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

TL;DR

This paper extends Witten's $SL(2,b{Z})$ action to higher-dimensional QFTs with $b{Z}_2$ symmetry, revealing an anomaly characterized by the Brown-Kervaire invariant linked to the bulk $b{Z}_2$ gauge theory.

Contribution

It introduces an $SL(2,b{Z})$ action on $2k$-dimensional QFTs with $b{Z}_2$ symmetry and identifies the associated anomaly as the Brown-Kervaire invariant.

Findings

$SL(2,b{Z})$ action closes up to a topological phase

The phase is given by the Brown-Kervaire invariant

The anomaly is interpreted via bulk $b{Z}_2$ gauge theory

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.