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\begin {equation} m=\int |F|^2 \sim \int _0^{\infty } r^2dr\left | \frac {1}{r^2}+\frac {1}{r^4} \right |^2=\infty . \end {equation} | \begin{equation*} m = \int \vert F \vert ^ { 2 } \sim \int _ { 0 } ^ { \infty } r ^ { 2 } d r { \vert \frac { 1 } { r ^ { 2 } } + \frac { 1 } { r ^ { 4 } } \vert } ^ { 2 } = \infty . \end{equation*} | 0001073_page004 | isolated |
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$\phi $ | $ \phi $ | 0001073_page005 | embedded |
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$\phi $ | $ \phi $ | 0001073_page005 | embedded |
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$\star $ | $ \star $ | 0001073_page005 | embedded |
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$U(1)$ | $ U ( 1 ) $ | 0001073_page005 | embedded |
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$\lambda =\lambda ^0+\lambda ^1+\cdots $ | $ \lambda = \lambda ^ { 0 } + \lambda ^ { 1 } + \cdots $ | 0001073_page005 | embedded |
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$\vec f =q\vec v\times \vec B$ | $ \vec { f } = q \vec { v } \times \vec { B } $ | 0001073_page005 | embedded |
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$\delta \vec f =0$ | $ \delta \vec { f } = 0 $ | 0001073_page005 | embedded |
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$\delta B=\ast \delta F$ | $ \delta B = \ast \delta F $ | 0001073_page005 | embedded |
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$\rho _{\rm V}$ | $ \rho _ { V } $ | 0001073_page005 | embedded |
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$\theta =1/B$ | $ \theta = 1 \slash B $ | 0001073_page005 | embedded |
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$B$ | $ B $ | 0001073_page005 | embedded |
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$\theta $ | $ \theta $ | 0001073_page005 | embedded |
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$\rho _{\rm V}$ | $ \rho _ { V } $ | 0001073_page005 | embedded |
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$B$ | $ B $ | 0001073_page005 | embedded |
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$B$ | $ B $ | 0001073_page005 | embedded |
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\begin {eqnarray} \delta F & \simeq & [i\lambda , F]_{\star } \\ &\simeq & 0-\theta _{mn}\partial _m \lambda ^0\partial _n F^0 -(\theta _{mn}\partial _m \lambda ^1\partial _n F^0 +\theta _{mn}\partial _m \lambda ^0\partial _n F^1), \end {eqnarray} | \begin{align*} \delta F & \sime & [ i \lambda , F ] _ { \star } \\ & \sime & 0 - \theta _ { m n } \partial _ { m } \lambda ^ { 0 } \partial _ { n } F ^ { 0 } - ( \theta _ { m n } \partial _ { m } \lambda ^ { 1 } \partial _ { n } F ^ { 0 } + \theta _ { m n } \partial _ { m } \lambda ^ { 0 } \partial _ { n } F ^ { 1 } ) , \end{align*} | 0001073_page005 | isolated |
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\begin {eqnarray} \rho _{\rm V} & = & \frac {1}{8\pi }B^2=\frac {1}{8\pi \theta ^2},\\ && \end {eqnarray} | \begin{align*} \rho _ { V } & = & \frac { 1 } { 8 \pi } B ^ { 2 } = \frac { 1 } { 8 \pi \theta ^ { 2 } } , \end{align*} | 0001073_page005 | isolated |
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$\rho _{\rm V}$ | $ \rho _ { V } $ | 0001073_page006 | embedded |
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\begin {equation} \sqrt {\theta }=\left (\frac {1}{8\pi \rho _{\rm V}}\right )^{1/4}\simeq \left (\frac {1}{8\pi \times 10^{-47}\,\,\mathrm {GeV}^4}\right )^{1/4} \simeq 5.0\times 10^{-3}\,\,\mathrm {cm}. \end {equation} | \begin{equation*} \sqrt { \theta } = { ( \frac { 1 } { 8 \pi \rho _ { V } } ) } ^ { 1 \slash 4 } \sime { ( \frac { 1 } { 8 \pi \times 1 0 ^ { - 4 7 } \, \, { G e V } ^ { 4 } } ) } ^ { 1 \slash 4 } \sime 5 . 0 \times 1 0 ^ { - 3 } \, \, c m . \end{equation*} | 0001073_page006 | isolated |
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$\mathcal {N}=2$ | $ N = 2 $ | 0001073_page007 | embedded |
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$R^4$ | $ R ^ { 4 } $ | 0001073_page007 | embedded |
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$p$ | $ p $ | 0001073_page007 | embedded |
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$^1$ | $ { } ^ { 1 } $ | 0001101_page001 | embedded |
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$^2$ | $ { } ^ { 2 } $ | 0001101_page001 | embedded |
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$^1$ | $ { } ^ { 1 } $ | 0001101_page001 | embedded |
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$^3$ | $ { } ^ { 3 } $ | 0001101_page001 | embedded |
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$^1$ | $ { } ^ { 1 } $ | 0001101_page001 | embedded |
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$^2$ | $ { } ^ { 2 } $ | 0001101_page001 | embedded |
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$^3$ | $ { } ^ { 3 } $ | 0001101_page001 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page001 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page001 | embedded |
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$S^1/Z_2$ | $ S ^ { 1 } \slash Z _ { 2 } $ | 0001101_page002 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page002 | embedded |
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$R$ | $ R $ | 0001101_page002 | embedded |
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$10^{16}$ | $ 1 0 ^ { 1 6 } $ | 0001101_page002 | embedded |
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$\rho $ | $ \rho $ | 0001101_page002 | embedded |
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$R$ | $ R $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$3+1$ | $ 3 + 1 $ | 0001101_page002 | embedded |
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$F_{ab}=F_{\bar a\bar b}=g^{a \bar b}F_{a \bar b}=0$ | $ F _ { a b } = F _ { \bar { a } \bar { b } } = g ^ { a \bar { b } } F _ { a \bar { b } } = 0 $ | 0001101_page002 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$G$ | $ G $ | 0001101_page002 | embedded |
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$G\times H \subseteq E_{8}$ | $ G \times H \subseteq E _ { 8 } $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$H$ | $ H $ | 0001101_page002 | embedded |
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$G$ | $ G $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$E_{6}$ | $ E _ { 6 } $ | 0001101_page002 | embedded |
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$SO(10)$ | $ S O ( 1 0 ) $ | 0001101_page002 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page002 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page002 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page002 | embedded |
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$E_{6}$ | $ E _ { 6 } $ | 0001101_page002 | embedded |
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$G=SU(3)$ | $ G = S U ( 3 ) $ | 0001101_page002 | embedded |
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$X$ | $ X $ | 0001101_page002 | embedded |
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$G\subseteq E_{8}$ | $ G \subseteq E _ { 8 } $ | 0001101_page002 | embedded |
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$G$ | $ G $ | 0001101_page003 | embedded |
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$X$ | $ X $ | 0001101_page003 | embedded |
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$SU(n)\subset E_{8}$ | $ S U ( n ) \subset E _ { 8 } $ | 0001101_page003 | embedded |
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$SU(n)$ | $ S U ( n ) $ | 0001101_page003 | embedded |
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$SU(n)$ | $ S U ( n ) $ | 0001101_page003 | embedded |
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$E_{6}$ | $ E _ { 6 } $ | 0001101_page003 | embedded |
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$SO(10)$ | $ S O ( 1 0 ) $ | 0001101_page003 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page003 | embedded |
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$SU(3)$ | $ S U ( 3 ) $ | 0001101_page003 | embedded |
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$SU(4)$ | $ S U ( 4 ) $ | 0001101_page003 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page003 | embedded |
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$E_{8}$ | $ E _ { 8 } $ | 0001101_page003 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page003 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page003 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page003 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page004 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page004 | embedded |
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$3+1$ | $ 3 + 1 $ | 0001101_page004 | embedded |
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$3+1$ | $ 3 + 1 $ | 0001101_page004 | embedded |
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$Z=X/\tau _{X}$ | $ Z = X \slash \tau _ { X } $ | 0001101_page004 | embedded |
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$\pi _{1}(Z)=\ZZ _{2}$ | $ \pi _ { 1 } ( Z ) = Z _ { 2 } $ | 0001101_page004 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page004 | embedded |
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$\pi _{1}(Z)=\ZZ _{2}$ | $ \pi _ { 1 } ( Z ) = Z _ { 2 } $ | 0001101_page004 | embedded |
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$SU(5)$ | $ S U ( 5 ) $ | 0001101_page004 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page004 | embedded |
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$SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page004 | embedded |
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$B=F_{2}$ | $ B = F _ { 2 } $ | 0001101_page004 | embedded |
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$B=dP_{3}$ | $ B = d P _ { 3 } $ | 0001101_page004 | embedded |
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$SU(n)$ | $ S U ( n ) $ | 0001101_page004 | embedded |
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$n$ | $ n $ | 0001101_page004 | embedded |
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$N=1$ | $ N = 1 $ | 0001101_page004 | embedded |
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$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ | $ S U ( 3 ) _ { C } \times S U ( 2 ) _ { L } \times U ( 1 ) _ { Y } $ | 0001101_page004 | embedded |
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$X$ | $ X $ | 0001101_page004 | embedded |
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$Z=X/\tau _X$ | $ Z = X \slash \tau _ { X } $ | 0001101_page004 | embedded |
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$E_8$ | $ E _ { 8 } $ | 0001101_page004 | embedded |
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$X$ | $ X $ | 0001101_page004 | embedded |
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$\sigma $ | $ \sigma $ | 0001101_page004 | embedded |
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$\xi $ | $ \xi $ | 0001101_page004 | embedded |
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$B$ | $ B $ | 0001101_page004 | embedded |
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