Tuesday, May 17, 2005

Matrix Big Bang

In the original article, I have described two talks from the conference at Columbia University. Savdeep Sethi chose a topic - his project with Ben Craps and Erik Verlinde - that was the most appropriate one for a conference about string cosmology. And it was a lot of fun:

  • The Matrix Big Bang

And because he allowed me to write about it, let's go ahead. It is likely that this article will include some mathematics. If you don't see it, download Techexplorer.

His plan was the following:
  • Background - for his talk
  • A background - a very simple background where strings can propagate
  • Review of BFSS matrix theory
  • Its holographic description

Sav explained that string theory has been extremely successful in dealing with a certain type of singularities - the timelike singularities such as orbifold and conifold singularities. Some of them even preserve supersymmetry - but most of them respect the finiteness of the effective Newton's gravitational constant.

On the other hand, string theory has so far been unsuccessful to resolve other singularities such as the spacelike singularities near the Big bang and in the middle of the black hole. Sav's description of the source of the problem is that the scattering quanta always "see" an infinite value of Newton's constant at some moment.

There has been a rather long discussion with Dan Kabat and others why there was really any difference in this respect between the spacelike and the timelike singularities, but Dan eventually kind of agreed. Whatever you do, perturbative string theory is guaranteed to fail near the spacelike singularity and a better description is needed.

Of course, Sav's proposed better description is BFSS matrix theory.

But before he got there, he discussed various singular spaces. One of the simplest singular spaces is the Misner space,

where we used the light-like coordinates and identified them via a boost. This orbifold behaves differently in the timelike and spacelike regions of the two-dimensional spacetime: in the timelike region it defines the Milne universe, while the spacelike part of the space is the Rindler space. (Note that the previous sentence contains inline math, and if it displays incorrectly in your browser, let me know.) This Misner space is locally flat, however. What is the simplest background in string theory after the trivial flat space? Well, it is the almost trivial flat space - which means flat space with a slightly non-trivial dilaton.

To make things really simple, we can take the dilaton gradient to be null.

where the parameter Q is fictitious because a boost changes its value. Note that this simple dilaton background does not spoil the equations of motion for the dilaton. It does not change the central charge either. The central charge of your two-dimensional conformal field theory is obtained from the correlator (OPE) of two copies of the stress energy tensor:

where the dots represent less singular terms. The only effect of the linear dilaton on the stress energy tensor is an extra term

where the second term measures the dilaton gradient. It also contributes to the central charge in the OPE

But if the vector V is null, you see that the contribution vanishes because its square is zero. Also, the spectrum of this theory is isomorphic to the spectrum of the trivial theory with constant dilaton. There is an extra term in the Virasoro generator

which can be combined with another term if we replace

Once again, the spectrum of a single string is not really affected by the linear null dilaton.

That's a fine and simple background. But let us study it in the superstringy context, namely type IIA string theory. In that case, we have enough supersymmetry for the metric to be exact and not corrected by quantum corrections. Also, it may be lifted to M-theory. One obtains a non-flat eleven-dimensional geometry in which ten coordinates shrink as we approach the Big Bang while the eleventh coordinate expands. We first rewrite the metric in Einstein frame where it's not flat anymore:

and then we convert it to an eleven-dimensional metric:

This is a cosmologically non-trivial background with a null singularity. Can we understand it? Can we evolve a state at finite time arbitrarily close to the singularity?

How could we ever do it without knowing what M-theory is? We know one definition of M-theory, namely the BFSS matrix model (Banks, Fischler, Shenker, Susskind). Sav reproduced Seiberg's argument why is the matrix model correct.

You want to describe M-theory in 11 dimensions. You artificially compactify the light-like coordinate

As long as R is taken large, physics is reproduced just fine. This null compactification is on the edge of creating deadly time-like curves, and we should rather define it as the limit of nearly null compactifications which are otherwise slightly spacelike. Any such nearly null (but spacelike) compactification can be boosted, using the Lorentz symmetry of M-theory, to a pure and manifestly spacelike compactification of M-theory. Because the identification was almost null, the new spacelike circle will be very short after the boost and we obtain type IIA string theory with coupling that goes to zero. One can prove that the limiting procedure and the energy regime we want to study (finite energies in 11D Planck units in the original picture) decouples all degrees of freedom from the theory except for massless open string states stretched between N D0-branes - well, N D0-branes is what the boost produces out of N units of momenta in the light-like circle "X minus":

Recall that momentum in the light-like direction which becomes the short 11th dimension of type IIA string theory has the perturbative interpretation of the D0-brane charge. Because we want to keep the momentum "p plus" finite and we want to send the light-like radius R to infinity, we must also send N to infinity - and therefore all of physics of M-theory is encoded in the U(N) matrix quantum mechanics which is the dimensional reduction of ten-dimensional Yang-Mills theory to 0+1 dimensions and whose Hamiltonian is (and now I use the tag "autosize='true'" to impress Jacques Distler, hopefully it will work for non-MSIE browsers, too)

Sav also recalled how one obtains matrix string theory - the baby of mine and DVV - by compactifying one more circle in the eleven-dimensional spacetime, thereby producing a matrix model for type IIA string theory which happens to be a maximally supersymmetric Yang-Mills theory in 1+1 dimensions compactified on a spatial circle.

What about Sav's background? He compactified one more background, repeated Seiberg's procedure and obtained matrix string theory defined on the Milne space, the same space that we discussed at the beginning because of other motivations. Surprisingly, the Milne compactification breaks all supersymmetries even though the background we want to describe should have 16 supercharges. I argued that in the large N limit, the full supersymmetry should be recovered in the matrix model and Sav disagreed.

It is not quite clear whether the correct matrix model is just pure Yang-Mills theory or whether excited open string modes remain important. At any rate, Sav's lesson is that we should study open string theories on non-trivial backgrounds because they can tell us - because of open-closed dualities such as matrix theory - non-trivial lessons about the way how quantum gravity resolves cosmological backgrounds.

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