Principal 2-group bundles and the Freed–Quinn line bundle

Geometry, Symmetry and Physics
Event time: 
Monday, February 19, 2024 - 4:30pm
KT 217
Emily Cliff
Speaker affiliation: 
Université de Sherbrooke
Event description: 

Abstract: A 2-group is a categorical generalization of a group: it's a category with a multiplication operation which satisfies the usual group axioms only up to coherent isomorphisms. The isomorphism classes of its objects form an ordinary group, G. Given a 2-group G with underlying group G, we can similarly define a categorical generalization of the notion of principal bundles over a manifold (or stack) X, and obtain a bicategory Bun_G(X), living over the category Bun_G(X) of ordinary G-bundles on X. For G finite and X a Riemann surface, we prove that this gives a categorification of the Freed--Quinn line bundle, a mapping-class group equivariant line bundle on Bun_G(X) which plays an important role in Dijkgraaf--Witten theory (i.e. Chern--Simons theory for the finite group G). I will not assume background knowledge on 2-groups, 2-group bundles, or the Freed--Quinn line bundle; I will provide the necessary definitions and an outline of the proof of the categorification result. Time permitting, I will discuss work-in-progress regarding applications of this result, and generalizations to Chern--Simons theory in the case that G is a compact group. This talk is based on joint work with Daniel Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.