SL(2, ℤ) action on QFTs with ℤ2 symmetry and the Brown-Kervaire invariants

A bstract We consider an analogue of Witten’s SL(2 , ℤ) action on three-dimensional QFTs with U(1) symmetry for 2 k -dimensional QFTs with ℤ 2 ( 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 (2 k + 1)-dimensional ℤ 2 gauge theory.