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.

The BFSS matrix model was argued to arise in several 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 worldvolume theory of the M2-brane in M-theory (Nicolai-Helling 98, Dasgupta-Nicolai-Plefka 02).

In any case, it 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 spacetime thus defined.

There is also the IKKT matrix model, which takes this one step further by reducing one dimension further down (D(-1)-branes). See also at membrane matrix model.

Assessment

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 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.

Then, even assuming that in this corner all the crucial cohomological aspects of branes (K-theory charges, etc.) are secretly encoded in the matrix model, magically, 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 hadron physics.

Furthermore, there are 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 (here).

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

A good start $[$on 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.

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)

See also

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:

Relation to super-membranes

The interpretation of the BFSS model as a regularized M2-brane worldvolume theory is discussed in

Analogous regularizations lead to matrix model descriptions of D-branes:

Ground state problem

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

For a new attempt and 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 (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)

Last revised on July 13, 2019 at 07:16:13. See the history of this page for a list of all contributions to it.