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We still have no fundamental formulation of “M-theory” - the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. [...]. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily.

But, ultimately, Physical Mathematics must return to this grand issue.

 

(G. Moore, "Physical Mathematics and the Future", talk at Strings 2014)

We still have no fundamental formulation of “M-theory” - the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. [...]. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily.

But, ultimately, Physical Mathematics must return to this grand issue.

(G. Moore, "Physical Mathematics and the Future", talk at Strings 2014)

The uncertainty as to how well this sigma-model construction is under constrol is the cause of the discrepancy in the perception of how many string vacua are known: A perturbative string vacuum is equivalently that 2d SCFT which gives the scattering matrix, so in principle the "landscape of perturbative string vacua" is the moduli space of 2d SCFTs. But since this is not understood, what people instead scan is the space of target space geometries that are thought to induces 2d SCFTs. This is a subtle business, where one imagines one may incrementally approximate that 2d SCFT by adding alpha-prime corrections, cancelling anomalies, etc. Therefore one finds authors who worry that too many string vacua are known, and other authors who worry that too few string vacua are known.

The uncertainty as to how well this sigma-model construction is under control is the cause of the discrepancy in the perception of how many string vacua are known: A perturbative string vacuum is equivalently that 2d SCFT which gives the scattering matrix, so in principle the "landscape of perturbative string vacua" is the moduli space of 2d SCFTs. But since this is not understood, what people instead scan is the space of target space geometries that are thought to induces 2d SCFTs. This is a subtle business, where one imagines one may incrementally approximate that 2d SCFT by adding alpha-prime corrections, cancelling anomalies, etc. Therefore one finds authors who worry that too many string vacua are known, and other authors who worry that too few string vacua are known.

This is the first point where physics parts with a systematic mathematical development. Namely while it is hard to really construct full 2d SCFTs, the folk lore of the path integral allows to think of solutions to supergravity equations of motion as inducing 2d SCFTs via quantization of the "nonlinear sigma-model" with these spacetimes as their targets. So now instead of deducing effective target space geometry from the SCFT algebra, one prescribes classical target space geometry to which one believes SCFTs may be associated. It is from here on that much of string theory is now phrased in terms of classical geometry with some extra stringy effects sprinkled in (modular ivariance, anomaly cancellation, brane instanton effects).

This is the first point where physics parts with a systematic mathematical development. Namely while it is hard to really construct full 2d SCFTs, the folk lore of the path integral allows to think of solutions to supergravity equations of motion as inducing 2d SCFTs via quantization of the "nonlinear sigma-model" with these spacetimes as their targets. So now instead of deducing effective target space geometry from the SCFT algebra, one prescribes classical target space geometry to which one believes SCFTs may be associated. It is from here on that much of string theory is now phrased in terms of classical geometry with some extra stringy effects sprinkled in (modular invariance, anomaly cancellation, brane instanton effects).