Anomaly-induced vanishing of brane partition functions

TL;DR

This work develops an anomaly-based framework for understanding when brane partition functions vanish due to 't Hooft anomalies for higher-form symmetries. By using differential cohomology and a mapping-torus construction, it recasts vanishing as a basepoint anomaly that constrains background fluxes and backgrounds, yielding precise nonvanishing conditions. The paper then applies these ideas across a range of theories—generalised Maxwell/BF theories, discrete Dijkgraaf–Witten theories, and brane systems including D3- and M5-branes—recovering and extending Freed-Witten-type anomaly cancellation conditions and demonstrating how sources or non-Abelian structures can restore nonvanishing partition functions in intricate backgrounds (notably S-folds and F-theory setups). The results provide a unifying, field-theoretic perspective on flux quantization and anomaly inflow, with implications for consistent brane dynamics in nonperturbative string/M-theory contexts. Overall, the approach furnishes a systematic toolkit for predicting when brane worldvolume theories decouple or remain well-defined in the presence of nontrivial background charges and higher-form symmetries.

Abstract

In the presence of 't Hooft anomalies, backgrounds for the symmetries of a quantum field theory can lead to non-conservation of Noether currents, or more generally, to the presence of charged insertions in the path integral. When there is a net background charge, the partition function evaluated on closed manifolds will vanish. For anomalous symmetries, this statement can also be understood as the anomaly theory giving rise to a non-trivial anomalous phase for the partition function even for "rigid" transformations which leave all background fields unchanged. We use the generalisation of this second viewpoint to the setting of anomalous higher-form symmetries in order to show vanishing of the partition function for a number of examples, both with and without a Lagrangian description. In particular, we show how to derive from these considerations the analogue of the Freed-Witten anomaly cancellation condition for the M5-brane, and also that for the D3-brane in S-fold backgrounds.

Figures (2)

• Figure 1: At the level of differential forms, the anomaly theory of a given QFT can be determined via inflow, and is supported on some $Y^{d+1}$ such that $\partial Y^{d+1}=X^d$. The connection $A_{p+1}$ is formally extended to $\underline{A}_{p+1}$ in the bulk. This statement can be promoted to the level of differential cochains by explicitly keeping track of the characteristic class of the gauge field, in which case the anomaly theory can be equivalently defined over the mapping cylinder $X^d \times I$. Here the bulk gauge field interpolates between two gauge-equivalent differential cocycles $\check{A},\check{A}-d\check{\lambda} \in \check{Z}^{p+2}(X^d)$. Crucially, there is an auxiliary degree of freedom $\check{a} \in \check{Z}_\text{flat}^{p+1}(X^d)$ corresponding to rigid gauge transformations of $\check{A}$. When $\check{\lambda}$ is trivial, we can glue the two ends of the cylinder to form the mapping torus $X^d \times S^1$. The anomaly $\mathcal{A}[\check{A},0,\check{a}]$ is then given by reducing the anomaly theory over $X^d \times S^1$. It is possible to "cancel" $\mathcal{A}[\check{A},0,\check{a}]$ by inserting a Wilson line $\mathcal{W}$ along $S^1$.
• Figure 2: The partition function $\mathcal{Z}[\check{A}]$ is a section of a line bundle $\mathcal{L}$ over the space of gauge fields, $\mathfrak{A} \simeq \check{Z}^{p+2}(X^d)$. A 't Hooft anomaly is a phase $e^{2\pi i \mathcal{A}[\check{A},\check{\lambda},0]}$ acquired by $\mathcal{Z}[\check{A}]$ as we move along a non-trivial path $\check{A} \to \check{A} - d\check{\lambda}$ for some $\check{\lambda} \in \check{C}_\text{flat}^{p+1}(X^d)$. If the phase is trivial, then $\mathcal{L}$ can be lifted to a line bundle over $\mathfrak{A}/\mathfrak{G} \simeq \check{H}^{p+2}(X^d)$, where points along a given gauge orbit on $\mathfrak{A}$ are identified. On the other hand, a basepoint anomaly is a phase $e^{2\pi i \mathcal{A}[\check{A}',0,\check{a}]}$ acquired by $\mathcal{Z}[\check{A}']$ as we act on a fixed $\check{A}'$ with an automorphism $\check{A}' \to \check{A}'-d\check{a} = \check{A}'$ for some $\check{a} \in \check{Z}_\text{flat}^{p+1}(X^d)$. If such a phase is non-trivial, then $\mathcal{Z}[\check{A}']$ must vanish at this point on $\mathfrak{A}$.