Contents

# Contents

## Idea

The BFSS matrix model (Banks-Fischler-Shenker-Susskind 96, Seiberg 97) is the description of the worldline dynamics of interacting D0-branes. In the large N limit of a large number of D0-branes this is supposed to encode the strong coupling limit of type IIA string theory known as M-theory at least in certain corners of its moduli space.

The BFSS model is a limiting case of the BMN matrix model, which improves on some of its shortcomings (see the Open problems below).

The BFSS matrix model was argued to arise in several seemingly rather different (but apparently secretly equivalent) ways:

1. as the worldline theory of a large number of D0-branes in type IIA string theory,

2. as the Kaluza-Klein compactification of 10d super Yang-Mills theory to 1+0 space dimensions,

3. as a certain non-commutative regularization of the light-cone gauge quantization of the Green-Schwarz sigma-model for the M2-brane (Nicolai-Helling 98, Dasgupta-Nicolai-Plefka 02).

In this picture matrix blocks around the diagonal correspond to blobs of membrane, while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs.

graphics grabbed from Dasgupta-Nicolai-Plefka 02

In any case, the BFSS matrix model ends up being a quantum mechanical system whose bosonic degrees of freedom are a set of 9+1 large matrices. These play the role of would-be coordinate functions and their eigenvalues may be in interpreted as points in a non-commutative spacetime thus defined.

There is also the IKKT matrix model, which takes this one step further by reducing one dimension further down to D(-1)-branes in type IIB string theory.

## Open problems

### General

In the 90s there was much excitement about the BFSS model, as people hoped it might provide a definition of M-theory. It is from these times that Edward Witten changed the original suggestion that “M” is for “magic, mystery and membrane” to the suggestion that it is for “magic, mystery and matrix”. (See Witten’s 2014 Kyoto prize speech, last paragraph.)

However, while the BFSS matrix model clearly sees something M-theoretic, just as clearly it is not the full answer. Notably it needs for its definition an ambient asympototic Minkowski background, a light cone limit and a peculiar scaling of string coupling over string length, all of which means that it pertains to a particular corner of a full theory.

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.

Then, even assuming that in this corner all the crucial cohomological aspects of D-brane and M-brane charges (in twisted differential K-theory, twisted cohomotopy etc.) are secretly encoded in the matrix model, somehow, none of this is manifest, making the matrix model spit out numbers about a conceptually elusive theory in close analogy to how lattice QCD produces numbers without informing us about the actual conceptual nature of confined hadron physics.

A similar assessment has been given by Greg Moore, from pages 43-44 of his Physical Mathematics and the Future (here):

A good start $[$with defining M-theory$]$ 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.

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.

### Ground state

There are furcher technical open issues, such as the open question whether the theory has a decent ground state the way it needs to have to make sense (see the references below).

## References

### General

The original articles are

Review includes

A review of further developments is in

• David Berenstein, Classical dynamics and thermalization in holographic matrix models, talk at Leiden, October 2012 (pdf)

Discussion as a solution to the open problem of defining M-theory is in

where it says:

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.

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.

Derivation from open string field theory is discussed in

• Taejin Lee, Covariant Open String Field Theory on Multiple D$p$-Branes (arXiv:1703.06402)

Relation to the 6d (2,0)-supersymmetric QFT:

### Quantization of the M2-brane sigma-model to a matrix model

The Poisson bracket-formulation of the classical light-cone gauge Hamiltonian for the bosonic relativistic membrane and the corresponding matrix commutator regularization is due to:

On the regularized light-cone gauge quantization of the Green-Schwarz sigma model for the M2-brane on (super) Minkowski spacetime, yielding the BFSS matrix model:

Original articles:

Observation that the spectrum is continuous:

Review:

The generalization to pp-wave spacetimes (leading to the BMN matrix model):

### Relation to M5-branes

Discussion of light cone longitudal M5-branes in the BFSS matrix model (for light cone transversal M5-s see at BMN matrix model):

### Ground state problem

There remains the problem of existence of a sensible ground state of the BFSS model.

For a new attempt at solving this problem, and for pointers to previous attempts see

• L. Boulton, M.P. Garcia del Moral, A. Restuccia, The ground state of the D=11 supermembrane and matrix models on compact regions, Nuclear Physics B Volume 910, September 2016, Pages 665-684 (arXiv:1504.04071)

• L. Boulton, M.P. Garcia del Moral, A. Restuccia, Measure of the potential valleys of the supermembrane theory, Physics Letters B Volume 797, 2019, 134873 (arXiv:1811.05758)

### Graviton scattering

Computation of graviton scattering amplitudes:

### Black holes

Relation to black holes in string theory:

### Relation to lattice gauge theory

Relation to lattice gauge theory and numerical tests of AdS/CFT:

• Anosh Joseph, Review of Lattice Supersymmetry and Gauge-Gravity Duality (arXiv:1509.01440)

• Veselin G. Filev, Denjoe O’Connor, The BFSS model on the lattice, JHEP 1605 (2016) 167 (arXiv:1506.01366)

• Masanori Hanada, What lattice theorists can do for superstring/M-theory, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (arXiv:1604.05421)

### Holography

On AdS/CFT in the form of AdS2/CFT1 with the BFSS matrix model on the CFT side and black hole-like solutions in type IIA supergravity on the AdS side:

and concerning the analog of its holographic entanglement entropy:

Last revised on March 14, 2020 at 03:05:31. See the history of this page for a list of all contributions to it.