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.Comment: 61 pages, LaTeX; v2,3: typos fixed; v4: writing improved in several
sections; v5: typos fixe
