After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the “loop braid group”. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the “braid permutation group” of Fenn, Rimányi, and Rourke. In the context of 4d BF theory, this group naturally acts on the moduli space of flat $G$-bundles on the complement of a collection of unlinked unknotted circles in $\mathbb{R}^3$. When $G$ is unimodular, this gives a unitary representation of the loop braid group. We also discuss “quandle field theory”, in which the gauge group $G$ is replaced by a quandle.

In this paper, we study conformal field theories (CFTs) associated to gerbes. These theories suffer from a lack of cluster decomposition, but this problem can be resolved: the CFTs are the same as CFTs 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 CFTs, 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.

Hitchin has shown that the moduli space $\mathcal{M}$ of the dimensionally reduced self-dual Yang-Mills equations has a hyperKähler structure. In this paper, we first explicitly show the hyperKähler structure, the details of which is missing in Hitchin’s paper. We show here that $\mathcal{M}$ admits three prequantum line bundles, corresponding to the three symplectic forms. We use Quillen’s determinant line bundle construction and show that the Quillen curvatures of these prequantum line bundles are proportional to each of the symplectic forms mentioned above. The prequantum line bundles are holomorphic with respect to their respective complex structures. We show how these prequantum line bundles can be derived from cocycle line bundles of Chern-Simons gauge theory with complex gauge group in the case when the moduli space is smooth.