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noncritical string theory, compactification may involve further conceptual problems.
The basic idea of [49] and [63] is that the two desired theories are limits of certain
situations in M theory. Since the matrix model gives us a definition of M theory in some
cases, we can try to use it to construct these theories. This may seem somewhat circular,
since we plan to use these theories to construct Matrix Theory! The point is that the
uncompactified version of e.g. the (0, 2) theory is obtained by studying the low energy
dynamics of closely spaced fivebranes in uncompactified M theory. If we choose a light
cone frame, and orient the fivebranes so that the longitudinal direction lies within them,
then Berkooz and Douglas [64] have given us a complete prescription for this system in
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e.g. a single longitudinal fivebrane is a single instanton on a torus.
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Matrix Theory. It is the theory of the low energy interactions of k D4 branes and N
D0 branes in IIA string theory. This is the dimensional reduction to 0 + 1 dimensions
of a six dimensional U(N) gauge theory with eight real supercharges. In addition to the
Yang Mills vector multiplet, we have a hypermultiplet in the adjoint representation and k
hypermultiplets in the fundamental. In the quantum mechanical reduction, the 5 spatial
components of the vector fields and the 4 real components of the adjoint hypermultiplet
represent the 9 (nonabelian) transverse coordinates of excitations in eleven dimensional
spacetime. The vector components are directions perpendicular to the fivebranes, while
the adjoint components are transverse light cone directions which are in the branes. There
is a U(k) global symmetry which acts on the fundamental hypermultiplets. The weight
lattice of this group is the charge lattice of the (0, 2) theory we are trying to construct. In
IIA string theory we would have a 4 + 1 dimensional U(k) SYM theory describing the self
interactions of the D4 branes, but in Matrix Theory these degrees of freedom are dropped
because they do not carry longitudinal momentum. The physics associated with these
gauge interactions should reappear automatically in the N ’! " limit.
The Berkooz-Douglas model describes longitudinal fivebranes in interaction with the
full content of M theory. The Coulomb branch of the space of fields, where the components
of the vector multiplet are large (but commuting so that the energy is low), represents
propagation away from the fivebranes, while the Higgs branch, on which hypermultiplet
components are large (but satisfy the D flatness condition so that the energy is low)
represents propagation within the fivebranes. Mathematically, the Higgs branch is the
moduli space of N SU(k) instantons. We would like to take a limit in which the Higgs and
Coulomb branches decouple from each other. Viewed as a dimensionally reduced gauge
theory, our model has only one parameter, the gauge coupling. The Coulomb and Higgs
branches describe subsets of zero frequency (the quantum mechanical analog of zero mass)
degrees of freedom which interact with each other via the agency of finite frequency degrees
of freedom. The gauge coupling is relevant, and if we take it to infinity all finite frequency
degrees of freedom go off to infinite frequency. Thus the Coulomb and Higgs branches of
the theory should decouple in this limit. The theory on the fivebranes, which we expect
to be the (0, 2) field theory, is thus argued to be the infinite coupling limit of quantum
mechanics on the Higgs branch.
In other words, the claim is that the light cone Hamiltonian of (0, 2) superconformal
field theory with U(k) charge lattice, is the large N limit of the supersymmetric quantum
mechanics on the moduli space of N SU(k) instantons. The latter form of the assertion
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invokes a nonrenormalization theorem. The instanton moduli space is Hyperkahler. Pre-
sumably (but I do not know a precise argument in quantum mechanics) the only relevant
supersymmetric lagrangians for these fields are just free propagation on some Hyperkahler
geometry. The metric on the moduli space is determined by the Hyperkahler quotient
construction in the limit of weak SYM coupling (that is, it is determined by plugging the
solution of the D flatness condition into the classical Lagrangian). Furthermore, the SYM
coupling can be considered to be a component of a vector superfield. Therefore the metric
of the hypermultiplets cannot be deformed and takes the same value when the coupling is
infinitely strong as it does when it is infinitely weak.
A stronger version of the assertion, which may be more amenable to checks, is that
the finite N instanton quantum mechanics is the DLCQ of the (0, 2) field theory.
A similar set of arguments can be made for the the theory of [53] , described as the
weak coupling limit of k NS fivebranes in IIA string theory [49] , [63] . We replay the
above analysis, for fivebranes in M theory compactified on a circle of small radius, taking
the fivebranes to be longitudinal, and orthogonal to the circle. This leads to a matrix
string theory which is just the dimensional reduction of the same six dimensional gauge
theory to 1 + 1 dimensions[65]. The gs ’! 0 limit of [53] is again the strong coupling limit
of the gauge theory, and the dynamics on the fivebrane is the two dimensional conformal
field theory of the Higgs branch. In this context there has been some confusion about the
appropriate Lagrangian describing the conformal field theory and the reader is referred to
the literature for more details. Even when this is sorted out, we will still be faced with the
problem of compactifying this nonlocal, noncritical string theory.
Although the arguments supporting this matrix model for matrix models approach
are quite beautiful and convincing, I would like to point out a possible loophole, and some
evidence that perhaps this construction fails. While the general logic of this construction
is impeccable, there is one point at which error could creep in. In taking the large coupling
limit one used renormalization group and symmetry arguments to determine the limiting
theory. These arguments seem perfectly sensible as long as it is true that the correct
low energy degrees of freedom have been completely identified. That is, the construction
assumes that the low energy degrees of freedom of the strongly coupled Higgs branch are
just the classical variables which parametrize that branch.
In ordinary quantum field theory, arguments like this can break down because of
the formation of bound states. The true low energy degrees of freedom in a regime not
amenable to perturbation theory are not simple combinations of the underlying degrees
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of freedom. Rather, in the infrared limit, the description of these states as composites of
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