$SL(2,\mathbb{Z})$ 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.

arXiv comment19 pages

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• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
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• Topological holography, quantum criticality, and boundary states
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• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Anomalies of Generalized Symmetries from Solitonic Defects
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• General anomaly matching by Goldstone bosons
• What bordism-theoretic anomaly cancellation can do for U
• Matching higher symmetries across Intriligator-Seiberg duality
• Lattice construction of exotic invertible topological phases
• Generalized Landau Paradigm for quantum phases and phase transitions
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations