Cluster decomposition, T-duality, and gerby CFT's

TL;DR

This work establishes a worldsheet duality—interpreted as T-duality—between strings propagating on gerbes and strings propagating on disjoint unions of spaces with varying B-fields. It formalizes a general decomposition conjecture that the CFT of a string on a G-gerbe [X/H] equals the CFT on an associated space Y, effectively a disjoint union of X-type components labeled by irreps of G, with flat B-field data determined by the gerbe. The authors provide extensive tests via partition functions, spectra, D-branes, mirror symmetry, and noncommutative geometry, and discuss implications for equivariant K-theory, Hochschild cohomology, and LG mirror models. They further explore consequences for quantum cohomology, Gromov-Witten theory, and aspects of the geometric Langlands program, illustrating a broadly applicable framework for gerbes in two-dimensional quantum field theory and beyond.

Abstract

In this paper we study CFT's associated to gerbes. These theories suffer from a lack of cluster decomposition, but this problem can be resolved: the CFT's are the same as CFT's for disconnected targets. Such theories also lack cluster decomposition, but in that form, the lack is manifestly not very problematic. In particular, we shall see that this matching of CFT's, this duality between noneffective gaugings and sigma models on disconnected targets, is a worldsheet duality related to T-duality. We perform a wide variety of tests of this claim, ranging from checking partition functions at arbitrary genus to D-branes to mirror symmetry. We also discuss a number of applications of these results, including predictions for quantum cohomology and Gromov-Witten theory and additional physical understanding of the geometric Langlands program.