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

TL;DR

This work extends Witten's $SL(2,\mathbb{Z})$ action from 3d QFTs with $U(1)$ symmetry to $2k$-dimensional QFTs with a $\bb{Z}_2^{(k-1)}$-form symmetry, showing the action closes only up to an invertible phase. The authors identify this phase with the Brown-Kervaire invariant of a quadratic refinement of the intersection form, yielding a projective $SL(2,\mathbb{Z})$ action whose obstruction is interpreted as a bulk $\mathbb{Z}_2$ gauge theory anomaly in $(2k+1)$ dimensions. They develop explicit definitions of the $S$ and $T$ operations, relate the obstruction $Y$ to BK invariants, and explore a broad class of examples across oriented, spin, and pin manifolds, including 4d and 2d cases, showing when the action can be genuine or remains projective. The bulk interpretation via domain-wall dualities clarifies how these anomalies arise from bulk-boundary consistency and connects to known results in 3d Maxwell theory and higher-form symmetry contexts, with a roadmap for computing the full anomaly in future work.

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.

Figures (5)

• Figure 1: The theory with $G$ symmetry lives on the boundary of the $G$ gauge theory
• Figure 2: The duality group $D$ in the bulk acts on boundary theories
• Figure 3: The duality group $D$ in the bulk can be projective
• Figure 4: The braiding of bulk operators.
• Figure 5: The bulk domain wall labeled by $g$ determines a matrix $\braket{B|g|B'}$.