Higher central charges and topological boundaries in 2+1-dimensional TQFTs

TL;DR

This paper explores conditions involving higher central charges that determine the existence of topological boundaries in 2+1D TQFTs, extending beyond the traditional focus on chiral central charge.

Contribution

It establishes necessary and sufficient conditions for topological boundaries in Abelian TQFTs using higher central charges, and offers a geometric approach for non-Abelian cases.

Findings

Necessary and sufficient conditions for Abelian TQFT boundaries identified.

A duality in Abelian TQFT partition functions discovered.

Conditions for non-Abelian TQFT boundaries discussed geometrically.

Abstract

A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge $c_-$ has to vanish. In this paper, we consider conditions associated with "higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.