SL(2, ℤ) action on QFTs with ℤ 2 symmetry and the Brown-Kervaire invariants

We consider an analogue of Witten’s SL(2, ℤ) action on three-dimensional QFTs with U(1) symmetry for 2k-dimensional QFTs with ℤ2 (k − 1)-form symmetry. We show that the SL(2, 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, ℤ) anomaly of the bulk (2k + 1)-dimensional ℤ2 gauge theory.