Loop Variables and Gauge Invariant Exact Renormalization Group Equations for Closed String Theory

TL;DR

This paper develops gauge-invariant exact renormalization group equations for closed string backgrounds, addressing subtleties like holomorphic factorization issues and exploring conditions for massless gravitons within a background field framework.

Contribution

It extends the ERG formulation from open to closed strings, incorporating gauge invariance and background metrics, and discusses conditions for massless gravitons in string theory.

Findings

Holomorphic factorization does not hold with a cutoff, affecting gauge invariance.

Massive graviton arises in naive open-to-closed string generalization.

Massless graviton possible with background metric and combined gauge transformations.

Abstract

We formulate the Exact Renormalization Group on the string world sheet for closed string backgrounds. The same techniques that were used for open strings is used here. There are some subtleties. One is that holomorphic factorization of the closed string vertex operators does not hold in the presence of a cutoff on the Euclidean world sheet. This introduces extra terms in the Lagrangian at the cutoff scale and they turn out to be crucial for implementing gauge invariance. This naive generalization from open string to closed strings requires a {\em massive} graviton and the gauge symmetry is Abelian, just as in open string theory. Interestingly, it turns out that if one introduces a non dynamical background metric (as in background field formalism) and combines a gauge transformation on the field with a transformation on the coordinates and background metric, the graviton can be massless. Some examples of background coordinate covariant equations are worked out explicitly. A preliminary discussion of massive modes, massive gauge transformations and the role of world sheet regulator terms is given. Some of the gauge transformations can be given a geometric meaning if space time is assumed to be complex at some level.