Anomaly inflow and $p$-form gauge theories

Abstract

Chiral and non-chiral $p$-form gauge fields have gravitational anomalies and anomalies of Green-Schwarz type. This means that they are most naturally realized as the boundary modes of bulk topological phases in one higher dimensions. We give a systematic description of the total bulk-boundary system which is analogous to the realization of a chiral fermion on the boundary of a massive fermion. The anomaly of the boundary theory is given by the partition function of the bulk theory, which we explicitly compute in terms of the Atiyah-Patodi-Singer $\eta$-invariant. We use our formalism to determine the $\mathrm{SL}(2,\mathbb{Z})$ anomaly of the 4d Maxwell theory. We also apply it to study the worldvolume theories of a single D-brane and an M5-brane in the presence of orientifolds, orbifolds, and S-folds in string, M, and F theories. In an appendix we also describe a simple class of non-unitary invertible topological theories whose partition function is not a bordism invariant, illustrating the necessity of the unitarity condition in the cobordism classification of the invertible phases.

127 pages. v2: minor addition, v3: minor correction

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• Gapless edge modes in (4+1)-dimensional topologically massive tensor gauge theory and anomaly inflow for subsystem symmetry
• General anomaly matching by Goldstone bosons
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• Geometric Approach to 3D Interfaces at Strong Coupling
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• Mod-two APS index and domain-wall fermion
• Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phase
• Some comments on 6d global gauge anomalies
• Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{\mu \nu }$
• Understanding the index theorems with massive fermions