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$H$ | $ H $ | 0001015_page005 | embedded |
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$\pi $ | $ \pi $ | 0001015_page005 | embedded |
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$P^{^{-}}H=0, P^{^{-}}\pi =0$ | $ P ^ { { } ^ { - } } H = 0 , P ^ { { } ^ { - } } \pi = 0 $ | 0001015_page005 | embedded |
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$P^{^{-}}{}^{\alpha \mu }_{\beta \nu } =\delta ^{\alpha }_{\beta }\delta ^{\mu }_{\nu }- \epsilon ^{\alpha }_{\beta }J^{\mu }_{\nu }$ | $ P ^ { { } ^ { - } } { } _ { \beta \nu } ^ { \alpha \mu } = \delta _ { \beta } ^ { \alpha } \delta _ { \nu } ^ { \mu } - \epsilon _ { \beta } ^ { \alpha } J _ { \nu } ^ { \mu } $ | 0001015_page005 | embedded |
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$\epsilon $ | $ \epsilon $ | 0001015_page005 | embedded |
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$\Sigma $ | $ \Sigma $ | 0001015_page005 | embedded |
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$J$ | $ J $ | 0001015_page005 | embedded |
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\begin {eqnarray} \Xmu _t &=& \Xmu + \Intot e^{\mu }_{a,s}\circ db^{a}_s , \End e^{\mu }_{a,t}&=& e^{\mu }_{a} +\Intot -e^{\nu }_{a,s} e^{\lambda }_{b,s} \Gam {\nu }{\lambda }{\mu }(x_s) \circ db^{b}_s \End \Etamu _t &=& \Etamu + \theta ^{a}_t e^{\mu }_{a,t} \End +&&\!\!\!\!\!\!\!\!\! \Intot \big ( - \eta ^{\nu }_s \Gam {\nu }{\lambda }{\mu } e^{\lambda }_{b,s} \circ db^{b}_s - \theta ^a_t de^{\mu }_{a,s} +\frac {1}{4}\eta ^{\nu }_s \Curv {\nu }{\lambda }{\kappa }{\mu } (x_s)\eta ^{\lambda }_s\rho ^{a}_s e^{\kappa }_{a,s} ds \big ), \end {eqnarray} | \begin{align*} x _ { t } ^ { \mu } & = & x ^ { \mu } + \int _ { 0 } ^ { t } e _ { a , s } ^ { \mu } \circ d b _ { s } ^ { a } , \\ e _ { a , t } ^ { \mu } & = & e _ { a } ^ { \mu } + \int _ { 0 } ^ { t } - e _ { a , s } ^ { \nu } e _ { b , s } ^ { \lambda } \Gamma _ { \nu \lambda } ^ { \mu } ( x _ { s } ) \circ d b _ { s } ^ { b } \\ \eta _ { t } ^ { \mu } & = & \eta ^ { \mu } + \theta _ { t } ^ { a } e _ { a , t } ^ { \mu } \\ + & & \! \! \! \! \! \! \! \! \! \int _ { 0 } ^ { t } ( - \eta _ { s } ^ { \nu } \Gamma _ { \nu \lambda } ^ { \mu } e _ { b , s } ^ { \lambda } \circ d b _ { s } ^ { b } - \theta _ { t } ^ { a } d e _ { a , s } ^ { \mu } + \frac { 1 } { 4 } \eta _ { s } ^ { \nu } R _ { \nu \lambda \kappa } { } ^ { \mu } ( x _ { s } ) \eta _ { s } ^ { \lambda } \rho _ { s } ^ { a } e _ { a , s } ^ { \kappa } d s ) , \end{align*} | 0001015_page005 | isolated |
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$M$ | $ M $ | 0001015_page006 | embedded |
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$\delta u^{\mu } = i \epsilon \eta ^{\mu }$ | $ \delta u ^ { \mu } = i \epsilon \eta ^ { \mu } $ | 0001015_page006 | embedded |
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$J$ | $ J $ | 0001015_page006 | embedded |
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$M$ | $ M $ | 0001015_page006 | embedded |
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$u:\Sigma \to M$ | $ u : \Sigma \rightarrow M $ | 0001015_page006 | embedded |
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$p^{\alpha }_{\mu }$ | $ p _ { \mu } ^ { \alpha } $ | 0001015_page006 | embedded |
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$H^{\alpha }_{\mu }$ | $ H _ { \mu } ^ { \alpha } $ | 0001015_page006 | embedded |
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${\cal {P}}_{\mu }^{\alpha }$ | $ P _ { \mu } ^ { \alpha } $ | 0001015_page006 | embedded |
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$\pi ^{\alpha }_{\mu }$ | $ \pi _ { \mu } ^ { \alpha } $ | 0001015_page006 | embedded |
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$J$ | $ J $ | 0001015_page006 | embedded |
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$U(1)$ | $ U ( 1 ) $ | 0001073_page001 | embedded |
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$\theta $ | $ \theta $ | 0001073_page001 | embedded |
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$B$ | $ B $ | 0001073_page002 | embedded |
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$U(2)$ | $ U ( 2 ) $ | 0001073_page002 | embedded |
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$B$ | $ B $ | 0001073_page002 | embedded |
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$O(\theta )$ | $ O ( \theta ) $ | 0001073_page002 | embedded |
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$U(1)$ | $ U ( 1 ) $ | 0001073_page002 | embedded |
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$\phi $ | $ \phi $ | 0001073_page002 | embedded |
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$\mathbb {R}^3\setminus \{0\}$ | $ R ^ { 3 } \setminus \{ 0 \} $ | 0001073_page002 | embedded |
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$S^2$ | $ S ^ { 2 } $ | 0001073_page002 | embedded |
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$\pi _1(U(1))=\mathbb {Z}$ | $ \pi _ { 1 } ( U ( 1 ) ) = Z $ | 0001073_page002 | embedded |
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$m\propto 1/g_{\rm YM}$ | $ m \propto 1 \slash g _ { Y M } $ | 0001073_page002 | embedded |
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$O(\theta ^2)$ | $ O ( \theta ^ { 2 } ) $ | 0001073_page002 | embedded |
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$U(1)$ | $ U ( 1 ) $ | 0001073_page002 | embedded |
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$A$ | $ A $ | 0001073_page002 | embedded |
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$\theta $ | $ \theta $ | 0001073_page002 | embedded |
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$U(1)$ | $ U ( 1 ) $ | 0001073_page002 | embedded |
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$\star $ | $ \star $ | 0001073_page003 | embedded |
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$O(\theta ^2)$ | $ O ( \theta ^ { 2 } ) $ | 0001073_page003 | embedded |
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$F$ | $ F $ | 0001073_page003 | embedded |
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$i$ | $ i $ | 0001073_page003 | embedded |
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$j$ | $ j $ | 0001073_page003 | embedded |
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$O(\theta ^2)$ | $ O ( \theta ^ { 2 } ) $ | 0001073_page003 | embedded |
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$\theta $ | $ \theta $ | 0001073_page003 | embedded |
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$F$ | $ F $ | 0001073_page003 | embedded |
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$i$ | $ i $ | 0001073_page003 | embedded |
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$F$ | $ F $ | 0001073_page003 | embedded |
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$A$ | $ A $ | 0001073_page003 | embedded |
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$A$ | $ A $ | 0001073_page003 | embedded |
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$A\simeq A^0+A^1+A^2$ | $ A \sime A ^ { 0 } + A ^ { 1 } + A ^ { 2 } $ | 0001073_page003 | embedded |
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$\theta $ | $ \theta $ | 0001073_page003 | embedded |
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$F_{ij}\simeq F_{ij}^0+F_{ij}^1+F_{ij}^2$ | $ F _ { i j } \sime F _ { i j } ^ { 0 } + F _ { i j } ^ { 1 } + F _ { i j } ^ { 2 } $ | 0001073_page003 | embedded |
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$f^{1, 2}=dA^{1, 2}$ | $ f ^ { 1 , 2 } = d A ^ { 1 , 2 } $ | 0001073_page003 | embedded |
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$g$ | $ g $ | 0001073_page003 | embedded |
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$\ast 1$ | $ \ast 1 $ | 0001073_page003 | embedded |
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$F$ | $ F $ | 0001073_page003 | embedded |
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$\theta $ | $ \theta $ | 0001073_page003 | embedded |
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$f(x)\star g(x)=\exp (\frac {i}{2}\theta _{ij}\partial _i\partial _j') f(x)g(x')|_{x=x'}$ | $ f ( x ) \star g ( x ) = e x p ( \frac { i } { 2 } \theta _ { i j } \partial _ { i } \partial _ { j } ^ { \prime } ) f ( x ) g ( x ^ { \prime } ) \vert _ { x = x ^ { \prime } } $ | 0001073_page003 | embedded |
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$[x_i, x_j]=i\theta _{ij}$ | $ [ x _ { i } , x _ { j } ] = i \theta _ { i j } $ | 0001073_page003 | embedded |
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$A_0=0$ | $ A _ { 0 } = 0 $ | 0001073_page003 | embedded |
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\begin {equation} F=dA-\frac {i}{2}[A, A]_{\star }. \end {equation} | \begin{equation*} F = d A - \frac { i } { 2 } [ A , A ] _ { \star } . \end{equation*} | 0001073_page003 | isolated |
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\begin {equation} \label {eq:F_ij} F_{ij}\simeq \partial _i A_j-\partial _j A_i+\theta _{mn}\partial _m A_i \partial _n A_j, \end {equation} | \begin{equation*} F _ { i j } \sime \partial _ { i } A _ { j } - \partial _ { j } A _ { i } + \theta _ { m n } \partial _ { m } A _ { i } \partial _ { n } A _ { j } , \end{equation*} | 0001073_page003 | isolated |
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\begin {eqnarray} F_{ij}^0 & = & \partial _i A_j^0-\partial _j A_i^0 \\ F_{ij}^1 & = & \partial _i A_j^1-\partial _j A_i^1+\theta _{mn}\partial _m A_i^0 \partial _n A_j^0 \\ F_{ij}^2 & = & \partial _i A_j^2-\partial _j A_i^2+\theta _{mn}\partial _m A_i^0 \partial _n A_j^1+\theta _{mn}\partial _m A_i^1 \partial _n A_j^0. \end {eqnarray} | \begin{align*} F _ { i j } ^ { 0 } & = & \partial _ { i } A _ { j } ^ { 0 } - \partial _ { j } A _ { i } ^ { 0 } \\ F _ { i j } ^ { 1 } & = & \partial _ { i } A _ { j } ^ { 1 } - \partial _ { j } A _ { i } ^ { 1 } + \theta _ { m n } \partial _ { m } A _ { i } ^ { 0 } \partial _ { n } A _ { j } ^ { 0 } \\ F _ { i j } ^ { 2 } & = & \partial _ { i } A _ { j } ^ { 2 } - \partial _ { j } A _ { i } ^ { 2 } + \theta _ { m n } \partial _ { m } A _ { i } ^ { 0 } \partial _ { n } A _ { j } ^ { 1 } + \theta _ { m n } \partial _ { m } A _ { i } ^ { 1 } \partial _ { n } A _ { j } ^ { 0 } . \end{align*} | 0001073_page003 | isolated |
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\begin {equation} \label {eq:DF} DF=4\pi g\delta ^3(\vec r\,)\ast \!1 \end {equation} | \begin{equation*} D F = 4 \pi g \delta ^ { 3 } ( \vec { r } \, ) \ast \! 1 \end{equation*} | 0001073_page003 | isolated |
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\begin {equation} DF=dF-i[A, F]_{\star }. \end {equation} | \begin{equation*} D F = d F - i [ A , F ] _ { \star } . \end{equation*} | 0001073_page003 | isolated |
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\begin {eqnarray} dF^0 & = & 4\pi g\delta ^3(\vec r\,)\ast \!1 \\ dF^1 & = & -\theta _{mn}\partial _m A^0\wedge \partial _n F^0 \\ dF^2 & = & -\theta _{mn}\partial _m A^1\wedge \partial _n F^0 -\theta _{mn}\partial _m A^0\wedge \partial _n F^1. \end {eqnarray} | \begin{align*} d F ^ { 0 } & = & 4 \pi g \delta ^ { 3 } ( \vec { r } \, ) \ast \! 1 \\ d F ^ { 1 } & = & - \theta _ { m n } \partial _ { m } A ^ { 0 } \wedge \partial _ { n } F ^ { 0 } \\ d F ^ { 2 } & = & - \theta _ { m n } \partial _ { m } A ^ { 1 } \wedge \partial _ { n } F ^ { 0 } - \theta _ { m n } \partial _ { m } A ^ { 0 } \wedge \partial _ { n } F ^ { 1 } . \end{align*} | 0001073_page003 | isolated |
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$\nabla \cdot \vec B^0=4\pi g\delta ^3(\vec r\,)$ | $ \nabla \cdot \vec { B } ^ { 0 } = 4 \pi g \delta ^ { 3 } ( \vec { r } \, ) $ | 0001073_page004 | embedded |
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$B^0=\ast F^0$ | $ B ^ { 0 } = \ast F ^ { 0 } $ | 0001073_page004 | embedded |
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$\vec B^0=g\vec r/r^3$ | $ \vec { B } ^ { 0 } = g \vec { r } \slash r ^ { 3 } $ | 0001073_page004 | embedded |
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$A^0$ | $ A ^ { 0 } $ | 0001073_page004 | embedded |
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$A^{1, 2}$ | $ A ^ { 1 , 2 } $ | 0001073_page004 | embedded |
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$A^0$ | $ A ^ { 0 } $ | 0001073_page004 | embedded |
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$i\to k$ | $ i \rightarrow k $ | 0001073_page004 | embedded |
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$j\to i$ | $ j \rightarrow i $ | 0001073_page004 | embedded |
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$k\to j$ | $ k \rightarrow j $ | 0001073_page004 | embedded |
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$m\leftrightarrow n$ | $ m \leftrightarrow n $ | 0001073_page004 | embedded |
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$d f^1=0$ | $ d f ^ { 1 } = 0 $ | 0001073_page004 | embedded |
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$f^1=0$ | $ f ^ { 1 } = 0 $ | 0001073_page004 | embedded |
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$A^1$ | $ A ^ { 1 } $ | 0001073_page004 | embedded |
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$A^1$ | $ A ^ { 1 } $ | 0001073_page004 | embedded |
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$A^0$ | $ A ^ { 0 } $ | 0001073_page004 | embedded |
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$F^1$ | $ F ^ { 1 } $ | 0001073_page004 | embedded |
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$-2\epsilon _{ijk}(\theta _{mn} \theta _{pq}\partial _m A^0_k\partial _q A^0_j \partial _n \partial _p A^0_i)$ | $ - 2 \epsilon _ { i j k } ( \theta _ { m n } \theta _ { p q } \partial _ { m } A _ { k } ^ { 0 } \partial _ { q } A _ { j } ^ { 0 } \partial _ { n } \partial _ { p } A _ { i } ^ { 0 } ) $ | 0001073_page004 | embedded |
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$j$ | $ j $ | 0001073_page004 | embedded |
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$k$ | $ k $ | 0001073_page004 | embedded |
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$d f^2=0$ | $ d f ^ { 2 } = 0 $ | 0001073_page004 | embedded |
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$A^2$ | $ A ^ { 2 } $ | 0001073_page004 | embedded |
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$f^{1, 2}$ | $ f ^ { 1 , 2 } $ | 0001073_page004 | embedded |
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$f^{1, 2}=dA^{1, 2}$ | $ f ^ { 1 , 2 } = d A ^ { 1 , 2 } $ | 0001073_page004 | embedded |
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$F^0=dA^0$ | $ F ^ { 0 } = d A ^ { 0 } $ | 0001073_page004 | embedded |
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$S^2$ | $ S ^ { 2 } $ | 0001073_page004 | embedded |
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$A^0$ | $ A ^ { 0 } $ | 0001073_page004 | embedded |
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$A$ | $ A $ | 0001073_page004 | embedded |
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$A$ | $ A $ | 0001073_page004 | embedded |
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$g\simeq g^0+g^1+g^2$ | $ g \sime g ^ { 0 } + g ^ { 1 } + g ^ { 2 } $ | 0001073_page004 | embedded |
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$|A^0|\sim 1/r$ | $ \vert A ^ { 0 } \vert \sim 1 \slash r $ | 0001073_page004 | embedded |
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$|F^0|\sim 1/r^2$ | $ \vert F ^ { 0 } \vert \sim 1 \slash r ^ { 2 } $ | 0001073_page004 | embedded |
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$|F^1|\sim 1/r^4$ | $ \vert F ^ { 1 } \vert \sim 1 \slash r ^ { 4 } $ | 0001073_page004 | embedded |
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$|F^2|=0$ | $ \vert F ^ { 2 } \vert = 0 $ | 0001073_page004 | embedded |
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\begin {eqnarray} \epsilon _{ijk}\partial _i f^1_{jk} & = & -\epsilon _{ijk}\partial _i (\theta _{mn}\partial _m A^0_j\partial _n A^0_k) -\epsilon _{ijk}\theta _{nm}\partial _n A^0_k\partial _m F^0_{ij} \\ &= & -\epsilon _{ijk}\theta _{mn}\Big (\partial _m \partial _i A^0_j \partial _n A^0_k +\partial _mA^0_j\partial _n\partial _i A^0_k -\partial _n A^0_k\partial _m (\partial _i A_j^0- \partial _j A_i^0)\Big ). \end {eqnarray} | \begin{align*} \epsilon _ { i j k } \partial _ { i } f _ { j k } ^ { 1 } & = & - \epsilon _ { i j k } \partial _ { i } ( \theta _ { m n } \partial _ { m } A _ { j } ^ { 0 } \partial _ { n } A _ { k } ^ { 0 } ) - \epsilon _ { i j k } \theta _ { n m } \partial _ { n } A _ { k } ^ { 0 } \partial _ { m } F _ { i j } ^ { 0 } \\ & = & - \epsilon _ { i j k } \theta _ { m n } ( \partial _ { m } \partial _ { i } A _ { j } ^ { 0 } \partial _ { n } A _ { k } ^ { 0 } + \partial _ { m } A _ { j } ^ { 0 } \partial _ { n } \partial _ { i } A _ { k } ^ { 0 } - \partial _ { n } A _ { k } ^ { 0 } \partial _ { m } ( \partial _ { i } A _ { j } ^ { 0 } - \partial _ { j } A _ { i } ^ { 0 } ) ) . \end{align*} | 0001073_page004 | isolated |
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\begin {eqnarray} \epsilon _{ijk}\partial _i f^2_{jk} & = & -\epsilon _{ijk}\partial _i \Big (\theta _{mn}(\partial _m A^0_j\partial _n A^1_k- \partial _m A^0_k\partial _n A^1_j)\Big ) \\ && -\epsilon _{ijk}\theta _{nm}\partial _n A^1_k\partial _m F^0_{ij} -\epsilon _{ijk}\theta _{mn}\partial _m A^0_k\partial _n F^1_{ij}. \end {eqnarray} | \begin{align*} \epsilon _ { i j k } \partial _ { i } f _ { j k } ^ { 2 } & = & - \epsilon _ { i j k } \partial _ { i } ( \theta _ { m n } ( \partial _ { m } A _ { j } ^ { 0 } \partial _ { n } A _ { k } ^ { 1 } - \partial _ { m } A _ { k } ^ { 0 } \partial _ { n } A _ { j } ^ { 1 } ) ) \\ & & - \epsilon _ { i j k } \theta _ { n m } \partial _ { n } A _ { k } ^ { 1 } \partial _ { m } F _ { i j } ^ { 0 } - \epsilon _ { i j k } \theta _ { m n } \partial _ { m } A _ { k } ^ { 0 } \partial _ { n } F _ { i j } ^ { 1 } . \end{align*} | 0001073_page004 | isolated |
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\begin {equation} \epsilon _{ijk}\partial _i f^2_{jk}=-\epsilon _{ijk}\theta _{mn}\theta _{pq} \partial _m A^0_k\partial _n (\partial _p A^0_i\partial _q A^0_j). \end {equation} | \begin{equation*} \epsilon _ { i j k } \partial _ { i } f _ { j k } ^ { 2 } = - \epsilon _ { i j k } \theta _ { m n } \theta _ { p q } \partial _ { m } A _ { k } ^ { 0 } \partial _ { n } ( \partial _ { p } A _ { i } ^ { 0 } \partial _ { q } A _ { j } ^ { 0 } ) . \end{equation*} | 0001073_page004 | isolated |
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