Wednesday, October 26, 2005

Spacetime dynamics and RG flow

A couple of days ago there appeared a paper by Freedman, Headrick, and Lawrence that I find highly original. It not just follows up on a number of other papers but actually answers a question that has been lurking around for quite a while but had not really been addressed so far (at least as far as I am aware of). I had asked myself the question before but attributed it to my lack of understanding of the field and never worried enough to try to work it out myself. At least, these gentlemen have and produced this beautiful paper.

It is set in the context of tachyon condensation (and this is of course where all this K-Theory stuff is located): You imagine setting up some arrangement of branes and (as far as this paper is concerned even more important as this is about closed strings) some spatial manifold (if you want with first fundamental form, that is the conjugated momentum to a spatial metric) with all the fields you like in terms of string theory and ask what happens.

In general, your setup will be unstable. There could be forces or you could be in some unstable equilibrium. The result is that typically your space-time goes BOOOOOOOOOOM as you had Planck scale energy densities all around but eventually the dust (i.e. gravitational and other radiation) settles and you ask: What will I find?

The K-Theory approach to this is to compute all the conserved charges before turning on dynamics and then predicting you will end up in the lowest energy state with the same value for all the charges (here one might worry that we are in a gravitational theory which does not really have local energy density but only different expansion rates but let's not do that tonight). Then K-Theory (rather than for example de Rham or some other cohomology) is the correct theory of charges.

The disadvantage of this approach is that it is potentially very crude and just knowing a couple of charges might not tell you a lot.

You can also try to approach the problem from the worldsheet perspective. There you start out with a CFT and perturb it by a relevant operator. This kicks off a renormalisation group flow and you will end up in some other CFT describing the IR fixed point. General lore tells you that this IR RG fixed point describes your space-time after the boom. The c-theorem tells you that the central charge decreases during the flow but of course you want a critical string theory before and after and this is compensated by the dilaton getting the appropriate slope.

The paper is addresses this lore and checks if it is true. The first concern is of course that proper space-time dynamics is expected to (classically) be given by some ordinary field equation in some effective theory with typically two time derivatives and time reversal symmetry where the beta functions play the role of force. In contrast, RG flow is a first order differential equation where the beta-functions point in the direction of the flow. And (not only because of the c-theorem) there is a preferred direction of time (downhill from UV to IR).

As it is shown in the paper, this general scheme is in fact true. And since we have to include the dilaton anyway, this also gets its equation of motion and (like the Hubble term in Friedman Robertson Walker cosmology) provides a damping term for the space-time fields. So, at least for large damping, the space-time theory is also effectively first order but at small (or negative which is possible and of course needed for time reversal) damping the dynamics is of different character.

What the two descriptions agree on is the set of possible end-points of this tachyon condensation, but in general the dynamics is different and because second order equations can overshoot at minima, the proper space-time dynamics can end up in a different minimum than predicted by RG flow.

All this (with all details and nice calculations) is in the paper and I can strongly recommend reading it!

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