Contents

Idea

The term M-theory refers to a conjectured non-perturbative UV-completion of 11-dimensional supergravity whose dimensional reduction yields string theory.

Keeping in mind that already string theory itself and in fact already quantum field theory itself have only partially been formulated in a precise way, the conjecture is motivated from the fact that with the available knowledge of these subjects, one can see indications that there is a kind of commuting diagram of the form

$\array{ ??? &\stackrel{effective QFT}{\to}& 11d Supergravity \\ \downarrow && \downarrow^{\mathrlap{dimensional reduction}} \\ StringTheory &\stackrel{effective QFT}{\to}& 10d Supergravity }$

in some sense. The unknown top left corner here has optimistically been given a name, and that is “M-theory”. But even the rough global structure of the top left corner has remained elusive.

Hints

The available evidence that there is something of interest consists of various facets of the bottom left and the top right entry of the above diagram, that seem to have a common origin in the top left corner.

Membranes

Notably, from the black brane-solution structure in 11-dimensional supergravity and from the brane scan one finds that it contains a 2-brane, called the M2-brane, and to the extent that one has this under control one can show that under “double dimensional reduction” this becomes the string. However, it is clear that this cannot quite give a definition of the top left corner by perturbation theory as the superstring sigma-model does for the bottom left corner, because by the very nature of the conjecture, the top left corner is supposed to be given by a non-perturbative strong-coupling limit of the bottom left corner.

Strongly coupled type IIA strings and D0-branes

There is a bunch of consistency checks on the statement that the KK-compactification of 11-dimensional supergravity on a circle gives the strong coupling refinement of type IIA string theory.

One aspect of this is that type IIA string theory with a condensate of D0-branes behaves like a 10-dimensional theory that develops a further circular dimension of radius scaling with the density of D0-branes. (Banks-Fischler-Shenker-Susskind 97, Polchinski 99). See also (FSS 13, section 4.2).

U-duality

Another hint comes from the fact that the U-duality-structure of supergravity theories forms a clear pattern in those dimensions where one understands it well, giving rise to a description of higher dimensional supergravity theories by exceptional generalized geometry. Now, this pattern, as a mathematical pattern, can be continued to the case that would correspond to the top left corner above, by passing to exceptional generalized geometry over hyperbolic Kac-Moody Lie algebras such as first E10 and then, ultimately E11. The references there show that these are huge algebraic structures inside which people incrementally find all kinds of relations that are naturally identified with various aspects of M-theory. This leads to the conjecture that M-theory somehow is $E_{11}$ in some way. But it all remains rather mysterious at the moment.

(…)

References

General

The original insight that gave rise to the conjecture is due to

A public talk announcing the conjecture that the strong-coupling limit of type IIA string theory is 11-dimensional supergravity KK-compactified on a circle is at 15:12 in

• Edward Witten, talk, 1995(?) (video)

19:33: “Ten years ago we had the embarrassment that there were five consistent string theories plus a close cousin, which was 11-dimensional supergravity.” (19:40): “I promise you that by the end of the talk we have just one big theory.”

The term “M-theory” occurs somewhere around

An early popular account for a general audience is

• Edward Witten, Magic, Mystery, and Matrix, Notices of the AMS, volume 45, number 9 (1998) (pdf)

More technical surveys include

Surveys of the discussion of E-series Kac-Moody algebras/Kac-Moody groups in the context of M-theory include

• Sophie de Buyl, Kac-Moody Algebras in M-theory, PhD thesis (pdf)

• Paul Cook, Connections between Kac-Moody algebras and M-theory PhD thesis (arXiv:0711.3498)

Relation to D0-brane mechanics

Discussion of M-theory as arising from type II string theory via the effect of D0-branes is in

In terms of higher geometry

Discussion of phenomena of M-theory in higher geometry and generalized cohomology is in