$SL(2,\mathbb{Z})$ action on QFTs with ${\mathbb{Z}}_2$ symmetry and the Brown-Kervaire invariants

Abstract

We consider an analogue of Witten's $SL(2,\mathbb{Z})$ action on three-dimensional QFTs with $U(1)$ symmetry for $2k$-dimensional QFTs with ${\mathbb{Z}}_2$ $(k-1)$-form symmetry. We show that the $SL(2,\mathbb{Z})$ action only closes up to a multiplication by an invertible topological phase whose partition function is the Brown-Kervaire invariant of the spacetime manifold. We interpret it as part of the $SL(2,\mathbb{Z})$ anomaly of the bulk $(2k+1)$-dimensional ${\mathbb{Z}}_2$ gauge theory.

19 pages