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2
Minkowski norm of the compactification direction, -Rs, is taken to zero. If we assume
that compactified M theory has a vacuum state invariant under Lorentz transformations
in the uncompactified directions, then this is equivalent to compactification on a spacelike
circle (0, Rs, 09). But M theory on a spacelike circle of zero radius is free Type IIA string
theory.
18
This prescription was independently invented by L. Susskind, and described to the author in
July of 1997.
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In DLCQ, we are instructed to study the light cone Hamiltonian (energy minus lon-
gitudinal momentum) in a sector of fixed positive longitudinal momentum. Using Lorentz
invariance, this translates into a sector with fixed positive momentum around the spacelike
circle. In Type IIA language, we are instructed to work in a sector with fixed D0 brane
number N and to subtract the zerobrane mass N/R from the Hamiltonian. We then obtain
a spectrum of zero brane kinetic energies which go to zero for fixed transverse momentum
(of order the eleven dimensional Planck scale), as Rs goes to zero. Seiberg shows that this
is the scale of energies which are finite in the original, almost lightlike, frame. We should
thus try to write down the Effective Hamiltonian for all states which have energies of this
order or lower. The work of [15] , and [16] tells us that we should then include the SYM
interactions between D0 branes, which have the same scaling as their kinetic energy.
In the compactified theory, with compactification radii of order the eleven dimensional
Planck scale (and thus small in string units), the cleanest way to isolate the relevant degrees
of freedom and interactions is to do a T duality transformation. On tori of dimension less
than or equal to three, this reproduces the SYM prescription for compactification. On the
four torus, the T dual theory is the theory of N D4 branes, but the T dual IIA coupling
is going to infinity. Thus, the proper way to view this system is as a set of N fivebranes
in M theory, wrapped around the large eleventh dimension, and the T dual four torus.
The scale of the (original picture) D0 brane kinetic energies is the same as that of the
tensionless strings on the T dual M theory fivebranes. Thus the effective theory is the
(0, 2) conformal field theory, with structure group U(N).
On the five torus the T dual theory is that of N D5 branes in strongly coupled IIB
string theory. By S duality, this is the same as the system of IIB Neveu-Schwarz fivebranes
2
at infinitely weak coupling and fixed string tension, MS, which is the proposal of Seiberg
described above. We note however, that Maldacena and Strominger [58] have recently
argued that this system also contains a new continuum of states above a gap of order
MS. These are abstracted from a SUGRA description of the excitations of the NS 5 brane
system as near extremal black holes. The string frame description of these geometries
contains an infinite tube which decouples from the bulk as the string coupling goes to
zero. If the deviation from extremality is taken to zero along with the string coupling, it
is argued that the coupling between the continuum of modes running up and down the
tube, and the excitations on the five branes, remains finite in the limit. It goes to zero
only when N is taken to infinity.
45
We will see that this continuum represents extremely bizarre physics from the Matrix
Theory point of view, but we reserve that discussion until we have described the even more
bizarre situation on the six torus. Here, the T dual theory is that of D6 branes in strongly
coupled Type IIA string theory. Again, the appropriate description is in terms of eleven
dimensional SUGRA. That is, the theory consists of M theory compactified on a circle
with radius RT ! ", with N Kaluza-Klein monopoles (the original D0 branes) wrapped
around a six torus whose size is the T dual eleven dimensional Planck scale, Lp,(this is
T dual to the original six torus whose size is the original eleven dimensional Planck scale
before T duality). In the limit we get the theory of AN-1 singularities interacting with
SUGRA and wrapped around a Planck scale six torus.
Certain aspects of the physics are best understood before taking the limit. Then
the Kaluza-Klein monopoles can be viewed as particles in the uncompactified spacetime.
2
They have mass RT L-3. If the T dual D0 branes are given finite momenta in the original
p
Planck units, then the KK monopoles must be given momenta RT L-2. Their energy
p
is then finite in Lp units. But the situation regarding momenta is strange. The KK
monopoles carry infinitely more uncompactified momentum in the RT ! " limit than
the supergravitons of comparable energy. Thus in the limit, KK monopole momentum is
conserved, while supergraviton momentum is not. Indeed, this is the only way that the
relativistic supergraviton dispersion relation could have been made compatible with the
Galilean invariance which we require for the light cone interpretation of the matrix model.
From the point of view of the original M theory which we are trying to model, the
physics of this system is completely bizarre. It says that M theory (or at least DLCQ M
theory) on a six torus contains a continuum of excitations in addition to that described by
the asymptotic multiparticle states in ordinary spacetime. These states carry finite light
cone energy, but no transverse or longitudinal momentum. Scattering of M theory particles
can create these states and the energy lost to them need never appear in the asymptotic
region of M theory. The asymptotic states (in the usual sense) of M theory are not
complete, and their S matrix is not unitary. The situation is analogous to a hypothetical
theory of black hole remnants, except that the remnants are zero momentum objects
which fill all of transverse and longitudinal space in the light cone frame. Furthermore,
it is clear that such a description is not Lorentz covariant under the lightplane rotating
transformations which we hope to recover in the large N limit. The excitations described
on the five torus by Maldacena and Strominger produce a very similar picture, the major
46
difference being that their excitations are separated from the spacetime continuum by a
finite gap, and are therefore invisible at sufficiently low energies.
There is reason to believe that both of these problems go away in the large N limit
(and I think they must if the theory is to be Lorentz invariant ). KK monopoles repel
supergravitons carrying nonzero momentum (the only ones which couple in the RT ! "
limit) around the KK circle, because the size of the circle goes to zero at the center of the
monopole. In the RT ! " limit, the circle has infinite radius everywhere apart from the
position of the singularity. In the large N limit, this repulsion becomes infinitely strong.
Supergravitons localized at any finite distance from an A "-1 singularity have infinite
energy19. Thus it is plausible, though not proven, that they decouple in this limit. Similar
remarks may be made on the five torus, where Maldacena and Strominger have argued
that the excitations propagating in the throat of the near extremal black hole decouple
because the Hawking radiation rate vanishes in the large N limit. We will expand further
on these remarks when we discuss DLCQ below.
The problems encountered on the five and six tori may, in a way which I do not yet
understand, be precursors of a more evident problem in lower dimensional compactifica-
tions. DLCQ of a theory with four noncompact dimensions is effectively a 2+1 dimensional
theory, and gravity compactified to 2 + 1 dimensions has infrared divergences if we require
the geometry to be static. This is the origin of the claim[59] that low dimensional string
theories with static geometries do not have many states. Zeroth order string perturbation
theory misses this effect and leads one to expect toroidal compactifications of any dimen-
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