For D = 11 large (uncompactified) spacetime dimensions, the only "string theory" vacuum is M-theory

For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, since type I is S-dual to $Spin(32)/\mathbb{Z}_2$ heterotic?

For D = 4 there is a monstrous number of vacua and there's no full classification

For which values of D a full classification exists, and what is it?

My guess is that the classification exists for $D \geq D_{min}$ but what is the value of $D_{min}$?

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    $\begingroup$ Maybe a more efficient organizing principle is number of supersymmetries, for generic string vacua the number of dimensions is not a sharp question. $\endgroup$
    – user566
    Jan 6 '12 at 20:45
  • $\begingroup$ @Moshe, I think the number of large (uncompactified) dimensions is well-defined since the large-scale limit is a QFT with a well-defined spacetime dimension $\endgroup$
    – Squark
    Jan 6 '12 at 21:13
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    $\begingroup$ Dear @Squark, but even if the number of noncompact dimensions is well-defined, it's still sensible to subclassify the options according to the number of supercharges. For low number of noncompact dimensions, the number of supercharges may be very small and not too constraining. So only for high dimensions such as $D=9$, the classification is potentially manageable. In $D=9$, one gets various M-theories on Klein Bottle, Mobius strip, and similar funny extra choices aside from the circular compactifications. There are also papers on the $D=6$ landscape etc. $\endgroup$ Jan 7 '12 at 7:59
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    $\begingroup$ A good example of the richness of the vacua even in 7 large dimensions and above. See Triples, Fluxes, and Strings: arxiv.org/abs/hep-th/0103170 - various disconnected components of commuting Wilson lines, dualities between these vacua etc. I think it's a great paper by 7 authors, a substantially undervalued one (with 100 cits or so). $\endgroup$ Jan 7 '12 at 11:19
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    $\begingroup$ @Dilaton Out of topic but here is a free video of Gates Jr. ww3.tvo.org/video/164065/… on similar tone and more recent. $\endgroup$
    – anna v
    Dec 28 '12 at 15:49

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