The BFSS matrix model is a quantum mechanical model – i.e. quantum field theory in 0+1 dimensions - that describes uncompactified M-theory in 11 dimensions assuming that we study the large $N$ limit of the model with the $U(N)$ symmetry.

As myself and later Susskind determined, one may also directly interpret the finite $N$ BFSS matrix model as describing M-theory with a light-like direction $x^- = (x^0+x^{10})/\sqrt{2}$ compactified on a circle of radius $R$. This compactification is equivalent to the statement that the complementary momentum coordinate, $p^-$, is quantized i.e. equal to $N/R$ with the same $R$ I mentioned in the previous sentence. This description in terms of light-like coordinates one of which is compactified has been named Discrete Light Cone Quantization, DLCQ.

Because light-like compactification is unfamiliar and just "marginally" consistent (it is in between spacelike compactification which is OK and timelike one which creates closed timelike curves which are inconsistent), it is often helpful to define the light-like compactification as a limit of a spacelike one. This is the limiting trick that Seiberg uses, in the paper you linked to, as the basis to prove that the BFSS matrix model is correct.

When we want to get rid of the light-like compactification, it's enough to study the $R\to \infty$ "decompactification" limit. To keep all the components of the 11D energy-momentum vector fixed, we need $N/R$ fixed which means that $N$ must be sent to infinity, too.

It is misleading to say that the BFSS Matrix theory describes D0-branes. The BFSS model describes M-theory which includes no D0-branes as physical objects. The description – the quantum mechanical model with matrices – may be defined from a limit of dynamics of D0-branes in type IIA string theory but the 10-dimensional spacetime of this type IIA string theory has no direct relationship to the actual spacetime whose physics we want to describe by the matrix model.

The BFSS martrix model describes M-theory i.e. the superselection sector of string/M-theory that approaches an 11D spacetime at infinity only. If we insist on a finite value of $N$, this spacetime must be equipped with the compactified null direction. All other states of string/M-theory, like 10D string theories or their compactifications, are inaccessible by the BFSS model. The moduli are "frozen" in the BFSS model. They have $p^-=0$ whose Hilbert space of the $U(0)$ BFSS model is trivial so there's no way to unfreeze them and go to different superselection sectors.

However, there are other "matrix models" related to the BFSS model but not identical to it that describe various compactifications of string/M-theory in the same way as BFSS describes 11D M-theory. The most familiar one is matrix string theory which describes type IIA string theory in 10D. A similar matrix string theory with an $O(N)$ gauge group is known for the $E_8\times E_8$ heterotic string.

Schematically, the matrix model for M-theory compactified on $T^k$ is the maximally supersymmetric $U(N)$ gauge theory in $k+1$ dimensions with all $k$ spatial dimensions compactified on a torus. For $k=4,5$, this theory is ill-defined in the UV and string theory completes them to the $(2,0)$ theory in $d=6$ or little string theory in $d=6$. Seiberg-Sen's derivation gives these proper UV completions automatically. The matrix models for $k\gt 5$ (where we expect the truly exceptional noncompact U-duality groups) are not known and the Seiberg-Sen derivation breaks down. In a similar way, there are no known matrix models for sectors of string/M-theory with too many compactified spacetime dimensions.

The BFSS model is a special case for $k=0$. However, it may also be derived as a limit for any allowed value of $k$ in which the radii of the spacetime circles are sent to infinity.

The case $k=1$ is type IIA matrix string theory. The matrix model is the $1+1$-dimensional maximally SUSic $U(N)$ gauge theory with the spatial dimension compactified on a circle. This model may be derived from the low-energy dynamics of D1-branes in type IIB string theory. Again, the type IIB string theory has no "direct" relationship to the type IIA spacetime that the model ultimately describes – the link between them involves a T-duality and a limiting compactification followed by a boost.

The appearance of several "type IIA" pictures was originally confusing not only for the OP but for many famous physicists, too. After I released the first paper establishing matrix string theory, I received a confused e-mail from a physicist named Edward Witten (who previously sent me an enthusiastic e-mail on another paper) claiming that I was trying to address some questions about D0-branes in the BFSS model (because I talked about type IIA). But I wasn't. Instead, I derived the right – different than BFSS - matrix model for type IIA string theory and brought the first evidence, including the existence proof for the "long strings", that the theory coincides with the known spectrum and interactions of type IIA string theory.

There are also known matrix models for a K3 compactification of M-theory and several models for heterotic string/M-theory – follow e.g. the citations to the paper linked in this sentence. Quite typically, the vacua with 1/2 of the original SUSY have the gauge group $USp(2N)$ or, more realistically, $O(N)$. It's actually $O(N)$ and not $SO(N)$ and $N$ is allowed to be even or odd; all these "details" play a role in the derivation of all the states that the heterotic theory should have.

M-theory on a space with a single boundary – supporting the $E_8$ gauge supermultiplet – is a version of the BFSS matrix model. However, the gauge group is $O(N)$ and not $U(N)$. Some fields in the theory are required to be symmetric matrices of $O(N)$, others are antisymmetric, and there are extra 16 real fermionic vectors $\lambda$ that are the source of the $E_8$ states we may derive on the Hořava-Witten boundary.

Some compactifications of the $E_8$-based heterotic strings have known matrix models, too. Generally, the maximum number of dimensions we're allowed to compactify is even more constrained than in the case of the maximally SUSic vacua (toroidal compactifications of M-theory).

Matrix string theory exists for type IIA string theory as well as type IIB string theory in $d=10$ (the latter is a limit of the $2+1$-dimensional gauge theory). All the states, including all the even- and odd-dimensional D-branes, respectively, are contained in the models. However, it's only easy to demonstrate their presence for D-branes of sufficiently low dimension that preserve some SUSY. If they're still extended over the whole space, which is often needed for SUSY, except for D0-branes in type IIA, one also has to add new degrees of freedom to the matrix model.

Matrix theory – by which I mean the union of the BFSS matrix model and all of its "mutations" designed to describe other vacua – is another dual description of some situations in string/M-theory, much like the CFT in AdS/CFT is one more description like that. In each description like that, it may be harder to see certain objects or interactions that are easier to see in another description.

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Luboš Motl