Anomaly inflow and $p$-form gauge theories

TL;DR

The paper develops a systematic framework to treat both chiral and non-chiral $p$-form gauge fields as boundary modes of bulk invertible phases, where the boundary anomaly is captured by the bulk partition function expressed through the Atiyah-Patodi-Singer $\eta$-invariant. Central to the construction is a quadratic refinement of the differential cohomology pairing, ensuring a well-defined bulk action and invertible bulk theory in dimensions $d+1=3,7,11$. The authors derive explicit anomaly formulas for chiral $p$-forms, connect them to gravitational and Green-Schwarz-type terms, and apply the results to M-theory branes, D-branes in orientifolds/orbifolds, and $SL(2,\mathbb Z)$ duality in Maxwell theory, including S-fold backgrounds. They also discuss non-unitary counterexamples to bordism classifications to illustrate the necessity of unitarity. Collectively, the work provides a unified, topological account of perturbative and global anomalies for higher-form fields across string/M-theory contexts and clarifies flux-quantization and Bianchi identities via anomaly inflow.

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 $η$-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.