Noncompact CFT and the g-theorem

We clarify recent conjectures in string theory that the vacuum degeneracy g of a noncompact worldsheet sigma model with a continuous spectrum of scaling dimensions is lowered under renormalization group flow while preserving the central charge. Starting with a pedagogical review of Affleck and Ludwig’s g-theorem, we clarify the precise distinctions necessary in applying this result from unitary and compact two-dimensional conformal field theory (CFT) to the description of worldsheet renormalization group (RG) flows in string theory. We point out that the so-called nonuniversal contribution from relevant operators to the free energy is actually unambiguous in critical string theory: there is a unique choice of worldsheet ultraviolet regulator, zeta-function regularization, preserving all of the two-dimensional gauge symmetries. As an illustration, we use the worldsheet RG to describe the asymptotic approach to the zero temperature, supersymmetric limit of the unstable finite temperature type II string theory in the absence of branes, or of a Yang-Mills sector. At intermediate points along the flow to the infrared stable fixed point, we can separate the universal contribution to the free energy from the marginal and irrelevant bosonic modes in the worldsheet sigma model, from the contribution due to relevant modes. We confirm that the vacuum degeneracy is indeed lowered under worldsheet RG flow, and that the infrared stable endpoint is the noncompact, and supersymmetric, zero temperature vacuum, in broad agreement with the recent conjectures. shyamoli@thphysed.org A spate of recent work has addressed the subject of tachyon condensation in string theory, using a variety of techniques drawn from both open string field theory and the worldsheet renormalization group [4, 8]. An interesting case is that of localized closed string tachyon condensation in the context of noncompact worldsheet sigma models [5]. By noncompact is meant a worldsheet conformal field theory with an operator spectrum characterized by a continuum of scaling dimensions. In such cases, the assumptions under which Zamolodchikov’s c-theorem [1], or Affleck and Ludwig’s g-theorem [2], holds are no longer satisfied [3, 5], and it is interesting to ask the question of what quantity other than the effective central charge is minimized under worldsheet renormalization group (RG) flow? We should note that this is a question of fundamental interest in string theory, far beyond the particular context in which it has been raised. In [6], the interesting proposal has been made that the appropriate quantity in these cases is the vacuum degeneracy of bosonic states in the worldsheet sigma model. In this paper, we begin by clarifying some subtleties in the application of Affleck and Ludwig’s g-theorem to perturbative string theory dynamics. We will then illustrate the recent conjectures [5, 6] with an example of asymptotic RG flow to a noncompact infrared fixed point of the worldsheet conformal field theory. At intermediate points along the RG flow, the perturbative string theory we study has one, or more, tachyonic states in its tree level mass spectrum. While the description we provide is lowest order in the coupling constant expansion of string perturbation theory, it will turn out, remarkably, to be an exact, and nonperturbative, description of flow in a particular relevant coupling of the worldsheet renormalization group. The example of worldsheet RG flow we discuss is based on a relevant operator in the thermal mass spectrum of the finite temperature type II string theory, in the absence of branes or of a Yang-Mills sector [14]. We will derive an explicit path integral expression for the logarithm of the vacuum degeneracy, lng, of the type II string ensemble which is valid at all intermediate points along the RG flow to the noncompact infrared fixed pointh. Recall that the vacuum degeneracy in string theory can be extracted from the factorization limit of the one-loop graph in the string perturbation series: the one-loop graph factorizes on states contributing to the tree-level mass spectrum [9, 6, 14]. Inspection of our explicit expression for the vacuum degeneracy shows that the contribution to lng from marginal and irrelevant