatrix} -\\eta & 1\\!\\!1\\\\ \\eta & 1\\!\\!1 \\end {pmatrix}, \\end {equation}",
"$$ R=S=\\begin {pmatrix} \\cosh \\alpha & \\sinh \\alpha \\\\ \\sinh \\alpha & \\cosh \\alpha \\end {pmatrix}$$",
"\\begin {equation} \\label {soluexbb} S^{-1}(\\Omega ^T_b)=\\begin {pmatrix} 1 & 0 & 0 & 0\\\\ 0 & \\cosh \\alpha & \\sinh \\alpha & 0\\\\ 0 & \\sinh \\alpha &\\cosh \\alpha & 0 \\\\ 0 & 0 & 0 & 1 \\end {pmatrix}. \\end {equation}",
"\\begin {gather} {\\tilde B}_{\\mu t}=B_{\\mu t}\\cosh \\alpha +B_{\\mu x}\\sinh \\alpha ,\\quad {\\tilde B}_{\\mu x}=B_{\\mu t}\\sinh \\alpha +B_{\\mu x}\\cosh \\alpha ,\\\\ {\\tilde C}_{\\mu \\dots \\nu t}=C_{\\mu \\dots \\nu t}\\cosh \\alpha +C_{\\mu \\dots \\nu x}\\sinh \\alpha ,\\quad {\\tilde C}_{\\mu \\dots \\nu x}=C_{\\mu \\dots \\nu t}\\sinh \\alpha +C_{\\mu \\dots \\nu x}\\cosh \\alpha ,\\\\ {\\tilde B}_{tx}=B_{tx},\\quad {\\tilde C}_{\\mu \\dots \\nu tx}=C_{\\mu \\dots \\nu tx},\\quad {\\tilde B}_{\\mu \\nu }=B_{\\mu \\nu },\\quad {\\tilde C}_{\\mu \\dots \\nu }=C_{\\mu \\dots \\nu }. \\end {gather}",
"\\begin {equation} \\label {soluex4} S=\\begin {pmatrix} \\cosh \\alpha & -\\sinh \\alpha \\\\ -\\sinh \\alpha & \\cosh \\alpha \\end {pmatrix}, \\quad R=\\begin {pmatrix} \\cosh \\alpha & \\sinh \\alpha \\\\ \\sinh \\alpha & \\cosh \\alpha \\end {pmatrix}. \\end {equation}",
"\\begin {equation} \\label {soluex4b} S^{-1}(\\Omega ^T_s)=\\begin {pmatrix} \\cosh \\alpha & 0 & 0 &\\sinh \\alpha \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ \\sinh \\alpha & 0 & 0 &\\cosh \\alpha \\end {pmatrix}. \\end {equation}"
],
"latex_norm": [
"$ \\Omega $",
"$ O ( 2 d ) $",
"$ ( \\Omega ^ { T } ) ^ { - 1 } = \\Omega $",
"$ R = - 1 \\! \\! 1 + 2 e _ { i } $",
"$ S = - 1 \\! \\! 1 $",
"$ \\Lambda _ { i } $",
"$ B $",
"$ D $",
"$ \\eta $",
"$ S $",
"$ R $",
"$ O ( d - 1 , 1 ) $",
"$ S \\eta S ^ { T } = \\eta $",
"$ R \\eta R ^ { T } = \\eta $",
"$ t $",
"$ x $",
"$ B $",
"\\begin{align*} \\Omega = R ^ { - 1 } ( \\begin{array}{cc} S & 0 \\\\ 0 & R \\end{array} ) R = \\frac { 1 } { 2 } ( \\begin{array}{cc} R + S & R - S \\\\ R - S & R + S \\end{array} ) . \\end{align*}",
"\\begin{align*} S = 1 \\! \\! 1 , \\qquad R = ( \\begin{array}{cc} \\operatorname { c o s } \\theta & \\operatorname { s i n } \\theta \\\\ - \\operatorname { s i n } \\theta & \\operatorname { c o s } \\theta \\end{array} ) , \\end{align*}",
"\\begin{align*} S ( \\Omega ) = ( \\begin{array}{cccc} \\operatorname { c o s } \\frac { \\theta } { 2 } & 0 & 0 & - \\operatorname { s i n } \\frac { \\theta } { 2 } \\\\ 0 & \\operatorname { c o s } \\frac { \\theta } { 2 } & \\operatorname { s i n } \\frac { \\theta } { 2 } & 0 \\\\ 0 & - \\operatorname { s i n } \\frac { \\theta } { 2 } & \\operatorname { c o s } \\frac { \\theta } { 2 } & 0 \\\\ \\operatorname { s i n } \\frac { \\theta } { 2 } & 0 & 0 & \\operatorname { c o s } \\frac { \\theta } { 2 } \\end{array} ) . \\end{align*}",
"\\begin{align*} \\Omega = \\frac { 1 } { 2 } ( \\begin{array}{cc} \\eta ( S + R ) \\eta & \\eta ( R - S ) \\\\ ( R - S ) \\eta & S + R \\end{array} ) , \\quad R = \\frac { 1 } { \\sqrt { 2 } } ( \\begin{array}{cc} - \\eta & 1 \\! \\! 1 \\\\ \\eta & 1 \\! \\! 1 \\end{array} ) , \\end{align*}",
"\\begin{align*} R = S = ( \\begin{array}{cc} \\operatorname { c o s h } \\alpha & \\operatorname { s i n h } \\alpha \\\\ \\operatorname { s i n h } \\alpha & \\operatorname { c o s h } \\alpha \\end{array} ) \\end{align*}",
"\\begin{align*} S ^ { - 1 } ( \\Omega _ { b } ^ { T } ) = ( \\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & \\operatorname { c o s h } \\alpha & \\operatorname { s i n h } \\alpha & 0 \\\\ 0 & \\operatorname { s i n h } \\alpha & \\operatorname { c o s h } \\alpha & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} ) . \\end{align*}",
"\\begin{align*} \\widetilde { B } _ { \\mu t } = B _ { \\mu t } \\operatorname { c o s h } \\alpha + B _ { \\mu x } \\operatorname { s i n h } \\alpha , \\quad \\widetilde { B } _ { \\mu x } = B _ { \\mu t } \\operatorname { s i n h } \\alpha + B _ { \\mu x } \\operatorname { c o s h } \\alpha , \\\\ \\widetilde { C } _ { \\mu \\ldots \\nu t } = C _ { \\mu \\ldots \\nu t } \\operatorname { c o s h } \\alpha + C _ { \\mu \\ldots \\nu x } \\operatorname { s i n h } \\alpha , \\quad \\widetilde { C } _ { \\mu \\ldots \\nu x } = C _ { \\mu \\ldots \\nu t } \\operatorname { s i n h } \\alpha + C _ { \\mu \\ldots \\nu x } \\operatorname { c o s h } \\alpha , \\\\ \\widetilde { B } _ { t x } = B _ { t x } , \\quad \\widetilde { C } _ { \\mu \\ldots \\nu t x } = C _ { \\mu \\ldots \\nu t x } , \\quad \\widetilde { B } _ { \\mu \\nu } = B _ { \\mu \\nu } , \\quad \\widetilde { C } _ { \\mu \\ldots \\nu } = C _ { \\mu \\ldots \\nu } . \\end{align*}",
"\\begin{align*} S = ( \\begin{array}{cc} \\operatorname { c o s h } \\alpha & - \\operatorname { s i n h } \\alpha \\\\ - \\operatorname { s i n h } \\alpha & \\operatorname { c o s h } \\alpha \\end{array} ) , \\quad R = ( \\begin{array}{cc} \\operatorname { c o s h } \\alpha & \\operatorname { s i n h } \\alpha \\\\ \\operatorname { s i n h } \\alpha & \\operatorname { c o s h } \\alpha \\end{array} ) . \\end{align*}",
"\\begin{align*} S ^ { - 1 } ( \\Omega _ { s } ^ { T } ) = ( \\begin{array}{cccc} \\operatorname { c o s h } \\alpha & 0 & 0 & \\operatorname { s i n h } \\alpha \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\operatorname { s i n h } \\alpha & 0 & 0 & \\operatorname { c o s h } \\alpha \\end{array} ) . \\end{align*}"
],
"latex_expand": [
"$ \\mupOmega $",
"$ \\mitO ( 2 \\mitd ) $",
"$ ( \\mupOmega ^ { \\mitT } ) ^ { - 1 } = \\mupOmega $",
"$ \\mitR = - 1 \\! \\! 1 + 2 \\mite _ { \\miti } $",
"$ \\mitS = - 1 \\! \\! 1 $",
"$ \\mupLambda _ { \\miti } $",
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"$ \\miteta $",
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"$ \\mitO ( \\mitd - 1 , 1 ) $",
"$ \\mitS \\miteta \\mitS ^ { \\mitT } = \\miteta $",
"$ \\mitR \\miteta \\mitR ^ { \\mitT } = \\miteta $",
"$ \\mitt $",
"$ \\mitx $",
"$ \\mitB $",
"\\begin{align*} \\mupOmega = \\mscrR ^ { - 1 } \\left( \\begin{array}{cc} \\mitS & 0 \\\\ 0 & \\mitR \\end{array} \\right) \\mscrR = \\frac { 1 } { 2 } \\left( \\begin{array}{cc} \\mitR + \\mitS & \\mitR - \\mitS \\\\ \\mitR - \\mitS & \\mitR + \\mitS \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mitS = 1 \\! \\! 1 , \\qquad \\mitR = \\left( \\begin{array}{cc} \\operatorname { c o s } \\mittheta & \\operatorname { s i n } \\mittheta \\\\ - \\operatorname { s i n } \\mittheta & \\operatorname { c o s } \\mittheta \\end{array} \\right) , \\end{align*}",
"\\begin{align*} \\mitS ( \\mupOmega ) = \\left( \\begin{array}{cccc} \\operatorname { c o s } \\frac { \\mittheta } { 2 } & 0 & 0 & - \\operatorname { s i n } \\frac { \\mittheta } { 2 } \\\\ 0 & \\operatorname { c o s } \\frac { \\mittheta } { 2 } & \\operatorname { s i n } \\frac { \\mittheta } { 2 } & 0 \\\\ 0 & - \\operatorname { s i n } \\frac { \\mittheta } { 2 } & \\operatorname { c o s } \\frac { \\mittheta } { 2 } & 0 \\\\ \\operatorname { s i n } \\frac { \\mittheta } { 2 } & 0 & 0 & \\operatorname { c o s } \\frac { \\mittheta } { 2 } \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mupOmega = \\frac { 1 } { 2 } \\left( \\begin{array}{cc} \\miteta ( \\mitS + \\mitR ) \\miteta & \\miteta ( \\mitR - \\mitS ) \\\\ ( \\mitR - \\mitS ) \\miteta & \\mitS + \\mitR \\end{array} \\right) , \\quad \\mscrR = \\frac { 1 } { \\sqrt { 2 } } \\left( \\begin{array}{cc} - \\miteta & 1 \\! \\! 1 \\\\ \\miteta & 1 \\! \\! 1 \\end{array} \\right) , \\end{align*}",
"\\begin{align*} \\mitR = \\mitS = \\left( \\begin{array}{cc} \\operatorname { c o s h } \\mitalpha & \\operatorname { s i n h } \\mitalpha \\\\ \\operatorname { s i n h } \\mitalpha & \\operatorname { c o s h } \\mitalpha \\end{array} \\right) \\end{align*}",
"\\begin{align*} \\mitS ^ { - 1 } ( \\mupOmega _ { \\mitb } ^ { \\mitT } ) = \\left( \\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & \\operatorname { c o s h } \\mitalpha & \\operatorname { s i n h } \\mitalpha & 0 \\\\ 0 & \\operatorname { s i n h } \\mitalpha & \\operatorname { c o s h } \\mitalpha & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\tilde { \\mitB } _ { \\mitmu \\mitt } = \\mitB _ { \\mitmu \\mitt } \\operatorname { c o s h } \\mitalpha + \\mitB _ { \\mitmu \\mitx } \\operatorname { s i n h } \\mitalpha , \\quad \\tilde { \\mitB } _ { \\mitmu \\mitx } = \\mitB _ { \\mitmu \\mitt } \\operatorname { s i n h } \\mitalpha + \\mitB _ { \\mitmu \\mitx } \\operatorname { c o s h } \\mitalpha , \\\\ \\tilde { \\mitC } _ { \\mitmu \\ldots \\mitnu \\mitt } = \\mitC _ { \\mitmu \\ldots \\mitnu \\mitt } \\operatorname { c o s h } \\mitalpha + \\mitC _ { \\mitmu \\ldots \\mitnu \\mitx } \\operatorname { s i n h } \\mitalpha , \\quad \\tilde { \\mitC } _ { \\mitmu \\ldots \\mitnu \\mitx } = \\mitC _ { \\mitmu \\ldots \\mitnu \\mitt } \\operatorname { s i n h } \\mitalpha + \\mitC _ { \\mitmu \\ldots \\mitnu \\mitx } \\operatorname { c o s h } \\mitalpha , \\\\ \\tilde { \\mitB } _ { \\mitt \\mitx } = \\mitB _ { \\mitt \\mitx } , \\quad \\tilde { \\mitC } _ { \\mitmu \\ldots \\mitnu \\mitt \\mitx } = \\mitC _ { \\mitmu \\ldots \\mitnu \\mitt \\mitx } , \\quad \\tilde { \\mitB } _ { \\mitmu \\mitnu } = \\mitB _ { \\mitmu \\mitnu } , \\quad \\tilde { \\mitC } _ { \\mitmu \\ldots \\mitnu } = \\mitC _ { \\mitmu \\ldots \\mitnu } . \\end{align*}",
"\\begin{align*} \\mitS = \\left( \\begin{array}{cc} \\operatorname { c o s h } \\mitalpha & - \\operatorname { s i n h } \\mitalpha \\\\ - \\operatorname { s i n h } \\mitalpha & \\operatorname { c o s h } \\mitalpha \\end{array} \\right) , \\quad \\mitR = \\left( \\begin{array}{cc} \\operatorname { c o s h } \\mitalpha & \\operatorname { s i n h } \\mitalpha \\\\ \\operatorname { s i n h } \\mitalpha & \\operatorname { c o s h } \\mitalpha \\end{array} \\right) . \\end{align*}",
"\\begin{align*} \\mitS ^ { - 1 } ( \\mupOmega _ { \\mits } ^ { \\mitT } ) = \\left( \\begin{array}{cccc} \\operatorname { c o s h } \\mitalpha & 0 & 0 & \\operatorname { s i n h } \\mitalpha \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ \\operatorname { s i n h } \\mitalpha & 0 & 0 & \\operatorname { c o s h } \\mitalpha \\end{array} \\right) . \\end{align*}"
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0002133_page08 | {
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"\\begin {gather} {\\tilde g}_{\\mu 0}={g_{\\mu 0} \\cosh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ {\\tilde g}_{\\mu 1}={g_{\\mu 1} \\cosh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ {\\tilde B}_{\\mu 0}={-g_{00}g_{\\mu 1}\\sinh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ {\\tilde B}_{\\mu 1}={g_{\\mu 0}\\sinh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha }, \\end {gather}",
"\\begin {gather} {\\tilde g}_{\\mu \\nu }=g_{\\mu \\nu }-{(g_{\\mu 0} g_{\\nu 0}+g_{00}g_{\\mu 1}g_{\\nu 1})\\sinh ^2\\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ {\\tilde B}_{\\mu \\nu }={(g_{\\mu 0}g_{\\nu 1} -g_{\\mu 1}g_{\\nu 0})\\sinh \\alpha \\,\\cosh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha }, \\end {gather}",
"\\begin {gather} {\\tilde C}=C\\cosh \\alpha -C_{01}\\sinh \\alpha ,\\\\ {\\tilde C}_0=C_0,\\quad {\\tilde C}_1=C_1,\\quad {\\tilde C}_\\mu =C_\\mu \\cosh \\alpha -C_{\\mu 01}\\sinh \\alpha ,\\\\ \\begin {split} {\\tilde C}_{01}=& {C_{01}[1+2(1+g_{00})\\sinh ^2\\alpha ]\\cosh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha }\\\\ &-{C[1+(1+g_{00})(\\sinh ^2\\alpha +\\cosh ^2\\alpha )]\\sinh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha }, \\end {split} \\end {gather}",
"\\begin {gather} {\\tilde C}_{\\mu 0}=C_{\\mu 0}+{g_{00}g_{\\mu 1}\\sinh \\alpha (C \\cosh \\alpha -C_{01}\\sinh \\alpha )\\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ {\\tilde C}_{\\mu 1}=C_{\\mu 1}-{C g_{\\mu 0} \\sinh \\alpha \\cosh \\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha } +{C_{01}g_{\\mu 0}\\sinh ^2\\alpha \\over 1+(1+g_{00})\\sinh ^2\\alpha },\\\\ \\begin {split} {\\tilde C}_{\\mu \\nu }=& C_{\\mu \\nu }\\cosh \\alpha -C_{\\mu \\nu 01}\\sinh \\alpha \\\\ &+{(C_{01}\\sinh \\alpha -C\\cosh \\alpha )(g_{\\mu 0}g_{\\nu 1} -g_{\\mu 1}g_{\\nu 0})\\sinh 2\\alpha \\over 2[1+(1+g_{00})\\sinh ^2\\alpha ]}, \\end {split}\\\\ e^{-2{\\tilde \\phi }}=e^{-2\\phi }[1+(1+g_{00})\\sinh ^2\\alpha ]. \\end {gather}"
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"\\begin{align*} \\widetilde { g } _ { \\mu 0 } = \\frac { g _ { \\mu 0 } \\operatorname { c o s h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ \\widetilde { g } _ { \\mu 1 } = \\frac { g _ { \\mu 1 } \\operatorname { c o s h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ \\widetilde { B } _ { \\mu 0 } = \\frac { - g _ { 0 0 } g _ { \\mu 1 } \\operatorname { s i n h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ \\widetilde { B } _ { \\mu 1 } = \\frac { g _ { \\mu 0 } \\operatorname { s i n h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\end{align*}",
"\\begin{align*} \\widetilde { g } _ { \\mu \\nu } = g _ { \\mu \\nu } - \\frac { ( g _ { \\mu 0 } g _ { \\nu 0 } + g _ { 0 0 } g _ { \\mu 1 } g _ { \\nu 1 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ \\widetilde { B } _ { \\mu \\nu } = \\frac { ( g _ { \\mu 0 } g _ { \\nu 1 } - g _ { \\mu 1 } g _ { \\nu 0 } ) \\operatorname { s i n h } \\alpha \\, \\operatorname { c o s h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\end{align*}",
"\\begin{align*} \\widetilde { C } = C \\operatorname { c o s h } \\alpha - C _ { 0 1 } \\operatorname { s i n h } \\alpha , \\\\ \\widetilde { C } _ { 0 } = C _ { 0 } , \\quad \\widetilde { C } _ { 1 } = C _ { 1 } , \\quad \\widetilde { C } _ { \\mu } = C _ { \\mu } \\operatorname { c o s h } \\alpha - C _ { \\mu 0 1 } \\operatorname { s i n h } \\alpha , \\\\ & \\\\ & \\end{align*}",
"\\begin{align*} \\widetilde { C } _ { \\mu 0 } = C _ { \\mu 0 } + \\frac { g _ { 0 0 } g _ { \\mu 1 } \\operatorname { s i n h } \\alpha ( C \\operatorname { c o s h } \\alpha - C _ { 0 1 } \\operatorname { s i n h } \\alpha ) } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ \\widetilde { C } _ { \\mu 1 } = C _ { \\mu 1 } - \\frac { C g _ { \\mu 0 } \\operatorname { s i n h } \\alpha \\operatorname { c o s h } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } + \\frac { C _ { 0 1 } g _ { \\mu 0 } { \\operatorname { s i n h } } ^ { 2 } \\alpha } { 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha } , \\\\ & \\\\ & \\\\ e ^ { - 2 \\widetilde { \\phi } } = e ^ { - 2 \\phi } [ 1 + ( 1 + g _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\alpha ] . \\end{align*}"
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"\\begin{align*} \\tilde { \\mitg } _ { \\mitmu \\mitnu } = \\mitg _ { \\mitmu \\mitnu } - \\frac { ( \\mitg _ { \\mitmu 0 } \\mitg _ { \\mitnu 0 } + \\mitg _ { 0 0 } \\mitg _ { \\mitmu 1 } \\mitg _ { \\mitnu 1 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } { 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } , \\\\ \\tilde { \\mitB } _ { \\mitmu \\mitnu } = \\frac { ( \\mitg _ { \\mitmu 0 } \\mitg _ { \\mitnu 1 } - \\mitg _ { \\mitmu 1 } \\mitg _ { \\mitnu 0 } ) \\operatorname { s i n h } \\mitalpha \\, \\operatorname { c o s h } \\mitalpha } { 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } , \\end{align*}",
"\\begin{align*} \\tilde { \\mitC } = \\mitC \\operatorname { c o s h } \\mitalpha - \\mitC _ { 0 1 } \\operatorname { s i n h } \\mitalpha , \\\\ \\tilde { \\mitC } _ { 0 } = \\mitC _ { 0 } , \\quad \\tilde { \\mitC } _ { 1 } = \\mitC _ { 1 } , \\quad \\tilde { \\mitC } _ { \\mitmu } = \\mitC _ { \\mitmu } \\operatorname { c o s h } \\mitalpha - \\mitC _ { \\mitmu 0 1 } \\operatorname { s i n h } \\mitalpha , \\\\ & \\\\ & \\end{align*}",
"\\begin{align*} \\tilde { \\mitC } _ { \\mitmu 0 } = \\mitC _ { \\mitmu 0 } + \\frac { \\mitg _ { 0 0 } \\mitg _ { \\mitmu 1 } \\operatorname { s i n h } \\mitalpha ( \\mitC \\operatorname { c o s h } \\mitalpha - \\mitC _ { 0 1 } \\operatorname { s i n h } \\mitalpha ) } { 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } , \\\\ \\tilde { \\mitC } _ { \\mitmu 1 } = \\mitC _ { \\mitmu 1 } - \\frac { \\mitC \\mitg _ { \\mitmu 0 } \\operatorname { s i n h } \\mitalpha \\operatorname { c o s h } \\mitalpha } { 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } + \\frac { \\mitC _ { 0 1 } \\mitg _ { \\mitmu 0 } { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } { 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha } , \\\\ & \\\\ & \\\\ \\mite ^ { - 2 \\tilde { \\mitphi } } = \\mite ^ { - 2 \\mitphi } [ 1 + ( 1 + \\mitg _ { 0 0 } ) { \\operatorname { s i n h } } ^ { 2 } \\mitalpha ] . \\end{align*}"
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0002133_page09 | {
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0002142_page01 | {
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0002142_page02 | {
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"${\\bf r}\\rightarrow {\\bf r}^{\\prime }=f({\\bf {r}})$",
"$d$",
"$\\Delta _{\\phi }$",
"$\\phi _{i}$",
"$\\Phi =\\phi _{1}$",
"$\\Psi =\\phi _{2}$",
"$d$",
"\\begin {equation} \\Phi ^{\\prime }({\\bf r}^{\\prime }) =\\left |\\frac {\\partial {{\\bf r}^{\\prime }}} {\\partial {\\bf r}}\\right |^{T}\\Phi ({\\bf r}) \\end {equation}",
"\\begin {equation} \\Phi =\\def \\temp {\\multicolumn {1}{c|}{1}} \\begin{pmatrix} \\phi _{1}\\\\ \\phi _{2} \\\\ \\vdots \\\\ \\phi _{n}\\\\ \\end{pmatrix} \\end {equation}",
"\\begin {equation} T = \\begin{pmatrix} -\\frac {\\Delta _{\\phi }}{d}&0&\\ldots &0\\\\ 1&-\\frac {\\Delta _{\\phi }}{d}&\\ldots &\\vdots \\\\ 0&1& \\ldots &0 \\\\ \\vdots &\\ldots &\\ddots &0 \\\\ 0&\\ldots &1&-\\frac {\\Delta _{\\phi }}{d}\\\\ \\end{pmatrix}~. \\end {equation}",
"\\begin {eqnarray} \\Phi ^{\\prime }({\\bf {r^{\\prime }}})&=& \\left |\\frac {\\partial {{\\bf {r^{\\prime }}}}} {\\partial {\\bf {r}}}\\right |^{-\\frac {\\Delta _{\\phi }}{d}}\\Phi ({\\bf {r}})\\\\ \\Psi ^{\\prime }({\\bf {r^{\\prime }}})&=& \\left |\\frac {\\partial {{\\bf {r^{\\prime }}}}}{\\partial {\\bf {r}}} \\right |^{-\\frac {\\Delta _{\\phi }}{d}}\\left (\\Psi ({\\bf {r}}) +\\log |\\frac {\\partial {\\bf {r^{\\prime }}}}{\\partial {\\bf {r}}} |\\Phi (\\bf {r})\\right ). \\end {eqnarray}"
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"\\begin{align*} \\Phi = ( \\begin{array}{c} \\phi _ { 1 } \\\\ \\phi _ { 2 } \\\\ \\vdots \\\\ \\phi _ { n } \\end{array} ) \\end{align*}",
"\\begin{align*} T = ( \\begin{array}{cccc} - \\frac { \\Delta _ { \\phi } } { d } & 0 & \\ldots & 0 \\\\ 1 & - \\frac { \\Delta _ { \\phi } } { d } & \\ldots & \\vdots \\\\ 0 & 1 & \\ldots & 0 \\\\ \\vdots & \\ldots & \\ddots & 0 \\\\ 0 & \\ldots & 1 & - \\frac { \\Delta _ { \\phi } } { d } \\end{array} ) ~ . \\end{align*}",
"\\begin{align*} \\Phi ^ { \\prime } ( r ^ { \\prime } ) & = & { \\vert \\frac { \\partial r ^ { \\prime } } { \\partial r } \\vert } ^ { - \\frac { \\Delta _ { \\phi } } { d } } \\Phi ( r ) \\\\ \\Psi ^ { \\prime } ( r ^ { \\prime } ) & = & { \\vert \\frac { \\partial r ^ { \\prime } } { \\partial r } \\vert } ^ { - \\frac { \\Delta _ { \\phi } } { d } } ( \\Psi ( r ) + \\operatorname { l o g } \\vert \\frac { \\partial r ^ { \\prime } } { \\partial r } \\vert \\Phi ( r ) ) . \\end{align*}"
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"\\begin{align*} \\displaystyle \\mupPhi = \\left( \\begin{array}{c} \\mitphi _ { 1 } \\\\ \\mitphi _ { 2 } \\\\ \\vdots \\\\ \\mitphi _ { \\mitn } \\end{array} \\right) \\end{align*}",
"\\begin{align*} \\displaystyle \\mitT = \\left( \\begin{array}{cccc} - \\frac { \\mupDelta _ { \\mitphi } } { \\mitd } & 0 & \\ldots & 0 \\\\ 1 & - \\frac { \\mupDelta _ { \\mitphi } } { \\mitd } & \\ldots & \\vdots \\\\ 0 & 1 & \\ldots & 0 \\\\ \\vdots & \\ldots & \\ddots & 0 \\\\ 0 & \\ldots & 1 & - \\frac { \\mupDelta _ { \\mitphi } } { \\mitd } \\end{array} \\right) ~ . \\end{align*}",
"\\begin{align*} \\mupPhi ^ { \\prime } ( \\mitr ^ { \\prime } ) & = & { \\left\\vert \\frac { \\mitpartial \\mitr ^ { \\prime } } { \\mitpartial \\mitr } \\right\\vert } ^ { - \\frac { \\mupDelta _ { \\mitphi } } { \\mitd } } \\mupPhi ( \\mitr ) \\\\ \\mupPsi ^ { \\prime } ( \\mitr ^ { \\prime } ) & = & { \\left\\vert \\frac { \\mitpartial \\mitr ^ { \\prime } } { \\mitpartial \\mitr } \\right\\vert } ^ { - \\frac { \\mupDelta _ { \\mitphi } } { \\mitd } } \\left( \\mupPsi ( \\mitr ) + \\operatorname { l o g } \\vert \\frac { \\mitpartial \\mitr ^ { \\prime } } { \\mitpartial \\mitr } \\vert \\mupPhi ( \\mitr ) \\right) . \\end{align*}"
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0002142_page03 | {
"latex": [
"$(d-1)$",
"$y=0$",
"$\\left <\\Phi ({\\bf {r}})\\right >=f_{1}(\\bf {r})$",
"$\\left <\\Psi (\\bf {r})\\right >=f_{2}(\\bf {r})$",
"$f_{1}$",
"$f_{2}$",
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"$f_{2}$",
"$G_{1}({\\bf {r}}_{1},{\\bf r}_{2}) =\\left <\\Phi ({\\bf r}_{1})\\Phi ({\\bf r}_{2})\\right >$",
"${\\bf r}_{1}$",
"${\\bf r}_{2}$",
"$(x_{1},y_{1})$",
"$(x_{2},y_{2})$",
"$G_{1}$",
"$x_{1}-x_{2}$",
"$y_{1}$",
"$y_{2}$",
"\\begin {equation} \\bf {r^{\\prime }}=\\bf {r}+\\epsilon \\bf {r} \\end {equation}",
"\\begin {eqnarray} f_{1}(y)&=&(1+\\epsilon )^{\\Delta _{\\phi }}f_{1}(y^{\\prime })~,\\\\ f_{2}(y)&=&(1+\\epsilon )^{\\Delta _{\\phi }}\\left (f_{2}(y^{\\prime }) +\\log (1+\\epsilon )^{d}f(y^{\\prime })\\right )~. \\end {eqnarray}",
"\\begin {eqnarray} y\\frac {\\partial {f_{1}}}{\\partial {y}}+\\Delta _{\\phi }f_{1}&=&0\\\\ y\\frac {\\partial {f_{2}}}{\\partial {y}}+\\Delta _{\\phi }f_{2}+d\\:f&=&0 \\end {eqnarray}",
"\\begin {eqnarray} \\left <\\Phi (\\bf {r})\\right >&=&\\frac {C_{1}}{y^{\\Delta _\\phi }} \\\\ \\left <\\Psi (\\bf {r})\\right >&=&\\frac {1}{y^{\\Delta _\\phi }}(C_{2}-d\\:C_{1}\\log {y}) \\end {eqnarray}",
"\\begin {equation} G_{1}(x_{1}-x_{2},y_{1},y_{2})=(1+\\epsilon )^{\\Delta _{\\phi }} (1+\\epsilon )^{\\Delta _{\\phi }}G(x_{1}^{\\prime }-x_{2}^{\\prime } ,y_{1}^{\\prime },y_{2}^{\\prime })~. \\end {equation}",
"\\begin {eqnarray} x^{\\prime }&=&x+\\epsilon (x^{2}-y^{2})\\\\ y^{\\prime }&=&y+2\\epsilon xy, \\end {eqnarray}",
"\\begin {equation} G_{1}(x_{1}-x_{2},y_{1},y_{2})= (1+2\\epsilon x_{1})^{\\Delta _{\\phi }}(1+2\\epsilon x_{2})^{\\Delta _{\\phi }}G(x_{1}^{\\prime }-x_{2}^{\\prime },y_{1}^{\\prime },y_{2}^{\\prime })~. \\end {equation}"
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"$ f _ { 1 } $",
"$ f _ { 2 } $",
"$ \\epsilon $",
"$ f _ { 1 } $",
"$ f _ { 2 } $",
"$ G _ { 1 } ( r _ { 1 } , r _ { 2 } ) = \\langle \\Phi ( r _ { 1 } ) \\Phi ( r _ { 2 } ) \\rangle $",
"$ r _ { 1 } $",
"$ r _ { 2 } $",
"$ ( x _ { 1 } , y _ { 1 } ) $",
"$ ( x _ { 2 } , y _ { 2 } ) $",
"$ G _ { 1 } $",
"$ x _ { 1 } - x _ { 2 } $",
"$ y _ { 1 } $",
"$ y _ { 2 } $",
"\\begin{equation*} r ^ { \\prime } = r + \\epsilon r \\end{equation*}",
"\\begin{align*} f _ { 1 } ( y ) & = & ( 1 + \\epsilon ) ^ { \\Delta _ { \\phi } } f _ { 1 } ( y ^ { \\prime } ) ~ , \\\\ f _ { 2 } ( y ) & = & ( 1 + \\epsilon ) ^ { \\Delta _ { \\phi } } ( f _ { 2 } ( y ^ { \\prime } ) + \\operatorname { l o g } ( 1 + \\epsilon ) ^ { d } f ( y ^ { \\prime } ) ) ~ . \\end{align*}",
"\\begin{align*} y \\frac { \\partial f _ { 1 } } { \\partial y } + \\Delta _ { \\phi } f _ { 1 } & = & 0 \\\\ y \\frac { \\partial f _ { 2 } } { \\partial y } + \\Delta _ { \\phi } f _ { 2 } + d \\> f & = & 0 \\end{align*}",
"\\begin{align*} \\langle \\Phi ( r ) \\rangle & = & \\frac { C _ { 1 } } { y ^ { \\Delta _ { \\phi } } } \\\\ \\langle \\Psi ( r ) \\rangle & = & \\frac { 1 } { y ^ { \\Delta _ { \\phi } } } ( C _ { 2 } - d \\> C _ { 1 } \\operatorname { l o g } y ) \\end{align*}",
"\\begin{equation*} G _ { 1 } ( x _ { 1 } - x _ { 2 } , y _ { 1 } , y _ { 2 } ) = ( 1 + \\epsilon ) ^ { \\Delta _ { \\phi } } ( 1 + \\epsilon ) ^ { \\Delta _ { \\phi } } G ( x _ { 1 } ^ { \\prime } - x _ { 2 } ^ { \\prime } , y _ { 1 } ^ { \\prime } , y _ { 2 } ^ { \\prime } ) ~ . \\end{equation*}",
"\\begin{align*} x ^ { \\prime } & = & x + \\epsilon ( x ^ { 2 } - y ^ { 2 } ) \\\\ y ^ { \\prime } & = & y + 2 \\epsilon x y , \\end{align*}",
"\\begin{equation*} G _ { 1 } ( x _ { 1 } - x _ { 2 } , y _ { 1 } , y _ { 2 } ) = ( 1 + 2 \\epsilon x _ { 1 } ) ^ { \\Delta _ { \\phi } } ( 1 + 2 \\epsilon x _ { 2 } ) ^ { \\Delta _ { \\phi } } G ( x _ { 1 } ^ { \\prime } - x _ { 2 } ^ { \\prime } , y _ { 1 } ^ { \\prime } , y _ { 2 } ^ { \\prime } ) ~ . \\end{equation*}"
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"$ \\mitf _ { 1 } $",
"$ \\mitf _ { 2 } $",
"$ \\mitepsilon $",
"$ \\mitf _ { 1 } $",
"$ \\mitf _ { 2 } $",
"$ \\mitG _ { 1 } ( \\mitr _ { 1 } , \\mitr _ { 2 } ) = \\left\\langle \\mupPhi ( \\mitr _ { 1 } ) \\mupPhi ( \\mitr _ { 2 } ) \\right\\rangle $",
"$ \\mitr _ { 1 } $",
"$ \\mitr _ { 2 } $",
"$ ( \\mitx _ { 1 } , \\mity _ { 1 } ) $",
"$ ( \\mitx _ { 2 } , \\mity _ { 2 } ) $",
"$ \\mitG _ { 1 } $",
"$ \\mitx _ { 1 } - \\mitx _ { 2 } $",
"$ \\mity _ { 1 } $",
"$ \\mity _ { 2 } $",
"\\begin{equation*} \\mitr ^ { \\prime } = \\mitr + \\mitepsilon \\mitr \\end{equation*}",
"\\begin{align*} \\mitf _ { 1 } ( \\mity ) & = & ( 1 + \\mitepsilon ) ^ { \\mupDelta _ { \\mitphi } } \\mitf _ { 1 } ( \\mity ^ { \\prime } ) ~ , \\\\ \\mitf _ { 2 } ( \\mity ) & = & ( 1 + \\mitepsilon ) ^ { \\mupDelta _ { \\mitphi } } \\left( \\mitf _ { 2 } ( \\mity ^ { \\prime } ) + \\operatorname { l o g } ( 1 + \\mitepsilon ) ^ { \\mitd } \\mitf ( \\mity ^ { \\prime } ) \\right) ~ . \\end{align*}",
"\\begin{align*} \\mity \\frac { \\mitpartial \\mitf _ { 1 } } { \\mitpartial \\mity } + \\mupDelta _ { \\mitphi } \\mitf _ { 1 } & = & 0 \\\\ \\mity \\frac { \\mitpartial \\mitf _ { 2 } } { \\mitpartial \\mity } + \\mupDelta _ { \\mitphi } \\mitf _ { 2 } + \\mitd \\> \\mitf & = & 0 \\end{align*}",
"\\begin{align*} \\left\\langle \\mupPhi ( \\mitr ) \\right\\rangle & = & \\frac { \\mitC _ { 1 } } { \\mity ^ { \\mupDelta _ { \\mitphi } } } \\\\ \\left\\langle \\mupPsi ( \\mitr ) \\right\\rangle & = & \\frac { 1 } { \\mity ^ { \\mupDelta _ { \\mitphi } } } ( \\mitC _ { 2 } - \\mitd \\> \\mitC _ { 1 } \\operatorname { l o g } \\mity ) \\end{align*}",
"\\begin{equation*} \\mitG _ { 1 } ( \\mitx _ { 1 } - \\mitx _ { 2 } , \\mity _ { 1 } , \\mity _ { 2 } ) = ( 1 + \\mitepsilon ) ^ { \\mupDelta _ { \\mitphi } } ( 1 + \\mitepsilon ) ^ { \\mupDelta _ { \\mitphi } } \\mitG ( \\mitx _ { 1 } ^ { \\prime } - \\mitx _ { 2 } ^ { \\prime } , \\mity _ { 1 } ^ { \\prime } , \\mity _ { 2 } ^ { \\prime } ) ~ . \\end{equation*}",
"\\begin{align*} \\mitx ^ { \\prime } & = & \\mitx + \\mitepsilon ( \\mitx ^ { 2 } - \\mity ^ { 2 } ) \\\\ \\mity ^ { \\prime } & = & \\mity + 2 \\mitepsilon \\mitx \\mity , \\end{align*}",
"\\begin{equation*} \\mitG _ { 1 } ( \\mitx _ { 1 } - \\mitx _ { 2 } , \\mity _ { 1 } , \\mity _ { 2 } ) = ( 1 + 2 \\mitepsilon \\mitx _ { 1 } ) ^ { \\mupDelta _ { \\mitphi } } ( 1 + 2 \\mitepsilon \\mitx _ { 2 } ) ^ { \\mupDelta _ { \\mitphi } } \\mitG ( \\mitx _ { 1 } ^ { \\prime } - \\mitx _ { 2 } ^ { \\prime } , \\mity _ { 1 } ^ { \\prime } , \\mity _ { 2 } ^ { \\prime } ) ~ . \\end{equation*}"
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|
0002142_page04 | {
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"$\\epsilon $",
"$u=x_{1}-x_{2}$",
"$G_{1}$",
"$2\\Delta _{\\phi }$",
"$\\alpha ={y_{1}}/{u}$",
"$\\beta ={y_{2}}/{u}$",
"$G_{2}({\\bf r}_{1},{\\bf r}_{2}) =\\left <\\Phi ({\\bf r}_{1})\\Psi ({\\bf r}_{2})\\right >$",
"$G_{3}({\\bf r}_{1},{\\bf r}_{2})=\\left <\\Psi ({\\bf r}_{1}) \\Psi ({\\bf r}_{2})\\right >$",
"$G_{3}({\\bf r}_{1},{\\bf r}_{2})=\\left <\\Psi ({\\bf r}_{1}) \\Psi ({\\bf r}_{2})\\right >$",
"$\\eta =[(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}]/{y_{1}y_{2}}$",
"$r=|{\\bf r}_{1}-{\\bf r}_{2}|$",
"$a, b$",
"\\begin {eqnarray} u \\frac {\\partial {G_{1}}}{\\partial {u}} + y_{1} \\frac {\\partial {G_{1}}}{\\partial {y_{1}}} + y_{2} \\frac {\\partial {G_{1}}}{\\partial {y_{2}}} + 2{\\Delta }_{\\phi }G_{1}&=&0\\\\ ({y_{1}^{2}}-{y_{2}^{2}}) \\frac {\\partial {G_{1}}}{\\partial {u}} + u\\left (y_{1} \\frac {\\partial {G_{1}}}{\\partial {y_{1}}} - {y_{2}} \\frac {\\partial {G_{1}}}{\\partial {y_{2}}}\\right )&=&0 \\end {eqnarray}",
"\\begin {equation} G_{1}=\\frac {1}{(u)^{2\\Delta _{\\phi }}}g_{1}(\\alpha ,\\beta ) \\end {equation}",
"\\begin {equation} \\left [\\alpha + \\frac {\\alpha }{\\alpha ^{2}-\\beta ^{2}}\\right ]\\frac {\\partial {g_{1}}}{\\partial {\\alpha }}+\\left [\\beta + \\frac {\\beta }{\\beta ^{2}-\\alpha ^{2}}\\right ] \\frac {\\partial {g_{1}}}{\\partial {\\beta }}+2\\Delta _{\\phi }g_{1}=0~. \\end {equation}",
"\\begin {equation} g_{1}(\\alpha ,\\beta )=\\frac {1}{(\\alpha \\beta )^{\\Delta _{\\phi }}}h_{1}\\left (\\frac {1+(\\alpha -\\beta )^{2}}{\\alpha \\beta }\\right )~. \\end {equation}",
"\\begin {equation} \\left <{\\Phi }({\\bf r}_{1})\\Phi ({\\bf r}_{2})\\right >=\\frac {1}{(y_{1}y_{2})^{\\Delta _{\\phi }}}h_{1}\\left (\\frac {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}{y_{1}y_{2}}\\right )~. \\end {equation}",
"\\begin {eqnarray} G_{2}({\\bf r}_{1},{\\bf r}_{2})&=& \\frac {1}{(y_{1}y_{2})^{\\Delta _{\\phi }}} \\left [h_{2}(\\eta )-d\\:\\log {y_{2}}\\:\\:h_{1}(\\eta )\\right ]\\\\ G_{3}({\\bf r}_{1},{\\bf r}_{2})&=& \\frac {1}{(y_{1}y_{2})^{\\Delta _{\\phi }}}\\left [h_{3}(\\eta )-d\\:\\log {y_{1}y_{2}} \\:\\:h_{2}(\\eta ) +d^{2}\\:\\log {y_{1}}\\log {y_{2}}\\:\\:h_{1}(\\eta )\\right ] \\end {eqnarray}",
"\\begin {eqnarray} \\left <\\Phi ({\\bf r}_{1})\\Phi ({\\bf r}_{2})\\right >&=&0\\\\ \\left <\\Phi ({\\bf r}_{1})\\Psi ({\\bf r}_{2})\\right >&=& \\frac {a}{r^{2\\Delta _{\\phi }}}\\\\ \\left <\\Psi ({\\bf r}_{1})\\Psi ({\\bf r}_{2})\\right >&=& \\frac {1}{r^{2\\Delta _{\\phi }}}(b-d\\:a\\log {r}) \\end {eqnarray}"
],
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"$ u = x _ { 1 } - x _ { 2 } $",
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"$ G _ { 3 } ( r _ { 1 } , r _ { 2 } ) = \\langle \\Psi ( r _ { 1 } ) \\Psi ( r _ { 2 } ) \\rangle $",
"$ G _ { 3 } ( r _ { 1 } , r _ { 2 } ) = \\langle \\Psi ( r _ { 1 } ) \\Psi ( r _ { 2 } ) \\rangle $",
"$ \\eta = [ ( x _ { 1 } - x _ { 2 } ) ^ { 2 } + ( y _ { 1 } - y _ { 2 } ) ^ { 2 } ] \\slash y _ { 1 } y _ { 2 } $",
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"\\begin{align*} u \\frac { \\partial G _ { 1 } } { \\partial u } + y _ { 1 } \\frac { \\partial G _ { 1 } } { \\partial y _ { 1 } } + y _ { 2 } \\frac { \\partial G _ { 1 } } { \\partial y _ { 2 } } + 2 \\Delta _ { \\phi } G _ { 1 } & = & 0 \\\\ ( y _ { 1 } ^ { 2 } - y _ { 2 } ^ { 2 } ) \\frac { \\partial G _ { 1 } } { \\partial u } + u ( y _ { 1 } \\frac { \\partial G _ { 1 } } { \\partial y _ { 1 } } - y _ { 2 } \\frac { \\partial G _ { 1 } } { \\partial y _ { 2 } } ) & = & 0 \\end{align*}",
"\\begin{equation*} G _ { 1 } = \\frac { 1 } { ( u ) ^ { 2 \\Delta _ { \\phi } } } g _ { 1 } ( \\alpha , \\beta ) \\end{equation*}",
"\\begin{equation*} [ \\alpha + \\frac { \\alpha } { \\alpha ^ { 2 } - \\beta ^ { 2 } } ] \\frac { \\partial g _ { 1 } } { \\partial \\alpha } + [ \\beta + \\frac { \\beta } { \\beta ^ { 2 } - \\alpha ^ { 2 } } ] \\frac { \\partial g _ { 1 } } { \\partial \\beta } + 2 \\Delta _ { \\phi } g _ { 1 } = 0 ~ . \\end{equation*}",
"\\begin{equation*} g _ { 1 } ( \\alpha , \\beta ) = \\frac { 1 } { ( \\alpha \\beta ) ^ { \\Delta _ { \\phi } } } h _ { 1 } ( \\frac { 1 + ( \\alpha - \\beta ) ^ { 2 } } { \\alpha \\beta } ) ~ . \\end{equation*}",
"\\begin{equation*} \\langle \\Phi ( r _ { 1 } ) \\Phi ( r _ { 2 } ) \\rangle = \\frac { 1 } { ( y _ { 1 } y _ { 2 } ) ^ { \\Delta _ { \\phi } } } h _ { 1 } ( \\frac { ( x _ { 1 } - x _ { 2 } ) ^ { 2 } + ( y _ { 1 } - y _ { 2 } ) ^ { 2 } } { y _ { 1 } y _ { 2 } } ) ~ . \\end{equation*}",
"\\begin{align*} G _ { 2 } ( r _ { 1 } , r _ { 2 } ) & = & \\frac { 1 } { ( y _ { 1 } y _ { 2 } ) ^ { \\Delta _ { \\phi } } } [ h _ { 2 } ( \\eta ) - d \\> \\operatorname { l o g } y _ { 2 } \\> \\> h _ { 1 } ( \\eta ) ] \\\\ G _ { 3 } ( r _ { 1 } , r _ { 2 } ) & = & \\frac { 1 } { ( y _ { 1 } y _ { 2 } ) ^ { \\Delta _ { \\phi } } } [ h _ { 3 } ( \\eta ) - d \\> \\operatorname { l o g } y _ { 1 } y _ { 2 } \\> \\> h _ { 2 } ( \\eta ) + d ^ { 2 } \\> \\operatorname { l o g } y _ { 1 } \\operatorname { l o g } y _ { 2 } \\> \\> h _ { 1 } ( \\eta ) ] \\end{align*}",
"\\begin{align*} \\langle \\Phi ( r _ { 1 } ) \\Phi ( r _ { 2 } ) \\rangle & = & 0 \\\\ \\langle \\Phi ( r _ { 1 } ) \\Psi ( r _ { 2 } ) \\rangle & = & \\frac { a } { r ^ { 2 \\Delta _ { \\phi } } } \\\\ \\langle \\Psi ( r _ { 1 } ) \\Psi ( r _ { 2 } ) \\rangle & = & \\frac { 1 } { r ^ { 2 \\Delta _ { \\phi } } } ( b - d \\> a \\operatorname { l o g } r ) \\end{align*}"
],
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"\\begin{align*} \\mitu \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mitu } + \\mity _ { 1 } \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mity _ { 1 } } + \\mity _ { 2 } \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mity _ { 2 } } + 2 \\mupDelta _ { \\mitphi } \\mitG _ { 1 } & = & 0 \\\\ ( \\mity _ { 1 } ^ { 2 } - \\mity _ { 2 } ^ { 2 } ) \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mitu } + \\mitu \\left( \\mity _ { 1 } \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mity _ { 1 } } - \\mity _ { 2 } \\frac { \\mitpartial \\mitG _ { 1 } } { \\mitpartial \\mity _ { 2 } } \\right) & = & 0 \\end{align*}",
"\\begin{equation*} \\mitG _ { 1 } = \\frac { 1 } { ( \\mitu ) ^ { 2 \\mupDelta _ { \\mitphi } } } \\mitg _ { 1 } ( \\mitalpha , \\mitbeta ) \\end{equation*}",
"\\begin{equation*} \\left[ \\mitalpha + \\frac { \\mitalpha } { \\mitalpha ^ { 2 } - \\mitbeta ^ { 2 } } \\right] \\frac { \\mitpartial \\mitg _ { 1 } } { \\mitpartial \\mitalpha } + \\left[ \\mitbeta + \\frac { \\mitbeta } { \\mitbeta ^ { 2 } - \\mitalpha ^ { 2 } } \\right] \\frac { \\mitpartial \\mitg _ { 1 } } { \\mitpartial \\mitbeta } + 2 \\mupDelta _ { \\mitphi } \\mitg _ { 1 } = 0 ~ . \\end{equation*}",
"\\begin{equation*} \\mitg _ { 1 } ( \\mitalpha , \\mitbeta ) = \\frac { 1 } { ( \\mitalpha \\mitbeta ) ^ { \\mupDelta _ { \\mitphi } } } \\Planckconst _ { 1 } \\left( \\frac { 1 + ( \\mitalpha - \\mitbeta ) ^ { 2 } } { \\mitalpha \\mitbeta } \\right) ~ . \\end{equation*}",
"\\begin{equation*} \\left\\langle \\mupPhi ( \\mitr _ { 1 } ) \\mupPhi ( \\mitr _ { 2 } ) \\right\\rangle = \\frac { 1 } { ( \\mity _ { 1 } \\mity _ { 2 } ) ^ { \\mupDelta _ { \\mitphi } } } \\Planckconst _ { 1 } \\left( \\frac { ( \\mitx _ { 1 } - \\mitx _ { 2 } ) ^ { 2 } + ( \\mity _ { 1 } - \\mity _ { 2 } ) ^ { 2 } } { \\mity _ { 1 } \\mity _ { 2 } } \\right) ~ . \\end{equation*}",
"\\begin{align*} \\mitG _ { 2 } ( \\mitr _ { 1 } , \\mitr _ { 2 } ) & = & \\frac { 1 } { ( \\mity _ { 1 } \\mity _ { 2 } ) ^ { \\mupDelta _ { \\mitphi } } } \\left[ \\Planckconst _ { 2 } ( \\miteta ) - \\mitd \\> \\operatorname { l o g } \\mity _ { 2 } \\> \\> \\Planckconst _ { 1 } ( \\miteta ) \\right] \\\\ \\mitG _ { 3 } ( \\mitr _ { 1 } , \\mitr _ { 2 } ) & = & \\frac { 1 } { ( \\mity _ { 1 } \\mity _ { 2 } ) ^ { \\mupDelta _ { \\mitphi } } } \\left[ \\Planckconst _ { 3 } ( \\miteta ) - \\mitd \\> \\operatorname { l o g } \\mity _ { 1 } \\mity _ { 2 } \\> \\> \\Planckconst _ { 2 } ( \\miteta ) + \\mitd ^ { 2 } \\> \\operatorname { l o g } \\mity _ { 1 } \\operatorname { l o g } \\mity _ { 2 } \\> \\> \\Planckconst _ { 1 } ( \\miteta ) \\right] \\end{align*}",
"\\begin{align*} \\left\\langle \\mupPhi ( \\mitr _ { 1 } ) \\mupPhi ( \\mitr _ { 2 } ) \\right\\rangle & = & 0 \\\\ \\left\\langle \\mupPhi ( \\mitr _ { 1 } ) \\mupPsi ( \\mitr _ { 2 } ) \\right\\rangle & = & \\frac { \\mita } { \\mitr ^ { 2 \\mupDelta _ { \\mitphi } } } \\\\ \\left\\langle \\mupPsi ( \\mitr _ { 1 } ) \\mupPsi ( \\mitr _ { 2 } ) \\right\\rangle & = & \\frac { 1 } { \\mitr ^ { 2 \\mupDelta _ { \\mitphi } } } ( \\mitb - \\mitd \\> \\mita \\operatorname { l o g } \\mitr ) \\end{align*}"
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0002142_page05 | {
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"$y_{1}$",
"$y_{2}$",
"$y_{1}-y_{2}$",
"$x_{1}-x_{2}$",
"$\\eta $",
"$h_{1}$",
"$h_{2}$",
"$h_{3}$",
"$\\eta \\rightarrow {0}$",
"$C_{1}$",
"$C_{2}$",
"$\\eta $",
"$x_{1}-x_{2}$",
"$y_{1}$",
"$y_{2}$",
"$h(\\eta )$",
"$z\\:\\rightarrow \\:w(z)$",
"$ \\bar {z}\\:\\rightarrow \\:w(\\bar {z})$",
"$\\Phi _{i}$",
"$T(z)=T_{zz}(z)$",
"$\\bar {T}(\\bar {z})=T_{\\bar {z}\\:\\bar {z}}(\\bar {z})$",
"$c$",
"$z_{k}$",
"$k$",
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"\\begin {eqnarray} h_{1}(\\eta )&=&\\frac {1}{\\eta ^{\\Delta _{\\phi }}}\\left (\\frac {4\\frac {a}{d}}{\\log {\\eta }}+\\frac {C_{1}}{(\\log {\\eta })^{2}}+\\ldots \\right )\\\\ h_{2}(\\eta )&=&\\frac {1}{\\eta ^{\\Delta _{\\phi }}}\\left (-a+\\frac {C_{2}}{(\\log {\\eta })}+\\ldots \\right )\\\\ h_{3}(\\eta )&=&\\frac {1}{\\eta ^{\\Delta _{\\phi }}}\\left (b-d\\:C_{2}-\\frac {d^{2}}{4}C_{1}+\\ldots \\right ) \\end {eqnarray}",
"\\begin {equation} \\Phi _{i}(z,\\bar {z})\\rightarrow \\Phi _{i}(z,\\bar {z})+\\left [\\alpha ^{\\prime }(z) \\Delta _{i}^{j}+\\delta _{i}^{j}\\alpha (z)\\frac {\\partial }{\\partial {z}}+ \\overline {\\alpha ^{\\prime }(z)}\\: {\\bar {\\Delta }}_{i}^{j}+\\delta _{i}^{j}\\overline {\\alpha (z)} \\frac {\\partial }{\\partial {\\bar {z}}}\\right ] \\Phi _{j}(z,\\bar {z}) \\end {equation}",
"\\begin {eqnarray} \\frac 1{2\\pi i}\\oint _{c}dz\\:\\alpha (z)\\left <T(z)\\Phi _{i_{1}}(z_{1},\\bar {z_{1}})\\ldots \\right > -\\frac 1{2\\pi i} \\oint _{c}d\\bar {z}\\:\\overline {\\alpha (z)}\\left <\\bar {T} (\\bar {z})\\Phi _{i_{1}}(z_{1},{\\bar {z}}_{1})\\ldots \\right >\\\\ =\\sum _{k}\\sum _{j_{k}}\\left [\\alpha ^{\\prime }(z_{k}) \\Delta ^{j_{k}}_{i_{k}}+\\delta ^{j_{k}}_{i_{k}}\\alpha (z_{k})\\frac {\\partial }{\\partial {z_{k}}}+ \\overline {\\alpha ^{\\prime }(z_{k})}{\\bar {\\Delta }}^{j_{k}}_{i_{k}} +\\delta ^{j_{k}}_{i_{k}}\\overline {\\alpha (z_{k})} \\frac {\\partial }{\\partial {\\bar {z}}_{k}}\\right ] \\left <\\Phi _{j_{1}}(z_{1},{\\bar {z}}_{1})\\ldots \\right > \\end {eqnarray}",
"\\begin {equation} \\frac 1{2\\pi i}\\oint _{c}dz\\:\\alpha (z)\\left <T(z)\\Phi _{i_{1}}(z_{1},{\\bar {z}}_{1}) \\ldots \\right > =\\sum _{k}\\sum _{j_{k}}\\left [\\alpha ^{\\prime }(z_{k})\\Delta ^{j_{k}}_{i_{k}}+ \\delta ^{j_{k}}_{i_{k}}\\alpha (z_{k}) \\frac {\\partial }{\\partial {z_{k}}}\\right ] \\left <\\Phi _{j_{1}}(z_{1},{\\bar {z}}_{1})\\ldots \\right > \\end {equation}"
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"$ \\eta $",
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"$ \\bar { z } \\> \\rightarrow \\> w ( \\bar { z } ) $",
"$ \\Phi _ { i } $",
"$ T ( z ) = T _ { z z } ( z ) $",
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"$ \\bar { \\alpha } $",
"$ z $",
"$ \\bar { z } $",
"\\begin{align*} h _ { 1 } ( \\eta ) & = & \\frac { 1 } { \\eta ^ { \\Delta _ { \\phi } } } ( \\frac { 4 \\frac { a } { d } } { \\operatorname { l o g } \\eta } + \\frac { C _ { 1 } } { ( \\operatorname { l o g } \\eta ) ^ { 2 } } + \\ldots ) \\\\ h _ { 2 } ( \\eta ) & = & \\frac { 1 } { \\eta ^ { \\Delta _ { \\phi } } } ( - a + \\frac { C _ { 2 } } { ( \\operatorname { l o g } \\eta ) } + \\ldots ) \\\\ h _ { 3 } ( \\eta ) & = & \\frac { 1 } { \\eta ^ { \\Delta _ { \\phi } } } ( b - d \\> C _ { 2 } - \\frac { d ^ { 2 } } { 4 } C _ { 1 } + \\ldots ) \\end{align*}",
"\\begin{equation*} \\Phi _ { i } ( z , \\bar { z } ) \\rightarrow \\Phi _ { i } ( z , \\bar { z } ) + [ \\alpha ^ { \\prime } ( z ) \\Delta _ { i } ^ { j } + \\delta _ { i } ^ { j } \\alpha ( z ) \\frac { \\partial } { \\partial z } + \\overline { \\alpha ^ { \\prime } ( z ) } \\> \\bar { \\Delta } _ { i } ^ { j } + \\delta _ { i } ^ { j } \\overline { \\alpha ( z ) } \\frac { \\partial } { \\partial \\bar { z } } ] \\Phi _ { j } ( z , \\bar { z } ) \\end{equation*}",
"\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { c } d z \\> \\alpha ( z ) \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , \\bar { z _ { 1 } } ) \\ldots \\rangle - \\frac { 1 } { 2 \\pi i } \\oint _ { c } d \\bar { z } \\> \\overline { \\alpha ( z ) } \\langle \\bar { T } ( \\bar { z } ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle \\\\ = \\sum _ { k } \\sum _ { j _ { k } } [ \\alpha ^ { \\prime } ( z _ { k } ) \\Delta _ { i _ { k } } ^ { j _ { k } } + \\delta _ { i _ { k } } ^ { j _ { k } } \\alpha ( z _ { k } ) \\frac { \\partial } { \\partial z _ { k } } + \\overline { \\alpha ^ { \\prime } ( z _ { k } ) } \\bar { \\Delta } _ { i _ { k } } ^ { j _ { k } } + \\delta _ { i _ { k } } ^ { j _ { k } } \\overline { \\alpha ( z _ { k } ) } \\frac { \\partial } { \\partial \\bar { z } _ { k } } ] \\langle \\Phi _ { j _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle \\end{align*}",
"\\begin{equation*} \\frac { 1 } { 2 \\pi i } \\oint _ { c } d z \\> \\alpha ( z ) \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle = \\sum _ { k } \\sum _ { j _ { k } } [ \\alpha ^ { \\prime } ( z _ { k } ) \\Delta _ { i _ { k } } ^ { j _ { k } } + \\delta _ { i _ { k } } ^ { j _ { k } } \\alpha ( z _ { k } ) \\frac { \\partial } { \\partial z _ { k } } ] \\langle \\Phi _ { j _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle \\end{equation*}"
],
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"$ \\mity _ { 2 } $",
"$ \\mity _ { 1 } - \\mity _ { 2 } $",
"$ \\mitx _ { 1 } - \\mitx _ { 2 } $",
"$ \\miteta $",
"$ \\Planckconst _ { 1 } $",
"$ \\Planckconst _ { 2 } $",
"$ \\Planckconst _ { 3 } $",
"$ \\miteta \\rightarrow 0 $",
"$ \\mitC _ { 1 } $",
"$ \\mitC _ { 2 } $",
"$ \\miteta $",
"$ \\mitx _ { 1 } - \\mitx _ { 2 } $",
"$ \\mity _ { 1 } $",
"$ \\mity _ { 2 } $",
"$ \\Planckconst ( \\miteta ) $",
"$ \\mitz \\> \\rightarrow \\> \\mitw ( \\mitz ) $",
"$ \\bar { \\mitz } \\> \\rightarrow \\> \\mitw ( \\bar { \\mitz } ) $",
"$ \\mupPhi _ { \\miti } $",
"$ \\mitT ( \\mitz ) = \\mitT _ { \\mitz \\mitz } ( \\mitz ) $",
"$ \\bar { \\mitT } ( \\bar { \\mitz } ) = \\mitT _ { \\bar { \\mitz } \\> \\bar { \\mitz } } ( \\bar { \\mitz } ) $",
"$ \\mitc $",
"$ \\mitz _ { \\mitk } $",
"$ \\mitk $",
"$ \\mitj _ { \\mitk } $",
"$ \\mupPhi _ { \\miti _ { \\mitk } } $",
"$ \\mitalpha ( \\mitz ) $",
"$ \\mitalpha $",
"$ \\bar { \\mitalpha } $",
"$ \\mitz $",
"$ \\bar { \\mitz } $",
"\\begin{align*} \\Planckconst _ { 1 } ( \\miteta ) & = & \\frac { 1 } { \\miteta ^ { \\mupDelta _ { \\mitphi } } } \\left( \\frac { 4 \\frac { \\mita } { \\mitd } } { \\operatorname { l o g } \\miteta } + \\frac { \\mitC _ { 1 } } { ( \\operatorname { l o g } \\miteta ) ^ { 2 } } + \\ldots \\right) \\\\ \\Planckconst _ { 2 } ( \\miteta ) & = & \\frac { 1 } { \\miteta ^ { \\mupDelta _ { \\mitphi } } } \\left( - \\mita + \\frac { \\mitC _ { 2 } } { ( \\operatorname { l o g } \\miteta ) } + \\ldots \\right) \\\\ \\Planckconst _ { 3 } ( \\miteta ) & = & \\frac { 1 } { \\miteta ^ { \\mupDelta _ { \\mitphi } } } \\left( \\mitb - \\mitd \\> \\mitC _ { 2 } - \\frac { \\mitd ^ { 2 } } { 4 } \\mitC _ { 1 } + \\ldots \\right) \\end{align*}",
"\\begin{equation*} \\mupPhi _ { \\miti } ( \\mitz , \\bar { \\mitz } ) \\rightarrow \\mupPhi _ { \\miti } ( \\mitz , \\bar { \\mitz } ) + \\left[ \\mitalpha ^ { \\prime } ( \\mitz ) \\mupDelta _ { \\miti } ^ { \\mitj } + \\mitdelta _ { \\miti } ^ { \\mitj } \\mitalpha ( \\mitz ) \\frac { \\mitpartial } { \\mitpartial \\mitz } + \\overline { \\mitalpha ^ { \\prime } ( \\mitz ) } \\> \\bar { \\mupDelta } _ { \\miti } ^ { \\mitj } + \\mitdelta _ { \\miti } ^ { \\mitj } \\overline { \\mitalpha ( \\mitz ) } \\frac { \\mitpartial } { \\mitpartial \\bar { \\mitz } } \\right] \\mupPhi _ { \\mitj } ( \\mitz , \\bar { \\mitz } ) \\end{equation*}",
"\\begin{align*} \\frac { 1 } { 2 \\mitpi \\miti } \\oint _ { \\mitc } \\mitd \\mitz \\> \\mitalpha ( \\mitz ) \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz _ { 1 } } ) \\ldots \\right\\rangle - \\frac { 1 } { 2 \\mitpi \\miti } \\oint _ { \\mitc } \\mitd \\bar { \\mitz } \\> \\overline { \\mitalpha ( \\mitz ) } \\left\\langle \\bar { \\mitT } ( \\bar { \\mitz } ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle \\\\ = \\sum _ { \\mitk } \\sum _ { \\mitj _ { \\mitk } } \\left[ \\mitalpha ^ { \\prime } ( \\mitz _ { \\mitk } ) \\mupDelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } + \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } \\mitalpha ( \\mitz _ { \\mitk } ) \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } } + \\overline { \\mitalpha ^ { \\prime } ( \\mitz _ { \\mitk } ) } \\bar { \\mupDelta } _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } + \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } \\overline { \\mitalpha ( \\mitz _ { \\mitk } ) } \\frac { \\mitpartial } { \\mitpartial \\bar { \\mitz } _ { \\mitk } } \\right] \\left\\langle \\mupPhi _ { \\mitj _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle \\end{align*}",
"\\begin{equation*} \\frac { 1 } { 2 \\mitpi \\miti } \\oint _ { \\mitc } \\mitd \\mitz \\> \\mitalpha ( \\mitz ) \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle = \\sum _ { \\mitk } \\sum _ { \\mitj _ { \\mitk } } \\left[ \\mitalpha ^ { \\prime } ( \\mitz _ { \\mitk } ) \\mupDelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } + \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } \\mitalpha ( \\mitz _ { \\mitk } ) \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } } \\right] \\left\\langle \\mupPhi _ { \\mitj _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle \\end{equation*}"
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0002142_page06 | {
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"$\\left <\\bar {T}\\Phi \\ldots \\right >$",
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"$(z_{1},\\ldots ,z_{n},\\newline {\\bar {z}}_{1},\\ldots ,{\\bar {z}}_{n})$",
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"\\begin {equation} \\left <T(z)\\Phi _{i_{1}}(z_{1},{\\bar {z}}_{1})\\ldots \\right >=\\sum _{k}\\sum _{j_{k}} \\left [\\frac {\\Delta ^{j_{k}}_{i_{k}}}{(z-z_{k})^{2}} +\\frac {\\delta ^{j_{k}}_{i_{k}}}{(z-z_{k})}\\frac {\\partial }{\\partial {z_{k}}}\\right ] \\left <\\Phi _{j_{1}}(z_{1},{\\bar {z}}_{1})\\ldots \\right >\\:. \\end {equation}",
"\\begin {equation} T(z):=\\bar {T}(z)\\:\\:\\:\\:\\:\\: Im(z)<0 \\end {equation}",
"\\begin {eqnarray} \\frac 1{2\\pi i}{\\oint }_{c}dz\\:\\alpha (z)\\left <T(z)\\Phi _{i_{1}}(z_{1}, z^{\\prime }_{1})\\ldots \\right >+\\frac 1{2\\pi i}\\oint _{\\:\\bar {c}} dz\\:\\alpha (z)\\left <T(z){\\Phi }_{i_{1}}(z_{1},z^{\\prime }_{1})\\ldots \\right > \\\\ =\\sum _{k}\\sum _{j_{k}}\\left [{\\alpha }^{\\prime }(z_{k})\\Delta ^{j_{k}}_{i_{k}}+ \\delta ^{j_{k}}_{i_{k}}\\alpha (z_{k})\\frac {\\partial }{\\partial {z_{k}}}+ {\\alpha }^{\\prime }(z^{\\prime }_{k}){\\bar {\\Delta }}^{j_{k}}_{i_{k}} +\\alpha (z^{\\prime }_{k})\\frac {\\partial }{\\partial {z^{\\prime }_{k}}}\\right ] \\left <\\Phi _{j_{1}}(z_{1},z^{\\prime }_{1})\\ldots \\right > , \\end {eqnarray}",
"\\begin {eqnarray} \\left <T(z)\\Phi _{i_{1}}(z_{1},{z'}_{1})\\ldots \\right >= \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\ \\sum _{k}\\sum _{j_{k}}\\left [\\frac {\\Delta ^{i_{k}}_{j_{k}}}{(z-z_{k})^{2}}+ \\frac {\\delta ^{i_{k}}_{j_{k}}}{(z-z_{k})}\\frac {\\partial }{\\partial {z_{k}}} +\\frac {{\\bar {\\Delta }}^{j_{k}}_{i_{k}}}{(z-z^{\\prime }_{k})^{2}} +\\frac {\\delta ^{j_{k}}_{i_{k}}}{(z-z^{\\prime }_{k})} \\frac {\\partial }{\\partial {z^{\\prime }_{k}}}\\right ] \\left <\\Phi _{j_{1}}(z_{1},z^{\\prime }_{1})\\ldots \\right >\\:. \\end {eqnarray}"
],
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"$ \\bar { c } $",
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"$ T = \\bar { T } $",
"$ T _ { x y } = 0 $",
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"$ ( z _ { 1 } , \\ldots , z _ { n } , \\\\ \\bar { z } _ { 1 } , \\ldots , \\bar { z } _ { n } ) $",
"$ ( z _ { 1 } , \\ldots , z _ { n } , \\\\ \\bar { z } _ { 1 } , \\ldots , \\bar { z } _ { n } ) $",
"$ \\langle \\Phi _ { i _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\Phi _ { i _ { 2 n } } ( z _ { 2 n } , \\bar { z } _ { 2 n } ) \\rangle $",
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"$ \\Phi _ { i _ { 2 n } } $",
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"$ 2 \\times 2 $",
"$ \\Phi $",
"$ \\Psi $",
"$ \\Psi $",
"$ L _ { 0 } , L _ { \\pm 1 } $",
"\\begin{equation*} \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle = \\sum _ { k } \\sum _ { j _ { k } } [ \\frac { \\Delta _ { i _ { k } } ^ { j _ { k } } } { ( z - z _ { k } ) ^ { 2 } } + \\frac { \\delta _ { i _ { k } } ^ { j _ { k } } } { ( z - z _ { k } ) } \\frac { \\partial } { \\partial z _ { k } } ] \\langle \\Phi _ { j _ { 1 } } ( z _ { 1 } , \\bar { z } _ { 1 } ) \\ldots \\rangle \\> . \\end{equation*}",
"\\begin{equation*} T ( z ) : = \\bar { T } ( z ) \\> \\> \\> \\> \\> \\> I m ( z ) < 0 \\end{equation*}",
"\\begin{align*} \\frac { 1 } { 2 \\pi i } { \\oint } _ { c } d z \\> \\alpha ( z ) \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , z _ { 1 } ^ { \\prime } ) \\ldots \\rangle + \\frac { 1 } { 2 \\pi i } \\oint _ { \\> \\bar { c } } d z \\> \\alpha ( z ) \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , z _ { 1 } ^ { \\prime } ) \\ldots \\rangle \\\\ = \\sum _ { k } \\sum _ { j _ { k } } [ \\alpha ^ { \\prime } ( z _ { k } ) \\Delta _ { i _ { k } } ^ { j _ { k } } + \\delta _ { i _ { k } } ^ { j _ { k } } \\alpha ( z _ { k } ) \\frac { \\partial } { \\partial z _ { k } } + \\alpha ^ { \\prime } ( z _ { k } ^ { \\prime } ) \\bar { \\Delta } _ { i _ { k } } ^ { j _ { k } } + \\alpha ( z _ { k } ^ { \\prime } ) \\frac { \\partial } { \\partial z _ { k } ^ { \\prime } } ] \\langle \\Phi _ { j _ { 1 } } ( z _ { 1 } , z _ { 1 } ^ { \\prime } ) \\ldots \\rangle , \\end{align*}",
"\\begin{align*} \\langle T ( z ) \\Phi _ { i _ { 1 } } ( z _ { 1 } , { z ^ { \\prime } } _ { 1 } ) \\ldots \\rangle = \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\\\ \\sum _ { k } \\sum _ { j _ { k } } [ \\frac { \\Delta _ { j _ { k } } ^ { i _ { k } } } { ( z - z _ { k } ) ^ { 2 } } + \\frac { \\delta _ { j _ { k } } ^ { i _ { k } } } { ( z - z _ { k } ) } \\frac { \\partial } { \\partial z _ { k } } + \\frac { \\bar { \\Delta } _ { i _ { k } } ^ { j _ { k } } } { ( z - z _ { k } ^ { \\prime } ) ^ { 2 } } + \\frac { \\delta _ { i _ { k } } ^ { j _ { k } } } { ( z - z _ { k } ^ { \\prime } ) } \\frac { \\partial } { \\partial z _ { k } ^ { \\prime } } ] \\langle \\Phi _ { j _ { 1 } } ( z _ { 1 } , z _ { 1 } ^ { \\prime } ) \\ldots \\rangle \\> . \\end{align*}"
],
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"$ \\left< \\bar \\mitT \\mupPhi \\ldots \\right> $",
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"$ \\langle \\mupPhi \\ldots \\rangle $",
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"$ ( \\mitz _ { 1 } , \\ldots , \\mitz _ { \\mitn } , \\\\ \\bar { \\mitz } _ { 1 } , \\ldots , \\bar { \\mitz } _ { \\mitn } ) $",
"$ ( \\mitz _ { 1 } , \\ldots , \\mitz _ { \\mitn } , \\\\ \\bar { \\mitz } _ { 1 } , \\ldots , \\bar { \\mitz } _ { \\mitn } ) $",
"$ \\left\\langle \\mupPhi { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\mupPhi _ { \\miti _ { 2 \\mitn } } ( \\mitz _ { 2 \\mitn } , \\bar { \\mitz } _ { 2 \\mitn } ) \\right\\rangle $",
"$ \\mitz _ { 1 } , \\ldots , \\mitz _ { 2 \\mitn } $",
"$ \\mupPhi _ { \\miti _ { \\mitn + 1 } } $",
"$ \\mupPhi _ { \\miti _ { 2 \\mitn } } $",
"$ \\bar { \\mupDelta } $",
"$ 2 \\times 2 $",
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"\\begin{equation*} \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle = \\sum _ { \\mitk } \\sum _ { \\mitj _ { \\mitk } } \\left[ \\frac { \\mupDelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ) ^ { 2 } } + \\frac { \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ) } \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } } \\right] \\left\\langle \\mupPhi _ { \\mitj _ { 1 } } ( \\mitz _ { 1 } , \\bar { \\mitz } _ { 1 } ) \\ldots \\right\\rangle \\> . \\end{equation*}",
"\\begin{equation*} \\mitT ( \\mitz ) : = \\bar { \\mitT } ( \\mitz ) \\> \\> \\> \\> \\> \\> \\mitI \\mitm ( \\mitz ) < 0 \\end{equation*}",
"\\begin{align*} \\frac { 1 } { 2 \\mitpi \\miti } { \\oint } _ { \\mitc } \\mitd \\mitz \\> \\mitalpha ( \\mitz ) \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\mitz _ { 1 } ^ { \\prime } ) \\ldots \\right\\rangle + \\frac { 1 } { 2 \\mitpi \\miti } \\oint _ { \\> \\bar { \\mitc } } \\mitd \\mitz \\> \\mitalpha ( \\mitz ) \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , \\mitz _ { 1 } ^ { \\prime } ) \\ldots \\right\\rangle \\\\ = \\sum _ { \\mitk } \\sum _ { \\mitj _ { \\mitk } } \\left[ \\mitalpha ^ { \\prime } ( \\mitz _ { \\mitk } ) \\mupDelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } + \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } \\mitalpha ( \\mitz _ { \\mitk } ) \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } } + \\mitalpha ^ { \\prime } ( \\mitz _ { \\mitk } ^ { \\prime } ) \\bar { \\mupDelta } _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } + \\mitalpha ( \\mitz _ { \\mitk } ^ { \\prime } ) \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } ^ { \\prime } } \\right] \\left\\langle \\mupPhi _ { \\mitj _ { 1 } } ( \\mitz _ { 1 } , \\mitz _ { 1 } ^ { \\prime } ) \\ldots \\right\\rangle , \\end{align*}",
"\\begin{align*} \\left\\langle \\mitT ( \\mitz ) \\mupPhi _ { \\miti _ { 1 } } ( \\mitz _ { 1 } , { \\mitz ^ { \\prime } } _ { 1 } ) \\ldots \\right\\rangle = \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\\\ \\sum _ { \\mitk } \\sum _ { \\mitj _ { \\mitk } } \\left[ \\frac { \\mupDelta _ { \\mitj _ { \\mitk } } ^ { \\miti _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ) ^ { 2 } } + \\frac { \\mitdelta _ { \\mitj _ { \\mitk } } ^ { \\miti _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ) } \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } } + \\frac { \\bar { \\mupDelta } _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ^ { \\prime } ) ^ { 2 } } + \\frac { \\mitdelta _ { \\miti _ { \\mitk } } ^ { \\mitj _ { \\mitk } } } { ( \\mitz - \\mitz _ { \\mitk } ^ { \\prime } ) } \\frac { \\mitpartial } { \\mitpartial \\mitz _ { \\mitk } ^ { \\prime } } \\right] \\left\\langle \\mupPhi _ { \\mitj _ { 1 } } ( \\mitz _ { 1 } , \\mitz _ { 1 } ^ { \\prime } ) \\ldots \\right\\rangle \\> . \\end{align*}"
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0002142_page07 | {
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"$u={(z_{1}-{\\bar {z}}_{1})(z_{2}-{\\bar {z}}_{2})}/ {(z_{1}-z_{2})({\\bar {z}}_{1}-{\\bar {z}}_{2}) (z_{1}-{\\bar {z}}_{2})({\\bar {z}}_{1}-z_{2})}$",
"$v=(z_{1}-z_{2})({\\bar {z}}_{1}-{\\bar {z}}_{2})/(z_{1}-{\\bar {z}}_{1}) (z_{2}-{\\bar {z}}_{2})\\:$",
"$f_{1},f_{2},f_{3}$",
"$f_{1},f_{2},f_{3}$",
"$\\phi $",
"$\\psi $",
"$w=\\nabla ^{2}\\phi $",
"$J=\\nabla ^{2}\\psi $",
"$\\mu $",
"$\\eta $",
"$\\phi $",
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"\\begin {eqnarray} \\left <\\Phi (z,\\bar {z})\\right >&=&\\frac {c}{(z-\\bar {z})^{2\\Delta }}\\:,\\\\ \\left <\\Psi (z,\\bar {z})\\right >&=&\\frac 1{(z-\\bar {z})^{2\\Delta }} \\left [c^{\\prime }-2c\\log {(z-\\bar {z})}\\right ]. \\end {eqnarray}",
"\\begin {eqnarray} \\left <\\Phi (z_{1})\\Phi (z_{2})\\right >&=&u^{2\\Delta }f_{1}(v)\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\^^M\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end {eqnarray}",
"\\begin {eqnarray} \\left <\\Phi (z_{1})\\Psi (z_{2})\\right >&=&u^{2\\Delta }\\left (f_{2}(v)-2\\log (z_{2}-\\bar {z_{2}})f_{1}(v)\\right )\\end {eqnarray}",
"\\begin {eqnarray} \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\left <\\Psi (z_{1})\\Psi (z_{2})\\right >~=~ u^{2\\Delta }(f_{3}(v)-2\\log \\left [(z_{1}-{\\bar {z}}_{1}) (z_{2}-{\\bar {z}}_{2})\\right ]f_{2}(v) \\\\ \\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: +4\\log (z_{1}-{\\bar {z}}_{1})\\log (z_{2}-{\\bar {z}}_{2})f_{1}(v)) \\end {eqnarray}",
"\\begin {eqnarray} \\frac {\\partial {w}}{\\partial {t}}&=&-\\epsilon _{\\alpha \\beta }\\partial _{\\alpha }{\\phi }\\partial _{\\beta }{w}+\\epsilon _{\\alpha \\beta }\\partial _{\\alpha }\\psi \\partial _{\\beta }{J}+\\mu \\nabla ^{2}w~\\\\ \\frac {\\partial {\\psi }}{\\partial {t}}&=&-\\epsilon _{\\alpha \\beta }\\partial _{\\alpha }{\\phi }\\partial _{\\beta }{\\psi }+\\eta J~, \\end {eqnarray}",
"\\begin {eqnarray} V_{\\alpha }&=&\\epsilon _{\\alpha \\beta }\\partial _{\\beta }{\\phi }~\\\\ B_{\\alpha }&=&\\epsilon _{\\alpha \\beta }\\partial _{\\beta }{\\psi }~, \\end {eqnarray}"
],
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"$ u = ( z _ { 1 } - \\bar { z } _ { 1 } ) ( z _ { 2 } - \\bar { z } _ { 2 } ) \\slash ( z _ { 1 } - z _ { 2 } ) ( \\bar { z } _ { 1 } - \\bar { z } _ { 2 } ) ( z _ { 1 } - \\bar { z } _ { 2 } ) ( \\bar { z } _ { 1 } - z _ { 2 } ) $",
"$ v = ( z _ { 1 } - z _ { 2 } ) ( \\bar { z } _ { 1 } - \\bar { z } _ { 2 } ) \\slash ( z _ { 1 } - \\bar { z } _ { 1 } ) ( z _ { 2 } - \\bar { z } _ { 2 } ) \\> $",
"$ f _ { 1 } , f _ { 2 } , f _ { 3 } $",
"$ f _ { 1 } , f _ { 2 } , f _ { 3 } $",
"$ \\phi $",
"$ \\psi $",
"$ w = \\nabla ^ { 2 } \\phi $",
"$ J = \\nabla ^ { 2 } \\psi $",
"$ \\mu $",
"$ \\eta $",
"$ \\phi $",
"$ \\psi $",
"\\begin{align*} \\langle \\Phi ( z , \\bar { z } ) \\rangle & = & \\frac { c } { ( z - \\bar { z } ) ^ { 2 \\Delta } } \\> , \\\\ \\langle \\Psi ( z , \\bar { z } ) \\rangle & = & \\frac { 1 } { ( z - \\bar { z } ) ^ { 2 \\Delta } } [ c ^ { \\prime } - 2 c \\operatorname { l o g } ( z - \\bar { z } ) ] . \\end{align*}",
"\\begin{align*} \\langle \\Phi ( z _ { 1 } ) \\Phi ( z _ { 2 } ) \\rangle & = & u ^ { 2 \\Delta } f _ { 1 } ( v ) \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> ~ \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\end{align*}",
"\\begin{align*} \\langle \\Phi ( z _ { 1 } ) \\Psi ( z _ { 2 } ) \\rangle & = & u ^ { 2 \\Delta } ( f _ { 2 } ( v ) - 2 \\operatorname { l o g } ( z _ { 2 } - \\bar { z _ { 2 } } ) f _ { 1 } ( v ) ) \\end{align*}",
"\\begin{align*} \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\langle \\Psi ( z _ { 1 } ) \\Psi ( z _ { 2 } ) \\rangle ~ = ~ u ^ { 2 \\Delta } ( f _ { 3 } ( v ) - 2 \\operatorname { l o g } [ ( z _ { 1 } - \\bar { z } _ { 1 } ) ( z _ { 2 } - \\bar { z } _ { 2 } ) ] f _ { 2 } ( v ) \\\\ \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> + 4 \\operatorname { l o g } ( z _ { 1 } - \\bar { z } _ { 1 } ) \\operatorname { l o g } ( z _ { 2 } - \\bar { z } _ { 2 } ) f _ { 1 } ( v ) ) \\end{align*}",
"\\begin{align*} \\frac { \\partial w } { \\partial t } & = & - \\epsilon _ { \\alpha \\beta } \\partial _ { \\alpha } \\phi \\partial _ { \\beta } w + \\epsilon _ { \\alpha \\beta } \\partial _ { \\alpha } \\psi \\partial _ { \\beta } J + \\mu \\nabla ^ { 2 } w ~ \\\\ \\frac { \\partial \\psi } { \\partial t } & = & - \\epsilon _ { \\alpha \\beta } \\partial _ { \\alpha } \\phi \\partial _ { \\beta } \\psi + \\eta J ~ , \\end{align*}",
"\\begin{align*} V _ { \\alpha } & = & \\epsilon _ { \\alpha \\beta } \\partial _ { \\beta } \\phi ~ \\\\ B _ { \\alpha } & = & \\epsilon _ { \\alpha \\beta } \\partial _ { \\beta } \\psi ~ , \\end{align*}"
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"$ \\mitu = ( \\mitz _ { 1 } - \\bar { \\mitz } _ { 1 } ) ( \\mitz _ { 2 } - \\bar { \\mitz } _ { 2 } ) \\slash ( \\mitz _ { 1 } - \\mitz _ { 2 } ) ( \\bar { \\mitz } _ { 1 } - \\bar { \\mitz } _ { 2 } ) ( \\mitz _ { 1 } - \\bar { \\mitz } _ { 2 } ) ( \\bar { \\mitz } _ { 1 } - \\mitz _ { 2 } ) $",
"$ \\mitv = ( \\mitz _ { 1 } - \\mitz _ { 2 } ) ( \\bar { \\mitz } _ { 1 } - \\bar { \\mitz } _ { 2 } ) \\slash ( \\mitz _ { 1 } - \\bar { \\mitz } _ { 1 } ) ( \\mitz _ { 2 } - \\bar { \\mitz } _ { 2 } ) \\> $",
"$ \\mitf _ { 1 } , \\mitf _ { 2 } , \\mitf _ { 3 } $",
"$ \\mitf _ { 1 } , \\mitf _ { 2 } , \\mitf _ { 3 } $",
"$ \\mitphi $",
"$ \\mitpsi $",
"$ \\mitw = \\nabla ^ { 2 } \\mitphi $",
"$ \\mitJ = \\nabla ^ { 2 } \\mitpsi $",
"$ \\mitmu $",
"$ \\miteta $",
"$ \\mitphi $",
"$ \\mitpsi $",
"\\begin{align*} \\left\\langle \\mupPhi ( \\mitz , \\bar { \\mitz } ) \\right\\rangle & = & \\frac { \\mitc } { ( \\mitz - \\bar { \\mitz } ) ^ { 2 \\mupDelta } } \\> , \\\\ \\left\\langle \\mupPsi ( \\mitz , \\bar { \\mitz } ) \\right\\rangle & = & \\frac { 1 } { ( \\mitz - \\bar { \\mitz } ) ^ { 2 \\mupDelta } } \\left[ \\mitc ^ { \\prime } - 2 \\mitc \\operatorname { l o g } ( \\mitz - \\bar { \\mitz } ) \\right] . \\end{align*}",
"\\begin{align*} \\left\\langle \\mupPhi ( \\mitz _ { 1 } ) \\mupPhi ( \\mitz _ { 2 } ) \\right\\rangle & = & \\mitu ^ { 2 \\mupDelta } \\mitf _ { 1 } ( \\mitv ) \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> ~ \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\end{align*}",
"\\begin{align*} \\left\\langle \\mupPhi ( \\mitz _ { 1 } ) \\mupPsi ( \\mitz _ { 2 } ) \\right\\rangle & = & \\mitu ^ { 2 \\mupDelta } \\left( \\mitf _ { 2 } ( \\mitv ) - 2 \\operatorname { l o g } ( \\mitz _ { 2 } - \\bar { \\mitz _ { 2 } } ) \\mitf _ { 1 } ( \\mitv ) \\right) \\end{align*}",
"\\begin{align*} \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\left\\langle \\mupPsi ( \\mitz _ { 1 } ) \\mupPsi ( \\mitz _ { 2 } ) \\right\\rangle ~ = ~ \\mitu ^ { 2 \\mupDelta } ( \\mitf _ { 3 } ( \\mitv ) - 2 \\operatorname { l o g } \\left[ ( \\mitz _ { 1 } - \\bar { \\mitz } _ { 1 } ) ( \\mitz _ { 2 } - \\bar { \\mitz } _ { 2 } ) \\right] \\mitf _ { 2 } ( \\mitv ) \\\\ \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> \\> + 4 \\operatorname { l o g } ( \\mitz _ { 1 } - \\bar { \\mitz } _ { 1 } ) \\operatorname { l o g } ( \\mitz _ { 2 } - \\bar { \\mitz } _ { 2 } ) \\mitf _ { 1 } ( \\mitv ) ) \\end{align*}",
"\\begin{align*} \\frac { \\mitpartial \\mitw } { \\mitpartial \\mitt } & = & - \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitalpha } \\mitphi \\mitpartial _ { \\mitbeta } \\mitw + \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitalpha } \\mitpsi \\mitpartial _ { \\mitbeta } \\mitJ + \\mitmu \\nabla ^ { 2 } \\mitw ~ \\\\ \\frac { \\mitpartial \\mitpsi } { \\mitpartial \\mitt } & = & - \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitalpha } \\mitphi \\mitpartial _ { \\mitbeta } \\mitpsi + \\miteta \\mitJ ~ , \\end{align*}",
"\\begin{align*} \\mitV _ { \\mitalpha } & = & \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitbeta } \\mitphi ~ \\\\ \\mitB _ { \\mitalpha } & = & \\mitepsilon _ { \\mitalpha \\mitbeta } \\mitpartial _ { \\mitbeta } \\mitpsi ~ , \\end{align*}"
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|
0002142_page08 | {
"latex": [
"$\\epsilon _{\\alpha \\beta }$",
"$\\epsilon _{12}=1$",
"$\\phi $",
"$\\psi $",
"$\\phi $",
"$\\psi $",
"$\\Delta =\\frac {-5}{7}$",
"$L$",
"\\begin {eqnarray} \\left <V_{x}(x,y)\\right >&=&\\partial _{y}\\left <\\phi (x,y)\\right >=-\\frac {2\\Delta C}{y^{2\\Delta +1}}~\\\\ \\left <V_{y}(x,y)\\right >&=&-\\partial _{x}\\left <\\phi (x,y)\\right >=0~, \\end {eqnarray}",
"\\begin {eqnarray} \\left <B_{x}(x,y)\\right >&=&\\partial _{y}\\left <\\psi (x,y)\\right >=-\\frac {2\\Delta }{y^{2\\Delta }}\\left [(C^{\\prime }+2C)-2C\\log y\\right ]~\\\\ \\left <B_{y}(x,y)\\right >&=&-\\partial _{x}\\left <\\psi (x,y)\\right >=0~. \\end {eqnarray}",
"\\begin {eqnarray} \\left <\\phi (x,y)\\right >&=&\\left (\\frac {\\pi }{L} \\right )^{2\\Delta }\\frac {C}{(\\sin {\\frac {\\pi }{L}}{y})^{2\\Delta }}\\\\ \\left <\\psi (x,y)\\right >&=&\\left (\\frac {\\pi }{L}\\right )^{2\\Delta } \\frac {1}{(\\sin {\\frac {\\pi }{L}}{y})^{2\\Delta }} \\left (C^{\\prime }+2C\\log {\\frac {\\pi }{L}}-2C\\log \\sin ({\\frac {\\pi }{L}}{y}) \\right ). \\end {eqnarray}"
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0002194_page05 | {
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0002194_page06 | {
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"\\begin{align*} & & \\langle \\prod _ { n = 1 } ^ { k } P _ { n } ( \\partial x ( \\tau _ { n } ) , \\partial ^ { 2 } x ( \\tau _ { n } ) , \\ldots ) e ^ { i p ^ { n } \\cdot x ( \\tau _ { n } ) } \\rangle _ { G , \\theta } \\\\ & & = \\operatorname { e x p } ( - \\frac { i } { 2 } \\sum _ { n > m } p _ { i } ^ { n } \\theta ^ { i j } p _ { j } ^ { m } \\epsilon ( \\tau _ { n } - \\tau _ { m } ) ) \\\\ & & ~ ~ ~ ~ ~ ~ ~ ~ \\times \\langle \\prod _ { n = 1 } ^ { k } P _ { n } ( \\partial x ( \\tau _ { n } ) , \\partial ^ { 2 } x ( \\tau _ { n } ) , \\ldots ) e ^ { i p ^ { n } \\cdot x ( \\tau _ { n } ) } \\rangle _ { G , \\theta = 0 } , \\end{align*}"
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0002194_page07 | {
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"\\begin {equation} f(x) \\ast g(x) = \\exp \\left . \\left ( \\frac {i}{2} \\theta ^{ij} \\frac {\\partial }{\\partial \\xi ^i} \\frac {\\partial }{\\partial \\zeta ^j} \\right ) f(x + \\xi ) g(x + \\zeta ) \\right |_{\\xi =\\zeta =0}, \\end {equation}",
"\\begin {eqnarray} \\hat {\\delta }_{\\hat {\\lambda }} \\hat {A}_i &=& \\partial _i \\hat {\\lambda } + i \\hat {\\lambda } \\ast \\hat {A}_i - i \\hat {A}_i \\ast \\hat {\\lambda }, \\\\ \\hat {F}_{ij} &=& \\partial _i \\hat {A}_j - \\partial _j \\hat {A}_i -i \\hat {A}_i \\ast \\hat {A}_j +i \\hat {A}_j \\ast \\hat {A}_i, \\\\ \\hat {\\delta }_{\\hat {\\lambda }} \\hat {F}_{ij} &=& i \\hat {\\lambda } \\ast \\hat {F}_{ij} - i \\hat {F}_{ij} \\ast \\hat {\\lambda }. \\end {eqnarray}"
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"\\begin{equation*} f ( x ) \\ast g ( x ) = \\operatorname { e x p } { ( \\frac { i } { 2 } \\theta ^ { i j } \\frac { \\partial } { \\partial \\xi ^ { i } } \\frac { \\partial } { \\partial \\zeta ^ { j } } ) f ( x + \\xi ) g ( x + \\zeta ) \\vert } _ { \\xi = \\zeta = 0 } , \\end{equation*}",
"\\begin{align*} \\hat { \\delta } _ { \\hat { \\lambda } } \\hat { A } _ { i } & = & \\partial _ { i } \\hat { \\lambda } + i \\hat { \\lambda } \\ast \\hat { A } _ { i } - i \\hat { A } _ { i } \\ast \\hat { \\lambda } , \\\\ \\hat { F } _ { i j } & = & \\partial _ { i } \\hat { A } _ { j } - \\partial _ { j } \\hat { A } _ { i } - i \\hat { A } _ { i } \\ast \\hat { A } _ { j } + i \\hat { A } _ { j } \\ast \\hat { A } _ { i } , \\\\ \\hat { \\delta } _ { \\hat { \\lambda } } \\hat { F } _ { i j } & = & i \\hat { \\lambda } \\ast \\hat { F } _ { i j } - i \\hat { F } _ { i j } \\ast \\hat { \\lambda } . \\end{align*}"
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|
0002194_page08 | {
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0002194_page09 | {
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0002194_page10 | {
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"\\begin{equation*} \\mitA _ { \\miti } \\mitB _ { \\miti } \\equiv ( \\mitg ^ { - 1 } ) ^ { \\miti \\mitj } \\mitA _ { \\miti } \\mitB _ { \\mitj } . \\end{equation*}",
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0002194_page11 | {
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"\\begin {eqnarray} G_s &=& g_s + O(\\alpha '), \\\\ \\hat {A}_i &=& A_i + O(\\alpha '). \\end {eqnarray}",
"\\begin {eqnarray} G_s &=& t g_s + O(\\alpha '), \\\\ \\hat {A}_i &=& \\sqrt {t} A_i + O(\\alpha '), \\end {eqnarray}",
"\\begin {equation} (G^{-1})^{ij} \\hat {F}_{jk} \\ast (G^{-1})^{kl} \\hat {F}_{li} = ( \\partial _i \\hat {A}_j - \\partial _j \\hat {A}_i ) ( \\partial _j \\hat {A}_i - \\partial _i \\hat {A}_j ) + O(\\alpha '^2). \\end {equation}",
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"\\begin{align*} \\displaystyle \\mitG _ { \\mits } & = & \\displaystyle \\mitg _ { \\mits } + \\mitO ( \\mitalpha ^ { \\prime } ) , \\\\ \\displaystyle \\hat { \\mitA } _ { \\miti } & = & \\displaystyle \\mitA _ { \\miti } + \\mitO ( \\mitalpha ^ { \\prime } ) . \\end{align*}",
"\\begin{align*} \\displaystyle \\mitG _ { \\mits } & = & \\displaystyle \\mitt \\mitg _ { \\mits } + \\mitO ( \\mitalpha ^ { \\prime } ) , \\\\ \\displaystyle \\hat { \\mitA } _ { \\miti } & = & \\displaystyle \\sqrt { \\mitt } \\mitA _ { \\miti } + \\mitO ( \\mitalpha ^ { \\prime } ) , \\end{align*}",
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"\\begin{equation*} \\mitf ( \\mitB + \\mitF ) = \\mitf ( \\mitF ) + \\mathrm { t o t a l ~ d e r i v a t i v e } , \\end{equation*}"
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0002194_page12 | {
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"$\\alpha '$",
"${\\rm Tr} F^2$",
"$F^4$",
"$\\alpha '$",
"$N$",
"$O(B, \\zeta ^3, k^3)$",
"${\\cal L} (B+F)$",
"$O(B, \\zeta ^3, k^3)$",
"${\\rm Tr} (B+F)^4$",
"$[{\\rm Tr} (B+F)^2]^2$",
"\\begin {eqnarray} && {\\rm Tr} (B+F)^2 \\\\ &=& {\\rm Tr} F^2 +2 {\\rm Tr} BF + {\\rm Tr} B^2 \\\\ &=& {\\rm Tr} F^2 + {\\rm total~derivative} + {\\rm const}. \\end {eqnarray}",
"\\begin {eqnarray} && \\frac {\\sqrt {\\det G}}{G_s} (G^{-1})^{ij} \\hat {F}_{jk} \\ast (G^{-1})^{kl} \\hat {F}_{li} \\\\ &=& \\frac {\\sqrt {\\det g}}{G_s} \\Biggl [ ( \\partial _i \\hat {A}_j - \\partial _j \\hat {A}_i ) ( \\partial _j \\hat {A}_i - \\partial _i \\hat {A}_j ) -4 (2 \\pi \\alpha ')^2 B_{kl} \\partial _k \\hat {A}_i \\partial _l \\hat {A}_j \\partial _j \\hat {A}_i \\\\ && +2 (2 \\pi \\alpha ')^2 (B^2)_{ij} ( \\partial _j \\hat {A}_k - \\partial _k \\hat {A}_j ) ( \\partial _k \\hat {A}_i - \\partial _i \\hat {A}_k ) \\\\ && -\\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 ( \\partial _i \\hat {A}_j - \\partial _j \\hat {A}_i ) ( \\partial _j \\hat {A}_i - \\partial _i \\hat {A}_j ) + O(\\alpha '^4) \\Biggr ]. \\end {eqnarray}",
"\\begin {eqnarray} A^{{\\rm asym}~a}_i (x) = \\zeta ^a_i e^{i k^a \\cdot x}, \\qquad a= 1, 2, \\ldots , N, \\\\ ( k^a )^2 =0, \\quad \\zeta ^a \\cdot k^a =0, \\quad \\sum _{a=1}^{N} k^a_i =0, \\end {eqnarray}",
"\\begin {eqnarray} {\\rm Tr} (B+F)^4 &=& {\\rm Tr} F^4 + 4 {\\rm Tr} BF^3 + O(B^2), \\\\ \\left [{\\rm Tr} (B+F)^2 \\right ]^2 &=& ( {\\rm Tr} F^2 )^2 + 4 {\\rm Tr} BF {\\rm Tr} F^2 + O(B^2). \\end {eqnarray}"
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"$ L ( B + F ) $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 3 } ) $",
"$ T r ( B + F ) ^ { 4 } $",
"$ [ T r ( B + F ) ^ { 2 } ] ^ { 2 } $",
"\\begin{align*} & & T r ( B + F ) ^ { 2 } \\\\ & = & T r F ^ { 2 } + 2 T r B F + T r B ^ { 2 } \\\\ & = & T r F ^ { 2 } + t o t a l ~ d e r i v a t i v e + c o n s t . \\end{align*}",
"\\begin{align*} & & \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } ( G ^ { - 1 } ) ^ { i j } \\hat { F } _ { j k } \\ast ( G ^ { - 1 } ) ^ { k l } \\hat { F } _ { l i } \\\\ & = & \\frac { \\sqrt { \\operatorname { d e t } g } } { G _ { s } } [ ( \\partial _ { i } \\hat { A } _ { j } - \\partial _ { j } \\hat { A } _ { i } ) ( \\partial _ { j } \\hat { A } _ { i } - \\partial _ { i } \\hat { A } _ { j } ) - 4 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\partial _ { k } \\hat { A } _ { i } \\partial _ { l } \\hat { A } _ { j } \\partial _ { j } \\hat { A } _ { i } \\\\ & & + 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( B ^ { 2 } ) _ { i j } ( \\partial _ { j } \\hat { A } _ { k } - \\partial _ { k } \\hat { A } _ { j } ) ( \\partial _ { k } \\hat { A } _ { i } - \\partial _ { i } \\hat { A } _ { k } ) \\\\ & & - \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } ( \\partial _ { i } \\hat { A } _ { j } - \\partial _ { j } \\hat { A } _ { i } ) ( \\partial _ { j } \\hat { A } _ { i } - \\partial _ { i } \\hat { A } _ { j } ) + O ( \\alpha ^ { \\prime 4 } ) ] . \\end{align*}",
"\\begin{align*} A _ { i } ^ { a s y m ~ a } ( x ) = \\zeta _ { i } ^ { a } e ^ { i k ^ { a } \\cdot x } , \\qquad a = 1 , 2 , \\ldots , N , \\\\ ( k ^ { a } ) ^ { 2 } = 0 , \\quad \\zeta ^ { a } \\cdot k ^ { a } = 0 , \\quad \\sum _ { a = 1 } ^ { N } k _ { i } ^ { a } = 0 , \\end{align*}",
"\\begin{align*} T r ( B + F ) ^ { 4 } & = & T r F ^ { 4 } + 4 T r B F ^ { 3 } + O ( B ^ { 2 } ) , \\\\ { [ T r ( B + F ) ^ { 2 } ] } ^ { 2 } & = & ( T r F ^ { 2 } ) ^ { 2 } + 4 T r B F T r F ^ { 2 } + O ( B ^ { 2 } ) . \\end{align*}"
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"$ \\mathrm { T r } \\mitF ^ { 2 } $",
"$ \\mitF ^ { 4 } $",
"$ \\mitalpha ^ { \\prime } $",
"$ \\mitN $",
"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 3 } ) $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 3 } ) $",
"$ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } $",
"$ [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } ] ^ { 2 } $",
"\\begin{align*} & & \\displaystyle \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\\\ & = & \\displaystyle \\mathrm { T r } \\mitF ^ { 2 } + 2 \\mathrm { T r } \\mitB \\mitF + \\mathrm { T r } \\mitB ^ { 2 } \\\\ & = & \\displaystyle \\mathrm { T r } \\mitF ^ { 2 } + \\mathrm { t o t a l ~ d e r i v a t i v e } + \\mathrm { c o n s t } . \\end{align*}",
"\\begin{align*} & & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } ( \\mitG ^ { - 1 } ) ^ { \\miti \\mitj } \\hat { \\mitF } _ { \\mitj \\mitk } \\ast ( \\mitG ^ { - 1 } ) ^ { \\mitk \\mitl } \\hat { \\mitF } _ { \\mitl \\miti } \\\\ & = & \\displaystyle \\frac { \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitG _ { \\mits } } \\Bigg [ ( \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } - \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } ) ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } ) - 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } \\mitpartial _ { \\mitl } \\hat { \\mitA } _ { \\mitj } \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } \\\\ & & \\displaystyle + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mitB ^ { 2 } ) _ { \\miti \\mitj } ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\mitk } - \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\mitj } ) ( \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitk } ) \\\\ & & \\displaystyle - \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } ( \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } - \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } ) ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } ) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) \\Bigg ] . \\end{align*}",
"\\begin{align*} \\displaystyle \\mitA _ { \\miti } ^ { \\mathrm { a s y m } ~ \\mita } ( \\mitx ) = \\mitzeta _ { \\miti } ^ { \\mita } \\mite ^ { \\miti \\mitk ^ { \\mita } \\cdot \\mitx } , \\qquad \\mita = 1 , 2 , \\ldots , \\mitN , \\\\ \\displaystyle ( \\mitk ^ { \\mita } ) ^ { 2 } = 0 , \\quad \\mitzeta ^ { \\mita } \\cdot \\mitk ^ { \\mita } = 0 , \\quad \\sum _ { \\mita = 1 } ^ { \\mitN } \\mitk _ { \\miti } ^ { \\mita } = 0 , \\end{align*}",
"\\begin{align*} \\displaystyle \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } & = & \\displaystyle \\mathrm { T r } \\mitF ^ { 4 } + 4 \\mathrm { T r } \\mitB \\mitF ^ { 3 } + \\mitO ( \\mitB ^ { 2 } ) , \\\\ \\displaystyle { \\left[ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } \\right] } ^ { 2 } & = & \\displaystyle ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } + 4 \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } + \\mitO ( \\mitB ^ { 2 } ) . \\end{align*}"
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|
0002194_page13 | {
"latex": [
"${\\rm Tr} BF^3$",
"${\\rm Tr} BF {\\rm Tr} F^2$",
"$F_{ij} = \\partial _i A_j - \\partial _j A_i$",
"$F_{ij} = \\partial _i A_j - \\partial _j A_i$",
"$\\partial ^2 A_i$",
"$\\partial _i A_i$",
"$k^2 =0$",
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"$h$",
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"$( k^a )^2 =0$",
"$f \\partial _i g \\partial _i h$",
"${\\rm Tr} BF^3$",
"${\\rm Tr} BF {\\rm Tr} F^2$",
"$O(B, \\zeta ^3, k^3)$",
"\\begin {equation} 0 = ( k^3 )^2 = ( k^1 + k^2 )^2 = 2 k^1 \\cdot k^2, \\end {equation}",
"\\begin {equation} f \\partial _i g \\partial _i h = \\frac {1}{2} (\\partial ^2 f g h - f \\partial ^2 g h - f g \\partial ^2 h) + \\frac {1}{2} \\partial ^2 (f g h) - \\partial _i ( \\partial _i f g h). \\label {formula} \\end {equation}",
"\\begin {eqnarray} {\\rm Tr} BF^3 &=& B_{ij} F_{jk} F_{kl} F_{li} \\\\ &=& 2 B_{ij} \\partial _j A_k \\partial _k A_l \\partial _l A_i -2 B_{ij} \\partial _j A_k \\partial _k A_l \\partial _i A_l \\\\ && + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative}, \\\\ {\\rm Tr} BF {\\rm Tr} F^2 &=& B_{ij} F_{ji} F_{kl} F_{lk} \\\\ &=& 4 B_{ij} \\partial _j A_i \\partial _k A_l \\partial _l A_k \\\\ &=& -8 B_{ij} A_i \\partial _k A_l \\partial _j \\partial _l A_k + {\\rm total~derivative} \\\\ &=& 8 B_{ij} \\partial _l A_i \\partial _k A_l \\partial _j A_k + {\\rm a~term~with~} \\partial _l A_l + {\\rm total~derivative}. \\end {eqnarray}",
"$$ {\\rm Tr} ( G^{-1} \\hat {F} \\ast G^{-1} \\hat {F} ) \\to -4 (2 \\pi \\alpha ')^2 B_{kl} \\partial _k \\hat {A}_i \\partial _l \\hat {A}_j \\partial _j \\hat {A}_i, $$",
"\\begin {eqnarray*} {\\rm Tr} BF^3 &\\to & 2 B_{ij} \\partial _j A_k \\partial _k A_l \\partial _l A_i -2 B_{ij} \\partial _i A_l \\partial _j A_k \\partial _k A_l, \\\\ {\\rm Tr} BF {\\rm Tr} F^2 &\\to & 8 B_{ij} \\partial _j A_k \\partial _k A_l \\partial _l A_i. \\end {eqnarray*}"
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"$ F _ { i j } = \\partial _ { i } A _ { j } - \\partial _ { j } A _ { i } $",
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"$ ( k ^ { a } ) ^ { 2 } = 0 $",
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"$ T r B F T r F ^ { 2 } $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 3 } ) $",
"\\begin{equation*} 0 = ( k ^ { 3 } ) ^ { 2 } = ( k ^ { 1 } + k ^ { 2 } ) ^ { 2 } = 2 k ^ { 1 } \\cdot k ^ { 2 } , \\end{equation*}",
"\\begin{equation*} f \\partial _ { i } g \\partial _ { i } h = \\frac { 1 } { 2 } ( \\partial ^ { 2 } f g h - f \\partial ^ { 2 } g h - f g \\partial ^ { 2 } h ) + \\frac { 1 } { 2 } \\partial ^ { 2 } ( f g h ) - \\partial _ { i } ( \\partial _ { i } f g h ) . \\end{equation*}",
"\\begin{align*} T r B F ^ { 3 } & = & B _ { i j } F _ { j k } F _ { k l } F _ { l i } \\\\ & = & 2 B _ { i j } \\partial _ { j } A _ { k } \\partial _ { k } A _ { l } \\partial _ { l } A _ { i } - 2 B _ { i j } \\partial _ { j } A _ { k } \\partial _ { k } A _ { l } \\partial _ { i } A _ { l } \\\\ & & + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e , \\\\ T r B F T r F ^ { 2 } & = & B _ { i j } F _ { j i } F _ { k l } F _ { l k } \\\\ & = & 4 B _ { i j } \\partial _ { j } A _ { i } \\partial _ { k } A _ { l } \\partial _ { l } A _ { k } \\\\ & = & - 8 B _ { i j } A _ { i } \\partial _ { k } A _ { l } \\partial _ { j } \\partial _ { l } A _ { k } + t o t a l ~ d e r i v a t i v e \\\\ & = & 8 B _ { i j } \\partial _ { l } A _ { i } \\partial _ { k } A _ { l } \\partial _ { j } A _ { k } + a ~ t e r m ~ w i t h ~ \\partial _ { l } A _ { l } + t o t a l ~ d e r i v a t i v e . \\end{align*}",
"\\begin{equation*} T r ( G ^ { - 1 } \\hat { F } \\ast G ^ { - 1 } \\hat { F } ) \\rightarrow - 4 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\partial _ { k } \\hat { A } _ { i } \\partial _ { l } \\hat { A } _ { j } \\partial _ { j } \\hat { A } _ { i } , \\end{equation*}",
"\\begin{align*} T r B F ^ { 3 } & \\rightarrow & 2 B _ { i j } \\partial _ { j } A _ { k } \\partial _ { k } A _ { l } \\partial _ { l } A _ { i } - 2 B _ { i j } \\partial _ { i } A _ { l } \\partial _ { j } A _ { k } \\partial _ { k } A _ { l } , \\\\ T r B F T r F ^ { 2 } & \\rightarrow & 8 B _ { i j } \\partial _ { j } A _ { k } \\partial _ { k } A _ { l } \\partial _ { l } A _ { i } . \\end{align*}"
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"$ \\mathrm { T r } \\mitB \\mitF ^ { 3 } $",
"$ \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } $",
"$ \\mitF _ { \\miti \\mitj } = \\mitpartial _ { \\miti } \\mitA _ { \\mitj } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } $",
"$ \\mitF _ { \\miti \\mitj } = \\mitpartial _ { \\miti } \\mitA _ { \\mitj } - \\mitpartial _ { \\mitj } \\mitA _ { \\miti } $",
"$ \\mitpartial ^ { 2 } \\mitA _ { \\miti } $",
"$ \\mitpartial _ { \\miti } \\mitA _ { \\miti } $",
"$ \\mitk ^ { 2 } = 0 $",
"$ \\mitzeta \\cdot \\mitk = 0 $",
"$ \\mitf \\mitpartial _ { \\miti } \\mitg \\mitpartial _ { \\miti } \\Planckconst $",
"$ \\mitf $",
"$ \\mitg $",
"$ \\Planckconst $",
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"$ \\mathrm { T r } \\mitB \\mitF ^ { 3 } $",
"$ \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } $",
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"\\begin{equation*} \\mitf \\mitpartial _ { \\miti } \\mitg \\mitpartial _ { \\miti } \\Planckconst = \\frac { 1 } { 2 } ( \\mitpartial ^ { 2 } \\mitf \\mitg \\Planckconst - \\mitf \\mitpartial ^ { 2 } \\mitg \\Planckconst - \\mitf \\mitg \\mitpartial ^ { 2 } \\Planckconst ) + \\frac { 1 } { 2 } \\mitpartial ^ { 2 } ( \\mitf \\mitg \\Planckconst ) - \\mitpartial _ { \\miti } ( \\mitpartial _ { \\miti } \\mitf \\mitg \\Planckconst ) . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mathrm { T r } \\mitB \\mitF ^ { 3 } & = & \\displaystyle \\mitB _ { \\miti \\mitj } \\mitF _ { \\mitj \\mitk } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\miti } \\\\ & = & \\displaystyle 2 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitl } \\mitA _ { \\miti } - 2 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\miti } \\mitA _ { \\mitl } \\\\ & & \\displaystyle + ~ \\mathrm { t e r m s } ~ \\mathrm { w i t h } ~ \\mitpartial ^ { 2 } \\mitA + \\mathrm { t o t a l ~ d e r i v a t i v e } , \\\\ \\displaystyle \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } & = & \\displaystyle \\mitB _ { \\miti \\mitj } \\mitF _ { \\mitj \\miti } \\mitF _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitk } \\\\ & = & \\displaystyle 4 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\miti } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitl } \\mitA _ { \\mitk } \\\\ & = & \\displaystyle - 8 \\mitB _ { \\miti \\mitj } \\mitA _ { \\miti } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitj } \\mitpartial _ { \\mitl } \\mitA _ { \\mitk } + \\mathrm { t o t a l ~ d e r i v a t i v e } \\\\ & = & \\displaystyle 8 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitl } \\mitA _ { \\miti } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } + \\mathrm { a } ~ \\mathrm { t e r m } ~ \\mathrm { w i t h } ~ \\mitpartial _ { \\mitl } \\mitA _ { \\mitl } + \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{align*}",
"\\begin{equation*} \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\ast \\mitG ^ { - 1 } \\hat { \\mitF } ) \\rightarrow - 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } \\mitpartial _ { \\mitl } \\hat { \\mitA } _ { \\mitj } \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mathrm { T r } \\mitB \\mitF ^ { 3 } & \\displaystyle \\rightarrow & \\displaystyle 2 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitl } \\mitA _ { \\miti } - 2 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\miti } \\mitA _ { \\mitl } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } , \\\\ \\displaystyle \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } & \\displaystyle \\rightarrow & \\displaystyle 8 \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitl } \\mitA _ { \\miti } . \\end{align*}"
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0002194_page14 | {
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"$B_{ij} \\partial _j A_k \\partial _k A_l \\partial _l A_i$",
"$B_{ij} \\partial _i A_l \\partial _j A_k \\partial _k A_l$",
"$\\hat {{\\cal L}} (\\hat {F})$",
"${\\cal L} (B+F)$",
"${\\cal L} (F)$",
"${\\cal L} (B+F)$",
"$\\alpha '$",
"$p$",
"$F^4$",
"$\\hat {A}_i$",
"$A_i$",
"\\begin {equation} 2 (2 \\pi \\alpha ')^2 {\\rm Tr} BF^3 -\\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} BF {\\rm Tr} F^2. \\label {BF^3} \\end {equation}",
"\\begin {eqnarray} {\\cal L}(F) = \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ {\\rm Tr} F^2 + (2 \\pi \\alpha ')^2 \\left [ \\frac {1}{2} {\\rm Tr} F^4 -\\frac {1}{8} ({\\rm Tr} F^2)^2 \\right ] \\\\ + ~O(\\alpha '^4) + {\\rm derivative~corrections~} \\Biggr ]. \\end {eqnarray}",
"\\begin {equation} {\\cal L}_{DBI} (F) = \\frac {1}{g_s (2 \\pi )^p (\\alpha ')^{(p+1)/2}} \\sqrt {\\det (g + 2 \\pi \\alpha ' F)}, \\label {DBI} \\end {equation}",
"\\begin {eqnarray} {\\cal L}(B+F) = \\frac {\\sqrt {\\det g}}{g_s} \\Biggl [ {\\rm Tr} (B+F)^2 + (2 \\pi \\alpha ')^2 \\left [ \\frac {1}{2} {\\rm Tr} (B+F)^4 -\\frac {1}{8} [ {\\rm Tr} (B+F)^2 ]^2 \\right ] \\\\ + ~O(\\alpha '^4) + {\\rm derivative~corrections~} \\Biggr ], \\end {eqnarray}",
"\\begin {eqnarray} \\hat {{\\cal L}}(\\hat {F}) = \\frac {\\sqrt {\\det G}}{G_s} \\Biggl [ {\\rm Tr} ( G^{-1} \\hat {F} \\ast G^{-1} \\hat {F} ) + (2 \\pi \\alpha ')^2 \\biggl [ \\frac {1}{2} {\\rm Tr} ( G^{-1} \\hat {F} )^4_{\\rm arbitrary} \\\\ -\\frac {1}{8} ( {\\rm Tr} ( G^{-1} \\hat {F} )^2 )^2_{\\rm arbitrary} \\biggr ] + ~O(\\alpha '^4) + {\\rm derivative~corrections~} \\Biggr ], \\end {eqnarray}"
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"$ \\hat { L } ( \\hat { F } ) $",
"$ L ( B + F ) $",
"$ L ( F ) $",
"$ L ( B + F ) $",
"$ \\alpha ^ { \\prime } $",
"$ p $",
"$ F ^ { 4 } $",
"$ \\hat { A } _ { i } $",
"$ A _ { i } $",
"\\begin{equation*} 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F T r F ^ { 2 } . \\end{equation*}",
"\\begin{align*} L ( F ) = \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ T r F ^ { 2 } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r F ^ { 4 } - \\frac { 1 } { 8 } ( T r F ^ { 2 } ) ^ { 2 } ] \\\\ + ~ O ( \\alpha ^ { \\prime 4 } ) + d e r i v a t i v e ~ c o r r e c t i o n s ~ ] . \\end{align*}",
"\\begin{equation*} L _ { D B I } ( F ) = \\frac { 1 } { g _ { s } ( 2 \\pi ) ^ { p } ( \\alpha ^ { \\prime } ) ^ { ( p + 1 ) \\slash 2 } } \\sqrt { \\operatorname { d e t } ( g + 2 \\pi \\alpha ^ { \\prime } F ) } , \\end{equation*}",
"\\begin{align*} L ( B + F ) = \\frac { \\sqrt { \\operatorname { d e t } g } } { g _ { s } } [ T r ( B + F ) ^ { 2 } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r ( B + F ) ^ { 4 } - \\frac { 1 } { 8 } [ T r ( B + F ) ^ { 2 } ] ^ { 2 } ] \\\\ + ~ O ( \\alpha ^ { \\prime 4 } ) + d e r i v a t i v e ~ c o r r e c t i o n s ~ ] , \\end{align*}",
"\\begin{align*} \\hat { L } ( \\hat { F } ) = \\frac { \\sqrt { \\operatorname { d e t } G } } { G _ { s } } [ T r ( G ^ { - 1 } \\hat { F } \\ast G ^ { - 1 } \\hat { F } ) + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ \\frac { 1 } { 2 } T r ( G ^ { - 1 } \\hat { F } ) _ { a r b i t r a r y } ^ { 4 } \\\\ - \\frac { 1 } { 8 } ( T r ( G ^ { - 1 } \\hat { F } ) ^ { 2 } ) _ { a r b i t r a r y } ^ { 2 } ] + ~ O ( \\alpha ^ { \\prime 4 } ) + d e r i v a t i v e ~ c o r r e c t i o n s ~ ] , \\end{align*}"
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"$ \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } \\mitpartial _ { \\mitl } \\mitA _ { \\miti } $",
"$ \\mitB _ { \\miti \\mitj } \\mitpartial _ { \\miti } \\mitA _ { \\mitl } \\mitpartial _ { \\mitj } \\mitA _ { \\mitk } \\mitpartial _ { \\mitk } \\mitA _ { \\mitl } $",
"$ \\hat { \\mitL } ( \\hat { \\mitF } ) $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\mitL ( \\mitF ) $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\mitalpha ^ { \\prime } $",
"$ \\mitp $",
"$ \\mitF ^ { 4 } $",
"$ \\hat { \\mitA } _ { \\miti } $",
"$ \\mitA _ { \\miti } $",
"\\begin{equation*} 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitL ( \\mitF ) = \\frac { \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\Bigg [ \\mathrm { T r } \\mitF ^ { 2 } + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 8 } ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\right] \\\\ \\displaystyle + ~ \\mitO ( \\mitalpha ^ { \\prime 4 } ) + \\mathrm { d e r i v a t i v e ~ c o r r e c t i o n s } ~ \\Bigg ] . \\end{align*}",
"\\begin{equation*} \\mitL _ { \\mitD \\mitB \\mitI } ( \\mitF ) = \\frac { 1 } { \\mitg _ { \\mits } ( 2 \\mitpi ) ^ { \\mitp } ( \\mitalpha ^ { \\prime } ) ^ { ( \\mitp + 1 ) \\slash 2 } } \\sqrt { \\operatorname { d e t } ( \\mitg + 2 \\mitpi \\mitalpha ^ { \\prime } \\mitF ) } , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitL ( \\mitB + \\mitF ) = \\frac { \\sqrt { \\operatorname { d e t } \\mitg } } { \\mitg _ { \\mits } } \\Bigg [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\left[ \\frac { 1 } { 2 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } - \\frac { 1 } { 8 } [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } ] ^ { 2 } \\right] \\\\ \\displaystyle + ~ \\mitO ( \\mitalpha ^ { \\prime 4 } ) + \\mathrm { d e r i v a t i v e ~ c o r r e c t i o n s } ~ \\Bigg ] , \\end{align*}",
"\\begin{align*} \\displaystyle \\hat { \\mitL } ( \\hat { \\mitF } ) = \\frac { \\sqrt { \\operatorname { d e t } \\mitG } } { \\mitG _ { \\mits } } \\Bigg [ \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } \\ast \\mitG ^ { - 1 } \\hat { \\mitF } ) + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\bigg [ \\frac { 1 } { 2 } \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) _ { \\mathrm { a r b i t r a r y } } ^ { 4 } \\\\ \\displaystyle - \\frac { 1 } { 8 } ( \\mathrm { T r } ( \\mitG ^ { - 1 } \\hat { \\mitF } ) ^ { 2 } ) _ { \\mathrm { a r b i t r a r y } } ^ { 2 } \\bigg ] + ~ \\mitO ( \\mitalpha ^ { \\prime 4 } ) + \\mathrm { d e r i v a t i v e ~ c o r r e c t i o n s } ~ \\Bigg ] , \\end{align*}"
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|
0002194_page15 | {
"latex": [
"$O(B, \\zeta ^3, k^3)$",
"$\\hat {F}^4$",
"$\\hat {{\\cal L}}(\\hat {F})$",
"$F^4$",
"${\\cal L}(B+F)$",
"$\\ast $",
"$O(\\alpha '^2)$",
"${\\cal L}(B+F)$",
"$O(B)$",
"$O(B^2)$",
"$\\hat {A}_i=A_i$",
"$O(B)$",
"$\\hat {A}_i$",
"$A_i$",
"\\begin {eqnarray} && \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} (B+F)^4 -\\frac {1}{8} (2 \\pi \\alpha ')^2 [ {\\rm Tr} (B+F)^2 ]^2 \\\\ &=& \\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} F^4 -\\frac {1}{8} (2 \\pi \\alpha ')^2 ( {\\rm Tr} F^2 )^2 \\\\ && +2 (2 \\pi \\alpha ')^2 {\\rm Tr} BF^3 -\\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} BF {\\rm Tr} F^2 \\\\ && +2 (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 F^2 -\\frac {1}{4} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 {\\rm Tr} F^2 + {\\rm total~derivative} + {\\rm const.}, \\end {eqnarray}",
"\\begin {equation} (2 \\pi \\alpha ')^2 \\left [ {\\rm Tr} ( B F )^2 - \\frac {1}{2} ( {\\rm Tr} B F )^2 \\right ] = {\\rm total~derivative}. \\end {equation}",
"\\begin {eqnarray} \\Delta {\\cal L} &\\equiv & 2 (2 \\pi \\alpha ')^2 {\\rm Tr} BF^3 -\\frac {1}{2} (2 \\pi \\alpha ')^2 {\\rm Tr} BF {\\rm Tr} F^2 - \\Bigl ( -4 (2 \\pi \\alpha ')^2 B_{kl} \\partial _k A_i \\partial _l A_j \\partial _j A_i \\Bigr ) \\\\ &=& 2 (2 \\pi \\alpha ')^2 B_{kl} F_{lj} F_{ji} F_{ik} +2 (2 \\pi \\alpha ')^2 B_{kl} A_k \\partial _l F_{ij} F_{ji} +2 (2 \\pi \\alpha ')^2 B_{kl} \\partial _k A_i \\partial _l A_j F_{ji} \\\\ && + {\\rm ~total~derivative} \\\\ &=& 2 (2 \\pi \\alpha ')^2 B_{kl} F_{ji} ( F_{lj} F_{ik} + A_k \\partial _l F_{ij} + \\partial _k A_i \\partial _l A_j ) + {\\rm total~derivative}. \\end {eqnarray}",
"\\begin {eqnarray} && B_{kl} F_{ji} \\partial _i [ A_k (\\partial _l A_j + F_{lj})] \\\\ &=& B_{kl} F_{ji} [ \\partial _i A_k (\\partial _l A_j + F_{lj}) + A_k (\\partial _l \\partial _i A_j + \\partial _i F_{lj})] \\\\ &=& B_{kl} F_{ji} \\left [ (F_{ik} + \\partial _k A_i) (F_{lj} + \\partial _l A_j) + A_k \\left ( \\frac {1}{2} \\partial _l F_{ij} + \\partial _i F_{lj} \\right ) \\right ] \\\\ &=& B_{kl} F_{ji} ( F_{lj} F_{ik} + A_k \\partial _l F_{ij} + \\partial _k A_i \\partial _l A_j ), \\end {eqnarray}"
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"\\begin{align*} & & \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r ( B + F ) ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ T r ( B + F ) ^ { 2 } ] ^ { 2 } \\\\ & = & \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r F ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } ( T r F ^ { 2 } ) ^ { 2 } \\\\ & & + 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F T r F ^ { 2 } \\\\ & & + 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } F ^ { 2 } - \\frac { 1 } { 4 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } T r F ^ { 2 } + t o t a l ~ d e r i v a t i v e + c o n s t . , \\end{align*}",
"\\begin{equation*} ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } [ T r ( B F ) ^ { 2 } - \\frac { 1 } { 2 } ( T r B F ) ^ { 2 } ] = t o t a l ~ d e r i v a t i v e . \\end{equation*}",
"\\begin{align*} \\Delta L & \\equiv & 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B F T r F ^ { 2 } - ( - 4 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\partial _ { k } A _ { i } \\partial _ { l } A _ { j } \\partial _ { j } A _ { i } ) \\\\ & = & 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } F _ { l j } F _ { j i } F _ { i k } + 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } A _ { k } \\partial _ { l } F _ { i j } F _ { j i } + 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\partial _ { k } A _ { i } \\partial _ { l } A _ { j } F _ { j i } \\\\ & & + ~ t o t a l ~ d e r i v a t i v e \\\\ & = & 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } F _ { j i } ( F _ { l j } F _ { i k } + A _ { k } \\partial _ { l } F _ { i j } + \\partial _ { k } A _ { i } \\partial _ { l } A _ { j } ) + t o t a l ~ d e r i v a t i v e . \\end{align*}",
"\\begin{align*} & & B _ { k l } F _ { j i } \\partial _ { i } [ A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } [ \\partial _ { i } A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) + A _ { k } ( \\partial _ { l } \\partial _ { i } A _ { j } + \\partial _ { i } F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } [ ( F _ { i k } + \\partial _ { k } A _ { i } ) ( F _ { l j } + \\partial _ { l } A _ { j } ) + A _ { k } ( \\frac { 1 } { 2 } \\partial _ { l } F _ { i j } + \\partial _ { i } F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } ( F _ { l j } F _ { i k } + A _ { k } \\partial _ { l } F _ { i j } + \\partial _ { k } A _ { i } \\partial _ { l } A _ { j } ) , \\end{align*}"
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"\\begin{align*} & & \\displaystyle \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } [ \\mathrm { T r } ( \\mitB + \\mitF ) ^ { 2 } ] ^ { 2 } \\\\ & = & \\displaystyle \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitF ^ { 4 } - \\frac { 1 } { 8 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } ( \\mathrm { T r } \\mitF ^ { 2 } ) ^ { 2 } \\\\ & & \\displaystyle + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } \\\\ & & \\displaystyle + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } \\mitF ^ { 2 } - \\frac { 1 } { 4 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB ^ { 2 } \\mathrm { T r } \\mitF ^ { 2 } + \\mathrm { t o t a l ~ d e r i v a t i v e } + \\mathrm { c o n s t } . , \\end{align*}",
"\\begin{equation*} ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\left[ \\mathrm { T r } ( \\mitB \\mitF ) ^ { 2 } - \\frac { 1 } { 2 } ( \\mathrm { T r } \\mitB \\mitF ) ^ { 2 } \\right] = \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mupDelta \\mitL & \\displaystyle \\equiv & \\displaystyle 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF ^ { 3 } - \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mathrm { T r } \\mitB \\mitF \\mathrm { T r } \\mitF ^ { 2 } - \\Big ( - 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } \\mitpartial _ { \\mitj } \\mitA _ { \\miti } \\Big ) \\\\ & = & \\displaystyle 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitl \\mitj } \\mitF _ { \\mitj \\miti } \\mitF _ { \\miti \\mitk } + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitA _ { \\mitk } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitj \\miti } + 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } \\mitF _ { \\mitj \\miti } \\\\ & & \\displaystyle + ~ \\mathrm { t o t a l ~ d e r i v a t i v e } \\\\ & = & \\displaystyle 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } ( \\mitF _ { \\mitl \\mitj } \\mitF _ { \\miti \\mitk } + \\mitA _ { \\mitk } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } ) + \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{align*}",
"\\begin{align*} & & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } \\mitpartial _ { \\miti } [ \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) ] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } [ \\mitpartial _ { \\miti } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) + \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitpartial _ { \\miti } \\mitA _ { \\mitj } + \\mitpartial _ { \\miti } \\mitF _ { \\mitl \\mitj } ) ] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } \\left[ ( \\mitF _ { \\miti \\mitk } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } ) ( \\mitF _ { \\mitl \\mitj } + \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } ) + \\mitA _ { \\mitk } \\left( \\frac { 1 } { 2 } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } + \\mitpartial _ { \\miti } \\mitF _ { \\mitl \\mitj } \\right) \\right] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } ( \\mitF _ { \\mitl \\mitj } \\mitF _ { \\miti \\mitk } + \\mitA _ { \\mitk } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } ) , \\end{align*}"
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0002194_page16 | {
"latex": [
"$\\Delta {\\cal L}$",
"$\\Delta {\\cal L}$",
"$\\hat {A}_i$",
"$\\Delta {\\cal L}$",
"$\\partial _i F_{ij}$",
"$\\hat {A}_i$",
"$A_i$",
"$\\theta $",
"$B$",
"\\begin {eqnarray} && B_{kl} F_{ji} \\partial _i [ A_k (\\partial _l A_j + F_{lj})] \\\\ &=& B_{kl} F_{ji} [ \\partial _i A_k (\\partial _l A_j + F_{lj}) + A_k (\\partial _l \\partial _i A_j + \\partial _i F_{lj})] \\\\ &=& B_{kl} F_{ji} \\left [ (F_{ik} + \\partial _k A_i) (F_{lj} + \\partial _l A_j) + A_k \\left ( \\frac {1}{2} \\partial _l F_{ij} + \\partial _i F_{lj} \\right ) \\right ] \\\\ &=& B_{kl} F_{ji} ( F_{lj} F_{ik} + A_k \\partial _l F_{ij} + \\partial _k A_i \\partial _l A_j ), \\end {eqnarray}",
"\\begin {equation} F_{ji} \\partial _i F_{lj} = \\frac {1}{2} F_{ji} \\partial _l F_{ij}, \\end {equation}",
"\\begin {equation} B_{kl} F_{ji} (F_{ik} \\partial _l A_j + \\partial _k A_i F_{lj}) =0. \\end {equation}",
"\\begin {eqnarray} \\Delta {\\cal L} &=& 2 (2 \\pi \\alpha ')^2 B_{kl} F_{ji} \\partial _i [ A_k (\\partial _l A_j + F_{lj})] + {\\rm total~derivative} \\\\ &=& 2 (2 \\pi \\alpha ')^2 B_{kl} \\partial _i F_{ij} A_k (\\partial _l A_j + F_{lj}) + {\\rm total~derivative}. \\end {eqnarray}",
"\\begin {equation} \\hat {A}_i = A_i + (2 \\pi \\alpha ')^2 \\Delta A_i + O(\\alpha '^4), \\end {equation}",
"\\begin {equation} (\\partial _i \\hat {A}_j - \\partial _j \\hat {A}_i) (\\partial _j \\hat {A}_i - \\partial _i \\hat {A}_j) = F_{ij} F_{ji} + 4 (2 \\pi \\alpha ')^2 \\partial _i F_{ij} \\Delta A_j + O(\\alpha '^4). \\end {equation}",
"\\begin {equation} 4 (2 \\pi \\alpha ')^2 \\partial _i F_{ij} \\Delta A_j = \\Delta {\\cal L}. \\end {equation}",
"\\begin {equation} \\Delta A_i = \\frac {1}{2} B_{kl} A_k (\\partial _l A_i + F_{li}), \\end {equation}",
"\\begin {equation} \\hat {A}_i = A_i + \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{kl} A_k (\\partial _l A_i + F_{li}) + O(\\alpha '^4). \\label {redefinition-1} \\end {equation}"
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"latex_norm": [
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"$ \\Delta L $",
"$ \\hat { A } _ { i } $",
"$ \\Delta L $",
"$ \\partial _ { i } F _ { i j } $",
"$ \\hat { A } _ { i } $",
"$ A _ { i } $",
"$ \\theta $",
"$ B $",
"\\begin{align*} & & B _ { k l } F _ { j i } \\partial _ { i } [ A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } [ \\partial _ { i } A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) + A _ { k } ( \\partial _ { l } \\partial _ { i } A _ { j } + \\partial _ { i } F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } [ ( F _ { i k } + \\partial _ { k } A _ { i } ) ( F _ { l j } + \\partial _ { l } A _ { j } ) + A _ { k } ( \\frac { 1 } { 2 } \\partial _ { l } F _ { i j } + \\partial _ { i } F _ { l j } ) ] \\\\ & = & B _ { k l } F _ { j i } ( F _ { l j } F _ { i k } + A _ { k } \\partial _ { l } F _ { i j } + \\partial _ { k } A _ { i } \\partial _ { l } A _ { j } ) , \\end{align*}",
"\\begin{equation*} F _ { j i } \\partial _ { i } F _ { l j } = \\frac { 1 } { 2 } F _ { j i } \\partial _ { l } F _ { i j } , \\end{equation*}",
"\\begin{equation*} B _ { k l } F _ { j i } ( F _ { i k } \\partial _ { l } A _ { j } + \\partial _ { k } A _ { i } F _ { l j } ) = 0 . \\end{equation*}",
"\\begin{align*} \\Delta L & = & 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } F _ { j i } \\partial _ { i } [ A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) ] + t o t a l ~ d e r i v a t i v e \\\\ & = & 2 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } \\partial _ { i } F _ { i j } A _ { k } ( \\partial _ { l } A _ { j } + F _ { l j } ) + t o t a l ~ d e r i v a t i v e . \\end{align*}",
"\\begin{equation*} \\hat { A } _ { i } = A _ { i } + ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } \\Delta A _ { i } + O ( \\alpha ^ { \\prime 4 } ) , \\end{equation*}",
"\\begin{equation*} ( \\partial _ { i } \\hat { A } _ { j } - \\partial _ { j } \\hat { A } _ { i } ) ( \\partial _ { j } \\hat { A } _ { i } - \\partial _ { i } \\hat { A } _ { j } ) = F _ { i j } F _ { j i } + 4 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } \\partial _ { i } F _ { i j } \\Delta A _ { j } + O ( \\alpha ^ { \\prime 4 } ) . \\end{equation*}",
"\\begin{equation*} 4 ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } \\partial _ { i } F _ { i j } \\Delta A _ { j } = \\Delta L . \\end{equation*}",
"\\begin{equation*} \\Delta A _ { i } = \\frac { 1 } { 2 } B _ { k l } A _ { k } ( \\partial _ { l } A _ { i } + F _ { l i } ) , \\end{equation*}",
"\\begin{equation*} \\hat { A } _ { i } = A _ { i } + \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { k l } A _ { k } ( \\partial _ { l } A _ { i } + F _ { l i } ) + O ( \\alpha ^ { \\prime 4 } ) . \\end{equation*}"
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"$ \\mupDelta \\mitL $",
"$ \\hat { \\mitA } _ { \\miti } $",
"$ \\mupDelta \\mitL $",
"$ \\mitpartial _ { \\miti } \\mitF _ { \\miti \\mitj } $",
"$ \\hat { \\mitA } _ { \\miti } $",
"$ \\mitA _ { \\miti } $",
"$ \\mittheta $",
"$ \\mitB $",
"\\begin{align*} & & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } \\mitpartial _ { \\miti } [ \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) ] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } [ \\mitpartial _ { \\miti } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) + \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitpartial _ { \\miti } \\mitA _ { \\mitj } + \\mitpartial _ { \\miti } \\mitF _ { \\mitl \\mitj } ) ] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } \\left[ ( \\mitF _ { \\miti \\mitk } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } ) ( \\mitF _ { \\mitl \\mitj } + \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } ) + \\mitA _ { \\mitk } \\left( \\frac { 1 } { 2 } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } + \\mitpartial _ { \\miti } \\mitF _ { \\mitl \\mitj } \\right) \\right] \\\\ & = & \\displaystyle \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } ( \\mitF _ { \\mitl \\mitj } \\mitF _ { \\miti \\mitk } + \\mitA _ { \\mitk } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } ) , \\end{align*}",
"\\begin{equation*} \\mitF _ { \\mitj \\miti } \\mitpartial _ { \\miti } \\mitF _ { \\mitl \\mitj } = \\frac { 1 } { 2 } \\mitF _ { \\mitj \\miti } \\mitpartial _ { \\mitl } \\mitF _ { \\miti \\mitj } , \\end{equation*}",
"\\begin{equation*} \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } ( \\mitF _ { \\miti \\mitk } \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitpartial _ { \\mitk } \\mitA _ { \\miti } \\mitF _ { \\mitl \\mitj } ) = 0 . \\end{equation*}",
"\\begin{align*} \\displaystyle \\mupDelta \\mitL & = & \\displaystyle 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitF _ { \\mitj \\miti } \\mitpartial _ { \\miti } [ \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) ] + \\mathrm { t o t a l ~ d e r i v a t i v e } \\\\ & = & \\displaystyle 2 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitpartial _ { \\miti } \\mitF _ { \\miti \\mitj } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\mitj } + \\mitF _ { \\mitl \\mitj } ) + \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{align*}",
"\\begin{equation*} \\hat { \\mitA } _ { \\miti } = \\mitA _ { \\miti } + ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mupDelta \\mitA _ { \\miti } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) , \\end{equation*}",
"\\begin{equation*} ( \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } - \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } ) ( \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\miti } - \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } ) = \\mitF _ { \\miti \\mitj } \\mitF _ { \\mitj \\miti } + 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitpartial _ { \\miti } \\mitF _ { \\miti \\mitj } \\mupDelta \\mitA _ { \\mitj } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{equation*}",
"\\begin{equation*} 4 ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitpartial _ { \\miti } \\mitF _ { \\miti \\mitj } \\mupDelta \\mitA _ { \\mitj } = \\mupDelta \\mitL . \\end{equation*}",
"\\begin{equation*} \\mupDelta \\mitA _ { \\miti } = \\frac { 1 } { 2 } \\mitB _ { \\mitk \\mitl } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\miti } + \\mitF _ { \\mitl \\miti } ) , \\end{equation*}",
"\\begin{equation*} \\hat { \\mitA } _ { \\miti } = \\mitA _ { \\miti } + \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitk \\mitl } \\mitA _ { \\mitk } ( \\mitpartial _ { \\mitl } \\mitA _ { \\miti } + \\mitF _ { \\mitl \\miti } ) + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{equation*}"
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0002194_page17 | {
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"\\begin {equation} \\frac {1}{4} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 {\\rm Tr} F^2, \\label {superfluous-B} \\end {equation}",
"\\begin {equation} G_s = g_s \\left [ 1 -\\frac {1}{4} (2 \\pi \\alpha ')^2 {\\rm Tr} B^2 + O(\\alpha '^4) \\right ]. \\label {G_s} \\end {equation}"
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"\\begin{equation*} G _ { s } = g _ { s } [ 1 - \\frac { 1 } { 4 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } T r B ^ { 2 } + O ( \\alpha ^ { \\prime 4 } ) ] . \\end{equation*}"
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|
0002194_page18 | {
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"$F^3$",
"$F \\partial ^2 F$",
"$\\partial F \\partial F$",
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"$T_2$",
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"$\\hat {F} \\hat {D}^2 \\hat {F}$",
"$\\hat {F}^3$",
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"$\\hat {D} \\hat {F} \\hat {D} \\hat {F}$",
"$\\hat {F}^3$",
"$\\alpha '$",
"$G_{ij}$",
"\\begin {eqnarray} T_1 \\equiv \\partial _i F_{ik} \\partial _j F_{jk}, \\qquad T_2 \\equiv \\partial _j F_{ik} \\partial _i F_{jk}, \\qquad T_3 \\equiv \\partial _k F_{ij} \\partial _k F_{ji}. \\end {eqnarray}",
"\\begin {equation} T_3 = -2 T_2, \\end {equation}",
"\\begin {eqnarray} T_1 &=& - F_{ik} \\partial _i \\partial _j F_{jk} + {\\rm total~derivative}, \\\\ T_2 &=& - F_{ik} \\partial _j \\partial _i F_{jk} + {\\rm total~derivative}. \\end {eqnarray}",
"\\begin {equation} \\hat {D}_i \\hat {F}_{jk} = \\partial _i \\hat {F}_{jk} -i \\hat {A}_i \\ast \\hat {F}_{jk} +i \\hat {F}_{jk} \\ast \\hat {A}_i. \\end {equation}",
"\\begin {eqnarray} && \\hat {T}_1 \\equiv \\hat {D}_i \\hat {F}_{ik} \\ast \\hat {D}_j \\hat {F}_{jk}, \\qquad \\hat {T}_2 \\equiv \\hat {D}_j \\hat {F}_{ik} \\ast \\hat {D}_i \\hat {F}_{jk}, \\qquad \\hat {T}_3 \\equiv \\hat {D}_k \\hat {F}_{ij} \\ast \\hat {D}_k \\hat {F}_{ji}, \\\\ && \\hat {T}_4 \\equiv i \\hat {F}_{ij} \\ast \\hat {F}_{jk} \\ast \\hat {F}_{ki}, \\end {eqnarray}"
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"$ \\alpha ^ { \\prime } $",
"$ G _ { i j } $",
"\\begin{equation*} T _ { 1 } \\equiv \\partial _ { i } F _ { i k } \\partial _ { j } F _ { j k } , \\qquad T _ { 2 } \\equiv \\partial _ { j } F _ { i k } \\partial _ { i } F _ { j k } , \\qquad T _ { 3 } \\equiv \\partial _ { k } F _ { i j } \\partial _ { k } F _ { j i } . \\end{equation*}",
"\\begin{equation*} T _ { 3 } = - 2 T _ { 2 } , \\end{equation*}",
"\\begin{align*} T _ { 1 } & = & - F _ { i k } \\partial _ { i } \\partial _ { j } F _ { j k } + t o t a l ~ d e r i v a t i v e , \\\\ T _ { 2 } & = & - F _ { i k } \\partial _ { j } \\partial _ { i } F _ { j k } + t o t a l ~ d e r i v a t i v e . \\end{align*}",
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"\\begin{align*} & & \\hat { T } _ { 1 } \\equiv \\hat { D } _ { i } \\hat { F } _ { i k } \\ast \\hat { D } _ { j } \\hat { F } _ { j k } , \\qquad \\hat { T } _ { 2 } \\equiv \\hat { D } _ { j } \\hat { F } _ { i k } \\ast \\hat { D } _ { i } \\hat { F } _ { j k } , \\qquad \\hat { T } _ { 3 } \\equiv \\hat { D } _ { k } \\hat { F } _ { i j } \\ast \\hat { D } _ { k } \\hat { F } _ { j i } , \\\\ & & \\hat { T } _ { 4 } \\equiv i \\hat { F } _ { i j } \\ast \\hat { F } _ { j k } \\ast \\hat { F } _ { k i } , \\end{align*}"
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"$ \\hat { \\mitD } \\hat { \\mitF } \\hat { \\mitD } \\hat { \\mitF } $",
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"$ \\mitalpha ^ { \\prime } $",
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"\\begin{equation*} \\mitT _ { 3 } = - 2 \\mitT _ { 2 } , \\end{equation*}",
"\\begin{align*} \\displaystyle \\mitT _ { 1 } & = & \\displaystyle - \\mitF _ { \\miti \\mitk } \\mitpartial _ { \\miti } \\mitpartial _ { \\mitj } \\mitF _ { \\mitj \\mitk } + \\mathrm { t o t a l ~ d e r i v a t i v e } , \\\\ \\displaystyle \\mitT _ { 2 } & = & \\displaystyle - \\mitF _ { \\miti \\mitk } \\mitpartial _ { \\mitj } \\mitpartial _ { \\miti } \\mitF _ { \\mitj \\mitk } + \\mathrm { t o t a l ~ d e r i v a t i v e } . \\end{align*}",
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|
0002194_page19 | {
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"\\begin {eqnarray} \\hat {T}_1 &=& - \\hat {F}_{ik} \\ast \\hat {D}_i \\hat {D}_j \\hat {F}_{jk} + {\\rm total~derivative}, \\\\ \\hat {T}_2 &=& - \\hat {F}_{ik} \\ast \\hat {D}_j \\hat {D}_i \\hat {F}_{jk} + {\\rm total~derivative}, \\end {eqnarray}",
"\\begin {eqnarray} \\hat {T}_1 - \\hat {T}_2 &=& - \\hat {F}_{ik} \\ast [ \\hat {D}_i, \\hat {D}_j] \\hat {F}_{jk} + {\\rm total~derivative} \\\\ &=& - \\hat {F}_{ik} \\ast (-i \\hat {F}_{ij} \\ast \\hat {F}_{jk} +i \\hat {F}_{jk} \\ast \\hat {F}_{ij}) + {\\rm total~derivative} \\\\ &=& -2 \\hat {T}_4 + {\\rm total~derivative}. \\end {eqnarray}",
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0002194_page20 | {
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"\\begin {equation} \\tilde {A}_i = A_i + f_i (\\partial F, \\partial ^2 F, \\ldots ), \\end {equation}",
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0002194_page21 | {
"latex": [
"$\\hat {T}_4$",
"$O(B)$",
"$\\hat {T}_4$",
"$\\hat {T}_4$",
"$i \\, {\\rm tr} {\\rm Tr} F^3$",
"$\\hat {F}^2$",
"$O(B, \\zeta ^3, k^3)$",
"$F^4$",
"${\\cal L} (B+F)$",
"$\\hat {T}_4$",
"$\\alpha '$",
"$\\hat {T}_4$",
"$\\alpha '$",
"$O(B, \\zeta ^3, k^5)$",
"$O(\\partial ^2 F^4)$",
"$F$",
"$B+F$",
"$O(\\partial ^2 F^4)$",
"\\begin {eqnarray} i \\, {\\rm tr} {\\rm Tr} (B+F)^3 &=& \\frac {i}{2} {\\rm tr} (B+F)_{ij} [ (B+F)_{jk}, (B+F)_{ki} ] \\\\ &=& \\frac {i}{2} {\\rm tr} F_{ij} [ F_{jk}, F_{ki} ] + \\frac {i}{2} B_{ij} {\\rm tr} [ F_{jk}, F_{ki} ] \\\\ &=& i \\, {\\rm tr} {\\rm Tr} F^3. \\end {eqnarray}",
"\\begin {equation} \\hat {T}_4 = \\frac {1}{2} (2 \\pi \\alpha ')^2 B_{nm} \\hat {F}_{ij} \\partial _n \\hat {F}_{jk} \\partial _m \\hat {F}_{ki} + O(\\alpha '^4). \\label {T_4-expansion} \\end {equation}",
"\\begin {eqnarray} \\hat {T}_4 &=& (2 \\pi \\alpha ')^2 B_{nm} \\partial _i \\hat {A}_j \\partial _n \\partial _j \\hat {A}_k \\partial _m \\partial _k \\hat {A}_i \\\\ && + {\\rm ~terms~with~} \\partial ^2 A + {\\rm total~derivative} + O(\\alpha '^4). \\end {eqnarray}"
],
"latex_norm": [
"$ \\hat { T } _ { 4 } $",
"$ O ( B ) $",
"$ \\hat { T } _ { 4 } $",
"$ \\hat { T } _ { 4 } $",
"$ i \\, t r T r F ^ { 3 } $",
"$ \\hat { F } ^ { 2 } $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 3 } ) $",
"$ F ^ { 4 } $",
"$ L ( B + F ) $",
"$ \\hat { T } _ { 4 } $",
"$ \\alpha ^ { \\prime } $",
"$ \\hat { T } _ { 4 } $",
"$ \\alpha ^ { \\prime } $",
"$ O ( B , \\zeta ^ { 3 } , k ^ { 5 } ) $",
"$ O ( \\partial ^ { 2 } F ^ { 4 } ) $",
"$ F $",
"$ B + F $",
"$ O ( \\partial ^ { 2 } F ^ { 4 } ) $",
"\\begin{align*} i \\, t r T r ( B + F ) ^ { 3 } & = & \\frac { i } { 2 } t r ( B + F ) _ { i j } [ ( B + F ) _ { j k } , ( B + F ) _ { k i } ] \\\\ & = & \\frac { i } { 2 } t r F _ { i j } [ F _ { j k } , F _ { k i } ] + \\frac { i } { 2 } B _ { i j } t r [ F _ { j k } , F _ { k i } ] \\\\ & = & i \\, t r T r F ^ { 3 } . \\end{align*}",
"\\begin{equation*} \\hat { T } _ { 4 } = \\frac { 1 } { 2 } ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { n m } \\hat { F } _ { i j } \\partial _ { n } \\hat { F } _ { j k } \\partial _ { m } \\hat { F } _ { k i } + O ( \\alpha ^ { \\prime 4 } ) . \\end{equation*}",
"\\begin{align*} \\hat { T } _ { 4 } & = & ( 2 \\pi \\alpha ^ { \\prime } ) ^ { 2 } B _ { n m } \\partial _ { i } \\hat { A } _ { j } \\partial _ { n } \\partial _ { j } \\hat { A } _ { k } \\partial _ { m } \\partial _ { k } \\hat { A } _ { i } \\\\ & & + ~ t e r m s ~ w i t h ~ \\partial ^ { 2 } A + t o t a l ~ d e r i v a t i v e + O ( \\alpha ^ { \\prime 4 } ) . \\end{align*}"
],
"latex_expand": [
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\mitO ( \\mitB ) $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\miti \\, \\mathrm { t r T r } \\mitF ^ { 3 } $",
"$ \\hat { \\mitF } ^ { 2 } $",
"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 3 } ) $",
"$ \\mitF ^ { 4 } $",
"$ \\mitL ( \\mitB + \\mitF ) $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\mitalpha ^ { \\prime } $",
"$ \\hat { \\mitT } _ { 4 } $",
"$ \\mitalpha ^ { \\prime } $",
"$ \\mitO ( \\mitB , \\mitzeta ^ { 3 } , \\mitk ^ { 5 } ) $",
"$ \\mitO ( \\mitpartial ^ { 2 } \\mitF ^ { 4 } ) $",
"$ \\mitF $",
"$ \\mitB + \\mitF $",
"$ \\mitO ( \\mitpartial ^ { 2 } \\mitF ^ { 4 } ) $",
"\\begin{align*} \\displaystyle \\miti \\, \\mathrm { t r T r } ( \\mitB + \\mitF ) ^ { 3 } & = & \\displaystyle \\frac { \\miti } { 2 } \\mathrm { t r } ( \\mitB + \\mitF ) _ { \\miti \\mitj } [ ( \\mitB + \\mitF ) _ { \\mitj \\mitk } , ( \\mitB + \\mitF ) _ { \\mitk \\miti } ] \\\\ & = & \\displaystyle \\frac { \\miti } { 2 } \\mathrm { t r } \\mitF _ { \\miti \\mitj } [ \\mitF _ { \\mitj \\mitk } , \\mitF _ { \\mitk \\miti } ] + \\frac { \\miti } { 2 } \\mitB _ { \\miti \\mitj } \\mathrm { t r } [ \\mitF _ { \\mitj \\mitk } , \\mitF _ { \\mitk \\miti } ] \\\\ & = & \\displaystyle \\miti \\, \\mathrm { t r T r } \\mitF ^ { 3 } . \\end{align*}",
"\\begin{equation*} \\hat { \\mitT } _ { 4 } = \\frac { 1 } { 2 } ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitn \\mitm } \\hat { \\mitF } _ { \\miti \\mitj } \\mitpartial _ { \\mitn } \\hat { \\mitF } _ { \\mitj \\mitk } \\mitpartial _ { \\mitm } \\hat { \\mitF } _ { \\mitk \\miti } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{equation*}",
"\\begin{align*} \\displaystyle \\hat { \\mitT } _ { 4 } & = & \\displaystyle ( 2 \\mitpi \\mitalpha ^ { \\prime } ) ^ { 2 } \\mitB _ { \\mitn \\mitm } \\mitpartial _ { \\miti } \\hat { \\mitA } _ { \\mitj } \\mitpartial _ { \\mitn } \\mitpartial _ { \\mitj } \\hat { \\mitA } _ { \\mitk } \\mitpartial _ { \\mitm } \\mitpartial _ { \\mitk } \\hat { \\mitA } _ { \\miti } \\\\ & & \\displaystyle + ~ \\mathrm { t e r m s } ~ \\mathrm { w i t h } ~ \\mitpartial ^ { 2 } \\mitA + \\mathrm { t o t a l ~ d e r i v a t i v e } + \\mitO ( \\mitalpha ^ { \\prime 4 } ) . \\end{align*}"
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