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$D=5$
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$ D = 5 $
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0304253_page019
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$^*$
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$ { } ^ { \ast } $
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$U(1)^3$
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$ U ( 1 ) ^ { 3 } $
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0304253_page019
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\begin {equation} ds^2=-H^{-2}(t,\vec x\,)dt^2+H^{2/(d-3)}(t,\vec x\,)\,a^2(t)d\vec x\,^2, \label {eq:rnmet} \end {equation}
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\begin{equation*} d s ^ { 2 } = - H ^ { - 2 } ( t , \vec { x } \, ) d t ^ { 2 } + H ^ { 2 \slash ( d - 3 ) } ( t , \vec { x } \, ) \, a ^ { 2 } ( t ) d \vec { x } \, { } ^ { 2 } , \end{equation*}
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0304253_page019
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\begin {equation} H(t,\vec x\,)=1+\sum _i\fft {q_i}{(a(t)|\vec x-\vec x_i|)^{d-3}}. \end {equation}
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\begin{equation*} H ( t , \vec { x } \, ) = 1 + \sum _ { i } \frac { q _ { i } } { ( a ( t ) \vert \vec { x } - \vec { x } _ { i } \vert ) ^ { d - 3 } } . \end{equation*}
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0304253_page019
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\begin {eqnarray} ds_5^2&=&-(H_1H_2H_3)^{-2/3}dt^2 + (H_1H_2H_3)^{1/3}e^{2\hat gt}d\vec x\,^2, \\ X_i&=&H_i^{-1}(H_1H_2H_3)^{1/3}, \\ A_{(1)}^{1,2}&=&i\left (1-\frac {1}{H_{1,2}}\right )dt,\qquad A_{(1)}^3 = \left (1-\frac {1}{H_3}\right )dt, \end {eqnarray}
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\begin{align*} d s _ { 5 } ^ { 2 } & = & - ( H _ { 1 } H _ { 2 } H _ { 3 } ) ^ { - 2 \slash 3 } d t ^ { 2 } + ( H _ { 1 } H _ { 2 } H _ { 3 } ) ^ { 1 \slash 3 } e ^ { 2 \hat { g } t } d \vec { x } \, { } ^ { 2 } , \\ X _ { i } & = & H _ { i } ^ { - 1 } ( H _ { 1 } H _ { 2 } H _ { 3 } ) ^ { 1 \slash 3 } , \\ A _ { ( 1 ) } ^ { 1 , 2 } & = & i ( 1 - \frac { 1 } { H _ { 1 , 2 } } ) d t , \qquad A _ { ( 1 ) } ^ { 3 } = ( 1 - \frac { 1 } { H _ { 3 } } ) d t , \end{align*}
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0304253_page019
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\begin {equation} H_i(t,\vec x\,)=1 + e^{-2\hat {g}t}\sum _j{\frac {q_i^j}{|\vec x-\vec x_j|^2}}. \end {equation}
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\begin{equation*} H _ { i } ( t , \vec { x } \, ) = 1 + e ^ { - 2 \hat { g } t } \sum _ { j } \frac { q _ { i } ^ { j } } { \vert \vec { x } - \vec { x } _ { j } \vert ^ { 2 } } . \end{equation*}
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$A^1_{(1)}$
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$ A _ { ( 1 ) } ^ { 1 } $
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$A^2_{(1)}$
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$ A _ { ( 1 ) } ^ { 2 } $
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$F_{4}$
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$ F _ { 4 } $
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$A^3_{(1)}$
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$ A _ { ( 1 ) } ^ { 3 } $
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$^*$
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$ { } ^ { \ast } $
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$\widetilde \Delta \equiv H^{2/3}\Delta =1+(1-H)\sinh ^2\alpha $
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$ \widetilde { \Delta } \equiv H ^ { 2 \slash 3 } \Delta = 1 + ( 1 - H ) { s i n h } ^ { 2 } \alpha $
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$H^5$
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$ H ^ { 5 } $
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$\hat g\tau =e^{\hat gt}$
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$ \hat { g } \tau = e ^ { \hat { g } t } $
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$\widetilde {\cal H}$
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$ \widetilde { H } $
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$^*$
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$ { } ^ { \ast } $
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$U(1)$
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$ U ( 1 ) $
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\begin {equation} ds_{10}^2 = \widetilde \Delta ^{1/2}(-H^{-1}dt^2+e^{2\hat gt} d\vec x\,^2)+\hat g^{-2}\widetilde \Delta ^{-1/2}ds^2(\widetilde H^5), \label {eq:bhten} \end {equation}
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\begin{equation*} d s _ { 1 0 } ^ { 2 } = \widetilde { \Delta } ^ { 1 \slash 2 } ( - H ^ { - 1 } d t ^ { 2 } + e ^ { 2 \hat { g } t } d \vec { x } \, { } ^ { 2 } ) + \hat { g } ^ { - 2 } \widetilde { \Delta } ^ { - 1 \slash 2 } d s ^ { 2 } ( \widetilde { H } ^ { 5 } ) , \end{equation*}
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0304253_page020
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\begin {equation} ds^2(\widetilde H^5)=d\alpha ^2+\sinh ^2\alpha \,d\Omega _3^2 +(1-H)\sinh ^2\alpha \,d\alpha ^2+H\cosh ^2\alpha (d\psi -\hat gH^{-1}(1-H)dt)^2 \end {equation}
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\begin{equation*} d s ^ { 2 } ( \widetilde { H } ^ { 5 } ) = d \alpha ^ { 2 } + { \operatorname { s i n h } } ^ { 2 } \alpha \, d \Omega _ { 3 } ^ { 2 } + ( 1 - H ) { \operatorname { s i n h } } ^ { 2 } \alpha \, d \alpha ^ { 2 } + H { \operatorname { c o s h } } ^ { 2 } \alpha ( d \psi - \hat { g } H ^ { - 1 } ( 1 - H ) d t ) ^ { 2 } \end{equation*}
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0304253_page020
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\begin {equation} ds^2(H^5)=d\alpha ^2+\sinh ^2\alpha \,d\Omega _3^2+\cosh ^2\alpha \,d\psi ^2. \end {equation}
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\begin{equation*} d s ^ { 2 } ( H ^ { 5 } ) = d \alpha ^ { 2 } + { \operatorname { s i n h } } ^ { 2 } \alpha \, d \Omega _ { 3 } ^ { 2 } + { \operatorname { c o s h } } ^ { 2 } \alpha \, d \psi ^ { 2 } . \end{equation*}
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0304253_page020
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\begin {equation} ds_{10}^2 = \widetilde {\cal H}_0^{-1/2}d\vec x\,^2 +\widetilde {\cal H}_0^{1/2}[-\widetilde \Delta H^{-1}d\tau ^2+\tau ^2ds^2(\widetilde H^5)], \label {eq:bhten2} \end {equation}
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\begin{equation*} d s _ { 1 0 } ^ { 2 } = \widetilde { H } _ { 0 } ^ { - 1 \slash 2 } d \vec { x } \, { } ^ { 2 } + \widetilde { H } _ { 0 } ^ { 1 \slash 2 } [ - \widetilde { \Delta } H ^ { - 1 } d \tau ^ { 2 } + \tau ^ { 2 } d s ^ { 2 } ( \widetilde { H } ^ { 5 } ) ] , \end{equation*}
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0304253_page020
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\begin {equation} \widetilde {\cal H}_0=\lim _{\tau \to 0}\widetilde {\cal H},\qquad \widetilde {\cal H}=1+\fft 1{\hat g^4\tau ^4\widetilde \Delta }. \end {equation}
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\begin{equation*} \widetilde { H } _ { 0 } = \underset { \tau \rightarrow 0 } { \operatorname { l i m } } \widetilde { H } , \qquad \widetilde { H } = 1 + \frac { 1 } { \hat { g } ^ { 4 } \tau ^ { 4 } \widetilde { \Delta } } . \end{equation*}
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0304253_page020
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\begin {equation} ds_{10}^2={\cal H}^{-1/2}d\vec x\,^2+{\cal H}^{1/2}(-dt^2+dr^2+r^2d\Omega _4^2), \end {equation}
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\begin{equation*} d s _ { 1 0 } ^ { 2 } = H ^ { - 1 \slash 2 } d \vec { x } \, { } ^ { 2 } + H ^ { 1 \slash 2 } ( - d t ^ { 2 } + d r ^ { 2 } + r ^ { 2 } d \Omega _ { 4 } ^ { 2 } ) , \end{equation*}
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0304253_page020
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\begin {equation} {\cal H}=1+\fft 1{\hat g^4|t^2-r^2|^2}. \end {equation}
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\begin{equation*} H = 1 + \frac { 1 } { \hat { g } ^ { 4 } \vert t ^ { 2 } - r ^ { 2 } \vert ^ { 2 } } . \end{equation*}
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$t\to r$
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$ t \rightarrow r $
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$t^2>r^2$
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$ t ^ { 2 } > r ^ { 2 } $
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$t=\tau \cosh \beta $
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$ t = \tau c o s h \beta $
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$r=\tau \sinh \beta $
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$ r = \tau s i n h \beta $
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$_5\times H^5$
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$ { } _ { 5 } \times H ^ { 5 } $
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0304253_page021
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${\cal H}_0=1/\hat g^4\tau ^4$
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$ H _ { 0 } = 1 \slash \hat { g } ^ { 4 } \tau ^ { 4 } $
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0304253_page021
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embedded
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$D=4$
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$ D = 4 $
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0304253_page021
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$U(1)^4$
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$ U ( 1 ) ^ { 4 } $
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$U(1)^4$
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$ U ( 1 ) ^ { 4 } $
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$^*$
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$ { } ^ { \ast } $
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0304253_page021
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$N=2$
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$ N = 2 $
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$A_{(1)}$
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$ A _ { ( 1 ) } $
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0304253_page021
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embedded
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\begin {equation} ds_{10}^2={\cal H}_0^{-1/2}d\vec x\,^2+{\cal H}_0^{1/2}[-d\tau ^2 +\tau ^2(d\beta ^2+\sinh ^2\beta \,d\Omega _4^2)], \end {equation}
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\begin{equation*} d s _ { 1 0 } ^ { 2 } = H _ { 0 } ^ { - 1 \slash 2 } d \vec { x } \, { } ^ { 2 } + H _ { 0 } ^ { 1 \slash 2 } [ - d \tau ^ { 2 } + \tau ^ { 2 } ( d \beta ^ { 2 } + { \operatorname { s i n h } } ^ { 2 } \beta \, d \Omega _ { 4 } ^ { 2 } ) ] , \end{equation*}
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0304253_page021
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\begin {eqnarray} ds_4^2&=&-(H_1H_2H_3H_4)^{-1/2}dt^2 + (H_1H_2H_3H_4)^{1/2}e^{2\hat gt} d\vec x\,^2,\\ X_i&=&H_i^{-1}(H_1H_2H_3H_4)^{1/4},\\ A_{(1)}^{i}&=&i\left (1-\frac {1}{H_i}\right )dt, \end {eqnarray}
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\begin{align*} d s _ { 4 } ^ { 2 } & = & - ( H _ { 1 } H _ { 2 } H _ { 3 } H _ { 4 } ) ^ { - 1 \slash 2 } d t ^ { 2 } + ( H _ { 1 } H _ { 2 } H _ { 3 } H _ { 4 } ) ^ { 1 \slash 2 } e ^ { 2 \hat { g } t } d \vec { x } \, { } ^ { 2 } , \\ X _ { i } & = & H _ { i } ^ { - 1 } ( H _ { 1 } H _ { 2 } H _ { 3 } H _ { 4 } ) ^ { 1 \slash 4 } , \\ A _ { ( 1 ) } ^ { i } & = & i ( 1 - \frac { 1 } { H _ { i } } ) d t , \end{align*}
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0304253_page021
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isolated
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\begin {equation} H_i(t,\vec x\,) = 1 + e^{-\hat {g}t}\sum _j{\frac {q_i^j}{|\vec x-\vec x_j|}}. \end {equation}
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\begin{equation*} H _ { i } ( t , \vec { x } \, ) = 1 + e ^ { - \hat { g } t } \sum _ { j } \frac { q _ { i } ^ { j } } { \vert \vec { x } - \vec { x } _ { j } \vert } . \end{equation*}
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0304253_page021
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isolated
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\begin {eqnarray} ds_4^2&=&-H^{-2}dt^2 + H^2e^{2\hat gt}d\vec x\,^2,\\ A_{(1)}&=&i\left (1-\frac {1}{H}\right )dt. \end {eqnarray}
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\begin{align*} d s _ { 4 } ^ { 2 } & = & - H ^ { - 2 } d t ^ { 2 } + H ^ { 2 } e ^ { 2 \hat { g } t } d \vec { x } \, { } ^ { 2 } , \\ A _ { ( 1 ) } & = & i ( 1 - \frac { 1 } { H } ) d t . \end{align*}
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$D=7$
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$ D = 7 $
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$U(1)^2$
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$ U ( 1 ) ^ { 2 } $
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$H_i=H$
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$ H _ { i } = H $
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$S^5$
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$ S ^ { 5 } $
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$SO(6)$
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$ S O ( 6 ) $
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$SO(p,6-p)$
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$ S O ( p , 6 - p ) $
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\begin {eqnarray} ds_7^2&=&-(H_1H_2)^{-4/5}dt^2 + (H_1H_2)^{1/5}e^{-2igt}d\vec x\,^2,\\ X_i&=&H_i^{-1}(H_1H_2)^{2/5}, \\ A_{(1)}^{1,2}&=&\left (1-\frac {1}{H_{1,2}}\right )dt, \end {eqnarray}
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\begin{align*} d s _ { 7 } ^ { 2 } & = & - ( H _ { 1 } H _ { 2 } ) ^ { - 4 \slash 5 } d t ^ { 2 } + ( H _ { 1 } H _ { 2 } ) ^ { 1 \slash 5 } e ^ { - 2 i g t } d \vec { x } \, { } ^ { 2 } , \\ X _ { i } & = & H _ { i } ^ { - 1 } ( H _ { 1 } H _ { 2 } ) ^ { 2 \slash 5 } , \\ A _ { ( 1 ) } ^ { 1 , 2 } & = & ( 1 - \frac { 1 } { H _ { 1 , 2 } } ) d t , \end{align*}
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0304253_page022
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isolated
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\begin {equation} H_i(t,\vec x\,)=1 + e^{4igt}\sum _j{\frac {q_i^j}{|\vec x-\vec x_j|^4}}. \end {equation}
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\begin{equation*} H _ { i } ( t , \vec { x } \, ) = 1 + e ^ { 4 i g t } \sum _ { j } \frac { q _ { i } ^ { j } } { \vert \vec { x } - \vec { x } _ { j } \vert ^ { 4 } } . \end{equation*}
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0304253_page022
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isolated
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\begin {eqnarray} ds_7^2&=&-H^{-8/5}dt^2 + H^{2/5}e^{-2igt}d\vec x\,^2,\\ X&=&H^{-1/5}, \\ A_{(1)}&=&\left (1-\frac {1}{H}\right )dt. \end {eqnarray}
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\begin{align*} d s _ { 7 } ^ { 2 } & = & - H ^ { - 8 \slash 5 } d t ^ { 2 } + H ^ { 2 \slash 5 } e ^ { - 2 i g t } d \vec { x } \, { } ^ { 2 } , \\ X & = & H ^ { - 1 \slash 5 } , \\ A _ { ( 1 ) } & = & ( 1 - \frac { 1 } { H } ) d t . \end{align*}
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0304253_page022
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$H^5$
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$ H ^ { 5 } $
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$g$
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$ g $
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$i\hat g$
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$D=5$
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$ D = 5 $
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$\psi ^a_\mu $
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$ \psi _ { \mu } ^ { a } $
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$A^{[ab]}_\mu $
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$ A _ { \mu } ^ { [ a b ] } $
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$1/2$
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$ 1 \slash 2 $
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$\lambda ^{abc}$
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$ \lambda ^ { a b c } $
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$\varphi ^{abcd}$
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$ \varphi ^ { a b c d } $
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$a,b,\ldots $
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$ a , b , \ldots $
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$USp(8)$
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$ U S p ( 8 ) $
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$E_{6(6)}/USp(8)$
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$ E _ { 6 ( 6 ) } \slash U S p ( 8 ) $
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0304253_page023
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$SO(5,1)\subset SL(6,R)\times SL(2,R)\subset E_{6(6)}$
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$ S O ( 5 , 1 ) \subset S L ( 6 , R ) \times S L ( 2 , R ) \subset E _ { 6 ( 6 ) } $
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0304253_page023
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${V_{AB}}^{cd}$
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$ { V _ { A B } } ^ { c d } $
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0304253_page023
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$SL(6,R)$
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$ S L ( 6 , R ) $
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0304253_page023
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$S$
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$ S $
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0304253_page023
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$SO(5,1)$
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$ S O ( 5 , 1 ) $
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0304253_page023
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$\{\Gamma _I,\Gamma _J\}=2\eta _{IJ}$
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$ \{ \Gamma _ { I } , \Gamma _ { J } \} = 2 \eta _ { I J } $
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0304253_page023
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$\eta _{IJ}={\rm diag}(+,+,+,+,+,-)$
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$ \eta _ { I J } = d i a g ( + , + , + , + , + , - ) $
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0304253_page023
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$\Gamma _{I\alpha }\equiv \Gamma _I$
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$ \Gamma _ { I \alpha } \equiv \Gamma _ { I } $
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0304253_page023
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$\alpha =1$
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$ \alpha = 1 $
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0304253_page023
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$\Gamma _{I\alpha }\equiv i\Gamma _I\Gamma _0$
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$ \Gamma _ { I \alpha } \equiv i \Gamma _ { I } \Gamma _ { 0 } $
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0304253_page023
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$\alpha =2$
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$ \alpha = 2 $
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0304253_page023
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$\Gamma _0$
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$ \Gamma _ { 0 } $
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0304253_page023
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$P_\mu {}^{abcd}$
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$ P _ { \mu } { } ^ { a b c d } $
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0304253_page023
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${Q_{\mu a}}^b$
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$ { Q _ { \mu a } } ^ { b } $
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0304253_page023
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$\tilde {V}_{cd}{}^{AB}$
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$ \widetilde { V } _ { c d } { } ^ { A B } $
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0304253_page023
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$V_{AB}{}^{cd}$
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$ V _ { A B } { } ^ { c d } $
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$S$
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$ S $
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0304253_page023
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$\hat T_{ij}$
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$ \hat { T } _ { i j } $
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0304253_page023
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$SL(6,R)/SO(5,1)$
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$ S L ( 6 , R ) \slash S O ( 5 , 1 ) $
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0304253_page023
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$g\to i\hat g$
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$ g \rightarrow i \hat { g } $
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0304253_page023
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$F_{(2)} \to -i\hat F_{(2)}$
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$ F _ { ( 2 ) } \rightarrow - i \hat { F } _ { ( 2 ) } $
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0304253_page023
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$1/2$
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$ 1 \slash 2 $
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0304253_page023
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\begin {eqnarray} U^{IJ}{}_{KL}&=&2{S_{[K}}^{[I}{S_{L]}}^{J]},\qquad V^{IJab}=\fft 18(\Gamma ^{KL})^{ab}{U^{IJ}}_{KL},\\ U_{I\alpha }{}^{J\beta }&=&S_I{}^J\delta _\alpha {}^\beta ,\qquad V_{I\alpha }{}^{ab}=\fft 1{2\sqrt {2}}(\Gamma _{K\beta })^{ab}U_{I\alpha }{}^{K\beta }, \end {eqnarray}
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\begin{align*} U ^ { I J } { } _ { K L } & = & 2 { S _ { [ K } } ^ { [ I } { S _ { L ] } } ^ { J ] } , \qquad V ^ { I J a b } = \frac { 1 } { 8 } ( \Gamma ^ { K L } ) ^ { a b } { U ^ { I J } } _ { K L } , \\ U _ { I \alpha } { } ^ { J \beta } & = & S _ { I } { } ^ { J } \delta _ { \alpha } { } ^ { \beta } , \qquad V _ { I \alpha } { } ^ { a b } = \frac { 1 } { 2 \sqrt { 2 } } ( \Gamma _ { K \beta } ) ^ { a b } U _ { I \alpha } { } ^ { K \beta } , \end{align*}
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0304253_page023
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\begin {equation} \tilde {V}_{cd}{}^{AB}D_{\mu }{V_{AB}}^{ab}=2{Q_{\mu [c}}^{[a}\delta _{d]}^{b]} + {P_{\mu }}^{ab}{}_{cd}, \end {equation}
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\begin{equation*} \widetilde { V } _ { c d } { } ^ { A B } D _ { \mu } { V _ { A B } } ^ { a b } = 2 { Q _ { \mu [ c } } ^ { [ a } \delta _ { d ] } ^ { b ] } + { P _ { \mu } } ^ { a b } { } _ { c d } , \end{equation*}
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0304253_page023
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isolated
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\begin {eqnarray} {T^a}_{bcd}&=&(2V^{IKae}\tilde {V}_{beJK}-V_{J\alpha }{}^{ae}\tilde V_{be}{}^{I\alpha })\eta ^{JL}\tilde {V}_{cdIL}, \\ T_{ab}&=&{T^c}_{abc}. \end {eqnarray}
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\begin{align*} { T ^ { a } } _ { b c d } & = & ( 2 V ^ { I K a e } \widetilde { V } _ { b e J K } - V _ { J \alpha } { } ^ { a e } \widetilde { V } _ { b e } { } ^ { I \alpha } ) \eta ^ { J L } \widetilde { V } _ { c d I L } , \\ T _ { a b } & = & { T ^ { c } } _ { a b c } . \end{align*}
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0304253_page023
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isolated
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\begin {equation} (\hat T^{-1})^{ij}=S_I{}^iS_J{}^j\eta ^{IJ}. \end {equation}
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\begin{equation*} ( \hat { T } ^ { - 1 } ) ^ { i j } = S _ { I } { } ^ { i } S _ { J } { } ^ { j } \eta ^ { I J } . \end{equation*}
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0304253_page023
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isolated
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\begin {eqnarray} \delta \psi _{\mu a}&=&D_{\mu }\epsilon _a - \fft {2i}{45}\hat g T_{ab}\gamma _{\mu }\epsilon ^b +\fft {i}6 (\gamma _\mu {}^{\nu \rho } -4\delta _\mu ^\nu \gamma ^{\rho })\hat F_{\nu \rho ab}\epsilon ^b,\\ \delta \lambda _{abc}&=&\sqrt {2}\gamma ^{\mu }P_{\mu \, abcd}\epsilon ^d -\fft {i}{\sqrt {2}}\hat g T_{d[abc]}\epsilon ^d +\fft {3i}{2\sqrt {2}}\gamma ^{\mu \nu }\hat F_{\mu \nu [ab}\epsilon _{c]}, \end {eqnarray}
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\begin{align*} \delta \psi _ { \mu a } & = & D _ { \mu } \epsilon _ { a } - \frac { 2 i } { 4 5 } \hat { g } T _ { a b } \gamma _ { \mu } \epsilon ^ { b } + \frac { i } { 6 } ( \gamma _ { \mu } { } ^ { \nu \rho } - 4 \delta _ { \mu } ^ { \nu } \gamma ^ { \rho } ) \hat { F } _ { \nu \rho a b } \epsilon ^ { b } , \\ \delta \lambda _ { a b c } & = & \sqrt { 2 } \gamma ^ { \mu } P _ { \mu \, a b c d } \epsilon ^ { d } - \frac { i } { \sqrt { 2 } } \hat { g } T _ { d [ a b c ] } \epsilon ^ { d } + \frac { 3 i } { 2 \sqrt { 2 } } \gamma ^ { \mu \nu } \hat { F } _ { \mu \nu [ a b } \epsilon _ { c ] } , \end{align*}
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0304253_page023
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isolated
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$^*$
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$ { } ^ { \ast } $
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0304253_page024
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embedded
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$\times $
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$ \times $
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0304253_page024
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embedded
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$^*$
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$ { } ^ { \ast } $
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0304253_page024
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embedded
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$^*$
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$ { } ^ { \ast } $
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0304253_page024
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embedded
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$D=5$
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$ D = 5 $
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0304253_page024
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embedded
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\begin {eqnarray} \hat F_{\mu \nu ab}&=&\hat F_{\mu \nu IJ}{V^{IJ}}_{ab}= \ft 14\hat F_{\mu \nu }^{ij}\eta _{im}\eta _{jn}S_K{}^mS_L{}^n(\Gamma _{KL})^{ab} ,\\ D_{\mu }\epsilon _a&=&\partial _{\mu } \epsilon _a + {Q_{\mu a}}^b\epsilon _b. \end {eqnarray}
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\begin{align*} \hat { F } _ { \mu \nu a b } & = & \hat { F } _ { \mu \nu I J } { V ^ { I J } } _ { a b } = \frac { 1 } { 4 } \hat { F } _ { \mu \nu } ^ { i j } \eta _ { i m } \eta _ { j n } S _ { K } { } ^ { m } S _ { L } { } ^ { n } ( \Gamma _ { K L } ) ^ { a b } , \\ D _ { \mu } \epsilon _ { a } & = & \partial _ { \mu } \epsilon _ { a } + { Q _ { \mu a } } ^ { b } \epsilon _ { b } . \end{align*}
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0304253_page024
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isolated
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$S^7$
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$ S ^ { 7 } $
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0304253_page026
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embedded
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$_7\times S^4$
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$ { } _ { 7 } \times S ^ { 4 } $
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0304253_page026
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embedded
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|
$_7\times S^4$
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$ { } _ { 7 } \times S ^ { 4 } $
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0304253_page026
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embedded
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$N=1$
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$ N = 1 $
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0304253_page026
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embedded
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