On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories

TL;DR

This paper explores the symmetry groups of abelian Dijkgraaf-Witten theories using gauge theory, revealing how these symmetries act on bulk Wilson lines and are generated by automorphisms and dualities.

Contribution

It provides a gauge theoretic realization of all symmetries in abelian Dijkgraaf-Witten theories, linking symmetry groups to automorphisms, dualities, and the structure of the Drinfeld center.

Findings

Symmetry groups are generated by automorphisms and dualities.

Transmission functors realize the bijection between bimodule categories and auto-equivalences.

Symmetries are characterized by their action on bulk Wilson lines.

Abstract

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group $A$, and with vanishing 3-cocycle, is generated by group automorphisms of $A$, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of $A$-bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center. The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.