# 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

$\begin{array}{ccc}???& \stackrel{\mathrm{effective}\mathrm{QFT}}{\to }& 11d\mathrm{Supergravity}\\ ↓& & {↓}^{\mathrm{dimensional}\mathrm{reduction}}\\ \mathrm{StringTheory}& \stackrel{\mathrm{effective}\mathrm{QFT}}{\to }& 10d\mathrm{Supergravity}\end{array}$\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 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, a partial sigma-model-description of this M2-brane has only very recently surfaced (see BLG model and ABJM theory) and even so it is clear that this cannot quite give a definition of the top left corner by perturbation theory (as the string 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.

### 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

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

The term “M-theory” occurs somewhere around