Contents

# 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 Horava-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)

The defining characteristic of M-theory is that it exhibits duality with type IIA string theory in the following way:

$\array{ M-Theory(?) &\stackrel{low\;energy\;limit}{\to}& 11d Supergravity \\ {}^{\mathllap{ \array{ small \\ coupling \\ limit}}}\downarrow && \downarrow^{\mathrlap{ \array{KK\;reduction \\ on S^1 }}} \\ type IIA string theory &\stackrel{low\;energy\;effective\;QFT}{\to}& 10d Supergravity }$

(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.

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.

## 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. 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. (Horava-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 D$0$-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. See at duality between M-theory and 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.

### 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-compactification of M-theory on $S^1_A$ yields a type IIA string coupling constant

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

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} \,.

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

M-theory on $S^1_A \times S^1_B$-elliptic fibrationKK-compactification on $S^1_A$type IIA string theoryT-dual KK-compactification on $S^1_B$type IIB string theorygeometrize the axio-dilatonF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping $S_A^1$double dimensional reduction $\mapsto$type IIA superstring$\mapsto$type IIB superstring$\mapsto$$\mapsto$heterotic superstring
M2-brane wrapping $S_B^1$$\mapsto$D2-brane$\mapsto$D1-brane$\mapsto$
M2-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ strings and $q$ D2-branes$\mapsto$(p,q)-string$\mapsto$
M5-brane wrapping $S_A^1$double dimensional reduction $\mapsto$D4-brane$\mapsto$D5-brane$\mapsto$
M5-brane wrapping $S_B^1$$\mapsto$NS5-brane$\mapsto$NS5-brane$\mapsto$$\mapsto$NS5-brane
M5-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ D4-brane and $q$ NS5-branes$\mapsto$(p,q)5-brane$\mapsto$
M5-brane wrapping $S_A^1 \times S_B^1$$\mapsto$$\mapsto$D3-brane$\mapsto$
KK-monopole/A-type ADE singularity (degeneration locus of $S^1_A$-circle fibration, Sen limit of $S^1_A \times S^1_B$ elliptic fibration)$\mapsto$D6-brane$\mapsto$D7-branes$\mapsto$A-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity$\mapsto$D6-brane with O6-planes$\mapsto$D7-branes with O7-planes$\mapsto$D-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity$\mapsto$$\mapsto$$\mapsto$exceptional ADE-singularity of elliptic fibration$\mapsto$E6-, 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).

### Kaluza-Klein compactifications

The AdS-CFT duality for the blackM5-brane may be turned around to deduce from the 6d (2,0)-superconformal QFT on the M5-brane scattering amplitudes in the 11-dimensional bulk-spacetime, hence in putative M-theory. While the 6d (2,0)-superconformal QFT is not completely known, conformal invariance and supersymmetry tightly constrains it (“conformal bootstrap”) and does allow to extract results.

This approach to computing putative M-theory scattering amplitudes is due to (ChesterPerlmutter18).

(…)

## The open problem of formulating M-theory

The tight web of hints and plausiblity checks notwithstanding, an actual formulation of M-theory as an actual theory remains an open problem.

This is not outrageous in itself: In mathematics there are good examples of cases where a collection of situations was or is suspected to be limiting cases of a single unified theory, without that theory itself having been or being known.

One example of this is the putative theory “absolute geometry over the field with one element”. In this analogy, the various perturbative string theories (HET, I, IIA, IIB and their KK-compactifications) correspond to arithmetic geometries over base prime field $\mathbb{F}_p$ for $p \geq 2$, and the would-be M-theory corresponds to a theory of a “field with one element” that unifies all this, by describing it at a deeper level (literally: a deeper base).

On the other hand, parts of the physics-minded literature tends to forget or downplay the conjectural nature of many assumptions or leaps of faiths that are being made when it comes to discussion of D-brane/M-brane dynamics and generally of non-perturbative effects in string theory.

The following is a collection of quotes from authors that highlight the open problem:

### General

The overriding problem in superunification in the coming years will be to take the Mystery out of M-theory, while keeping the Magic and the Membranes.

the still rather mysterious M theory governs many aspects of the lower dimensional string theories. What little is known of M theory is powerful enough to lead us to new phenomena in string theory and indeed new string theories. In particular the M theory fivebrane is strongly believed hold a new kind of self-dual string theory on its worldvolume. This new and somewhat elusive theory has also appeared in other contexts such as the compactification of type IIB string theory on K3 , M(atrix) theory on $\mathbb{T}^5$ and the S-duality of $N = 4$, $D = 4$ super-Yang-Mills. Thus one may hope that a greater understanding of this self-dual string may lead directly to a greater understanding of duality, string theory and M theory.

Despite all these successes, physicists are glimpsing only small corners of M-theory; the big picture is still lacking.

$[...]$

Indeed future historians may judge the late 20th century as a time when theorists were like children playing on the seashore, diverting themselves with the smoother pebbles or prettier shells of superstrings while the great ocean of M-theory lay undiscovered before them.

we are only just beginning to scratch the surface of the ultimate meaning of M-theory, and for the time being therefore, M stands for Magic and Mystery too.

Despite the recent excitement, however, we do not think that M(atrix) theory and the d= 11 supermembrane in their present incarnation are already the final answer in the search for M-Theory, even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine flat transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity.

future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them.

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.

A good start was given by the Matrix theory approach of Banks, Fischler, Shenker and Susskind. We have every reason to expect that this theory produces the correct scattering amplitudes of modes in the 11-dimensional supergravity multiplet in 11-dimensional Minkowski space - even at energies sufficiently large that black holes should be created. (This latter phenomenon has never been explicitly demonstrated). But Matrix theory is only a beginning and does not give us the whole picture of M-theory. The program ran into increasing technical difficulties when more complicated compactifications were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. $[...]$). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics.

given our utter lack of a complete description of M-theory, the bulk is not terribly useful for determining finite aspects of the dual CFT. However, we can turn this problem around using the modern perspective of the conformal bootstrap, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work.

I actually believe that string/M-theory is on the right track toward a deeper explanation. But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is missing or where to find it. The reason I am not is that in hindsight it is clear the view given in the 1980s of what is missing was too narrow. Instead of discovering what we thought was missing, we broadened the picture in the 90s, in unexpected directions.

The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what’s lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don’t know what it is.

In a certain sense, and this is not a popular statement, I think it’s premature to be asking: “What are the empirical consequences”, because it’s not yet in a mature enough state, where we can sensibly make falsifiable prediction.

(06:59) But the matrix model itself was not all of M-theory; it was a corner of M-theory, and it told us certain interesting things, but there were interesting things about M-theory that it didn’t tell us.

(07:13) I think we are still looking, in fact, for what M-theory really is.

(07:19) We have a patchwork picture. We understand various corners. But the overarching big picture of M-theory is still waiting to be discovered, in my view.

(08:12) M-theory in 1995 was very promising, and it’s taught us a lot about the fundamental interactions; but the final theory is still not with us.

(12:46) I wouldn’t like to predict what the ultimate picture of M-theory will be; I imagine it will be something quite different from what we can imagine now.

(16:36) That’s why I think M-theory is not yet in a mature enough stage for us to make falsifiable predictions.

(16:44) We don’t understand the theory sufficiently well yet to do so.

### Non-abelian DBI-action

A key ingredient of M-theory is supposedly the physics of intersecting branes with gauge enhancement in their worldvolume super Yang-Mills theory/DBI field theory. But the actual derivation or even formulation of the expected non-abelian DBI action remains open:

For a system of multiple D$p$-branes, the world-volume action is much less well understood than for a single brane. The extension of $[$ the super Yang-Mills action $]$ to a full super-symmetric or κ-symmetric nonabelian Born-Infeld action is not known.

$[$the proposal by Tseytlin 97$]$ has not yet been derived from any more fundamental principles

The explicit construction of such an action is a difficult problem that has been studied extensively (starting with [17]), but is not yet completely settled.

Chemissany 04, p. 5 (7 of 84)

Several attempts to generalize the Born-Infeld action describing one D-brane to non-Abelian action describing a stack of them have been made. The proper (perhaps closed) form of it is however not known up to date.

### M5-Brane anomaly cancellation

Another key ingredient of M-theory is the M5-brane. The argument for anomaly cancellation has a convoluted history (see there).

$[$$]$ the solution is not so clear. $[$The established procedure$]$ will not work for the M5-brane. $[$$]$ something new is required. What this something new is, is not a priori obvious. $[$$]$ $[$This is$]$ a daunting task. To my knowledge no serious attempts have been made to study the problem.

$[$The proposal of FHMM 98$]$ probably should not be viewed as a final understanding of the problem. One would eventually hope for a microscopic formulation of M-theory which makes some of the manipulations $[$proposed in FHMM 98$]$ appear more natural.

### Coincident M5-branes

Formulating the D=6 N=(2,0) SCFT expected on coincident M5-branes (and thus the gauge enhancement on coincident D4-branes under duality between M-theory and type IIA string theory) remains an open problem:

The IR dynamics of $N$ coincident M5 branes $[...]$ remains largely mysterious.

The M5-brane theory remains an important open problem.

The low energy effective theory on multiple M2 branes has been well understood in recent years. However, the low energy effective theory on multiple M5 branes is still an open problem.

3.6 Open problems and wishes

Let me close this discussion of M5-branes with some open problems and wish list of results:

i) Provide a field definition/construction of the (2,0) theory $[$$]$

ii) Find the mathematical structures that best capture aspects of the (2,0) theory $[$$]$

## References

### General

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 at University of Southern California, 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” originates 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)

and

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 (Horava-Witten 95)❛ see at Horava-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)

Relation to AdS/CFT and conformal bootstrap:

• Shai M. Chester, Eric Perlmutter, M-Theory Reconstruction from $(2,0)$ CFT and the Chiral Algebra Conjecture (arXiv:1805.00892)

### 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 D$0$-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