There are various hints (originally observed in Witten 95) that all perturbative superstring theories (type II (A and B), type I, heterotic (SO(32)SO(32) and E 8×E 8E_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 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. On the other hand, later it was argued that there is a regularization of the M2-brane worldvolume theory, which makes it becomes the BFSS matrix model (Nicolai-Helling 98, Dasgupta-Nicolai-Plefka 02). In reaction to these developments it was suggested that “M-theory” could be read as “matrix theory”.

Later, the membranes were interpreted in terms of matrices. Purely by chance, the word “matrix” also starts with “m”, so for a while I would say that the M stands for magic, mystery, or matrix. (Witten 14, last paragraph)

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 M-theory 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

??? lowenergylimit 11dSupergravity dimensionalreduction StringTheory lowenergyeffectiveQFT 10dSupergravity \array{ ??? &\stackrel{low\;energy\;limit}{\to}& 11d Supergravity \\ \downarrow && \downarrow^{\mathrlap{dimensional\;reduction}} \\ StringTheory &\stackrel{low\;energy\;effective\;QFT}{\to}& 10d Supergravity }

in some sense (see also e.g. (Obers-Pioline 98, p. 12)). 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.


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.


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. At the bottom of it, this is simply 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. But one may also see that the evident guess for a would-be membrane analog of the string perturbation series fails

Mike Duff, Paul Townsend, and other physicists working on supermembranes had spent a couple of years in the mid-1980s saying that there should be a theory of fundamental membranes analogous to the theory of fundamental strings. That wasn’t convincing for a large number of reasons. For one thing, a three-manifold doesn’t have an Euler characteristic, so there isn’t a topological expansion as there is in string theory. Moreover, in three dimensions there is no conformal invariance to help us make sense of membrane theory; membrane theory is nonrenormalizable just like general relativity.

(Edward Witten in interview by Hirosi Ooguri, Notices Amer. Math. Soc, May 2015 p491 (pdf))

This issue is the very root of the abbreviation “M-theory”:

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. (Hořava-Witten 95)

M-theory was meant as a temporary name pending a better understanding. Some colleagues thought that the theory should be understood as a membrane theory. Though I was skeptical, I decided to keep the letter “m” from “membrane” and call the theory M–theory, with time to tell whether the M stands for magic, mystery, or membrane. Later, the membranes were interpreted in terms of matrices. Purely by chance, the word “matrix” also starts with “m”, so for a while I would say that the M stands for magic, mystery, or matrix. (Witten 14, last paragraph)

Strongly coupled type IIA strings and D00-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).

Discussion of the relation of gauge enhancement of M-theory at ADE singularities and the corresponding coincident D-brane geometries in type IIA string theory is in (Sen 97).

More on the decomposition of the supergravity C-field in 11d to the RR-fields and the NS-fields in type IIA is in (Mathai-Sati 03, section 4).

For survey of how the components maps see also the table at Relation to F-theory.


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 11E_{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 A 1×S B 1T^2 = S^1_A\times S^1_B for the compactification torus of M-theory, where contracting the first S A 1S^1_A-factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory, also the second S B 1S^1_B is to be compactified (since T-duality sends the radius r Ar_A of S A 1S^1_A to the inverse radius r B= s 2/R Ar_B = \ell_s^2 / R_A of S B 1S^1_B). Therefore type IIB sugra in d=10d = 10 is obtained from 11d sugra compactified on the torus S A 1×S B 1S^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=11d = 11F-theory in d=12d = 12
\downarrow KK-reduction along elliptic fibration\downarrow axio-dilaton is modulus of elliptic fibration
IIA string theory in d=9d = 9\leftarrowT-duality\rightarrowIIB string theory in d=10d = 10

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

Im(τ)=R AR B. 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 A 1S^1_A yields a type IIA string coupling

g IIA=R A s. g_{IIA} = \frac{R_A}{\ell_s} \,.

Then the T-duality operation along S B 1S^1_B divides this by R BR_B:

g IIB =g IIA sR B =R AR B =Im(τ). \begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,.

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on S A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype IIB string theoryF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping S A 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branesA-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planesD-type nodal curve cycle degenertion locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

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.


Many more




First indications for M-theory came from the supermembrane Green-Schwarz sigma-model now called the M2-brane. A comprehensive collection of early articles is in

For some time though the success of string theory in 10-dimensions caused resistence to the idea of a theory of membranes in 11-dimensions, an account is in (Duff 99) and in brevity on the first pages of

The article that convinced the community of M-theory was

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” seems to originate in

as a “non-committed” shorthand for “membrane theory”

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. (Hořava-Witten 95, p. 2)


which coined the association

the eleven-dimensional “M-theory” (where M stands for magic, mystery, or membrane, according to taste) (Witten 95, p. 1)

that later gained much publicity:

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

The argument that the regularized M2-brane worldvolume theory is the BFSS matrix model is discussed in

  • Hermann Nicolai, Robert Helling, Supermembranes and M(atrix) Theory, Lectures given by H. Nicolai at the Trieste Spring School on Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theories, 23 - 31 March 1998 (arXiv:hep-th/9809103)

  • Arundhati Dasgupta, Hermann Nicolai, Jan Plefka, An Introduction to the Quantum Supermembrane, Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003 (arXiv:hep-th/0201182)

Recollections include the last paragraph of

The term became fully established with surveys including

Despite the magic and mystery, the relation to the original abbreviation for membrane-theory was highlighted again for instance in

More recent review includes

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 in (Hořava-Witten 95)❛ see at Hořava-Witten theory.

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

More complete discussion of the decomposition of the supergravity C-field as one passes from 11d to 10d is in

Relation to D00-brane mechanics

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

More on the relation to type IIA string theory

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 December 12, 2016 20:06:02 by Urs Schreiber (