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Let me elaborate on Ryan's correct comments. The flat background makes all components of the spinors covariantly constant; so the geometry is compatible with all of SUSY. A generic curved 6-real-dimensional manifold has an $O(6)$ holonomy or $SO(6)\sim SU(4)$ if it is orientable. The $SU(3)$ subgroup preserves 1/4 of the original supercharges – it is the ...

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Say we have a supercharge $Q$ in $\mathbb{R}^{10}$. To turn this into a supercharge on the $\mathbb{R}^4$ effective theory obtained by compactifying on $X$, we need to contract $Q$ with a covariantly constant spinor on $X$. The reason why we want it to be covariantly constant is because we want to take the size of $X$ to zero. Covariant constant spinors are ...

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The value $R=\alpha^{\prime 1/2}$ is the self-dual radius under T-duality. One may indeed extract the massless spectrum – the spectrum of all fields much lighter than $\alpha^{\prime -1/2}$. Because the CFT has an $SU(2)\times SU(2)$ symmetry, as can be seen from the OPEs of the currents, the spacetime physics has this symmetry, too. Because one finds ...

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Ok, this question requires a more careful answer than what was presented here. First, extra-dimensions appear in string theories or M-theory (which is in fact not a well defined or well known theory, if any). Considering only the bosonic string we have the Weyl invariance. If you calculate the energy momentum tensor then the Weyl invariance implies that its ...

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