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I had read about "string theory" having a large number of solutions (~$10^{500}$). Does this apply to the 11 dimensional M-theory or only to it's five 10-dimensional limiting cases?

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M-theory surely has a similarly large number of vacua, and these should be regarded as a subset of the larger set of vacua of string theory (namely, these are the 11-dimensional vacua).

The observation that string theory probably has a very large number of vacua dates back to the late 1980s, but it came unavoidably into view as "the landscape" after 1998, when dark energy was discovered, and string theorists realized they needed to find vacua with a small but nonzero cosmological constant, rather than vacua where it is exactly zero.

The construction of such "de Sitter vacua" proved difficult, until papers known as KKLT and KKLMMT (initials of their authors) produced a recipe in which a stable vacuum with negative cosmological constant could be uplifted to a metastable one with positive cosmological constant. These vacua consisted of geometries populated with branes and fluxes, and the number of choices appears to be enormous. (The googol-like number that is often quoted (10500) comes from saying there are about 500 topologically distinct hypersurfaces in the compact dimensions of an average geometry, and about 10 options for the amount of flux along each hypersurface. These numbers are just order-of-magnitude estimates and should not be taken too literally.)

My point is that this was when the multitude of string vacua became a "landscape" of practical significance. The empirical challenge of explaining dark energy forced string theorists to think about the possibility that the details of our universe are far more random than many would have wished, and drawn from an enormous ensemble of possibilities. Incidentally, it is not absolutely proven that all these vacua actually exist - they are constructed in an approximation which falls short of the full complexity of string theory, and there is a lively debate at the moment, about whether these constructions might be immediately unstable, rather than long-lasting, when all stringy effects are taken into account.

Another detail is that this landscape was constructed in Type IIB string theory. The higher-dimensional version of Type IIB is 12-dimensional "F-theory"; M-theory corresponds to Type IIA string theory. So if one regards the landscape very narrowly, as referring to the KKLT-type constructions, they were not part of M-theory. But in the broader sense of string vacua in general, there is certainly an M-theory landscape. The "G2-MSSM" approach to particle physics, includes consideration of a kind of M-theory landscape.

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