# Contents

## Idea

There are various hints (originally observed in Witten 95) that all perturbative superstring theories (type II (A and B), type I, heterotic ($SO(32)$ and $E_8 \times E_8$)) have a joint strong coupling non-perturbative limit whose low energy effective field theory description is 11-dimensional supergravity and which reduces to the various string theories by Kaluza-Klein compactification on an orientifold torus bundle, followed by various string dualities. Since the string itself is thought to arise from a membrane/M2-brane in 11-dimensions after double dimensional reduction this hypothetical theory has been called “M-theory” short for “membrane theory”; e.g. in Hořava-Witten 95:

As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes.

The “reasons to doubt” that interpretation is that the M2-brane certainly does not support a perturbation theory the way that the superstring does. This is part of the reason why the actual nature of “M-theory” remains mysterious.

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 – particularly from duality in string theory – 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.

### Relation to F-theory

The compactification of M-theory on a torus yields type II string theory – directly type IIA, and then type IIB after T-dualizing. It turns out that the axio-dilaton of the resulting type II-B string theory is equivalently the complex structure-modulus of this elliptic fibration by the compactification torus. This gives a description of non-perturbative aspects of type II which has come to be known as F-theory (see e.g. Johnson 97).

In slightly more detail, write, topologically, $T^2 = S^1_A\times S^1_B$ for the compactification torus of M-theory, where contracting the first $S^1_A$-factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory, also the second $S^1_B$ is to be compactified (since T-duality sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$). Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the torus $S^1_A \times S^1_B$. More generally, this torus may be taken to be an elliptic curve and this may vary over the 9d base space as an elliptic fibration.

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity vacuum is the input data that F-theory is concerned with.

A schematic depiction of this story is the following:

M-theory in $d = 11$F-theory in $d = 12$
$\downarrow$ KK-reduction along elliptic fibration$\downarrow$ axio-dilaton is modulus of elliptic fibration
IIA string theory in $d = 9$$\leftarrow$T-duality$\rightarrow$IIB string theory in $d = 10$

In the simple case where the elliptic fiber is indeed just $S^1_A \times S^1_B$, the imaginary part of its complex modulus is

$Im(\tau) = \frac{R_A}{R_B} \,.$

By following through the above diagram, one finds how this determines the coupling constant in the type II string theory:

First, the KK-reduction of M-theory on $S^1_A$ yields a type IIA string coupling

$g_{IIA} = \frac{R_A}{\ell_s} \,.$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

\begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.

### Cohomological properties

A derivation of D-brane charge, RR-fields and other K-theory structure in type II superstring theory from M-theory was argued in (FMW 00).

Seet also at cubical structure in M-theory.

(…)

## 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

For more original references see also at M2-brane.

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)

Early articles clarifying the relation to type II string theory now known as F-theory include

The relation also to the heterotic string was understood (see at Hořava-Witten theory) in

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)

### Cohomological considerations

Discussion of the cohomological charge quantization in type II (RR-fields as cocycles in KR-theory) in relation to the M-theory supergravity C-field is in

For more on this perspective as 10d type II as a self-dual higher gauge theory in the boudnary of a kind of 11-d Chern-Simons theory is in

### 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

See also the references at exceptional generalized geometry.

In fact, much of the broad structure of M-theory and its relation to the various string theory limits can be seen from the classification of exceptional super L-∞ algebras (such as the supergravity Lie 3-algebra and the supergravity Lie 6-algebra), as discussed in

By passing to automorphism algebras, this reproduces the polyvector extensions of the super Poincaré Lie algebra, which enter the traditional discussion of M-theory, such as the M-theory super Lie algebra (which arises as the symmetries of the M5-brane ∞-Wess-Zumino-Witten theory).

Revised on May 19, 2014 00:37:42 by Urs Schreiber (217.39.7.253)