Electromagnetic Duality and Entanglement Anomalies

TL;DR

This work analyzes electromagnetic duality for abelian $p$-form gauge theories on general $D$-manifolds, showing duality is exact in odd $D$ but contains a purely topological anomaly in even $D$ proportional to the Euler characteristic via $\log\sqrt{\frac{q}{\tilde q}}$. It computes the duality ratio of partition functions by carefully assembling oscillator, zero-mode, and instanton sectors, invoking Ray-Singer and Reidemeister torsions and the Cheeger-Müller theorem to show the anomaly vanishes in odd dimensions while reproducing the known $D=4$ result in even dimensions. The paper then connects duality to entanglement, deriving an entanglement anomaly $\Delta S_A = (-1)^{p+1}\chi(\partial A)\log\sqrt{\frac{q}{\tilde q}}$ in even $D$, interpretable as the duality of an edge-mode theory living on the entangling surface; it also analyzes $\theta$-terms and shows the thermal entropy is duality-invariant. Collectively, the results reconcile previous conflicting claims and provide a universal, regulator-insensitive account of how duality acts on quantum entanglement in gauge theories, with a clear edge-mode interpretation that links bulk topological data to surface degrees of freedom.

Abstract

Duality is an indispensable tool for describing the strong-coupling dynamics of gauge theories. However, its actual realization is often quite subtle: quantities such as the partition function can transform covariantly, with degrees of freedom rearranged in a nonlocal fashion. We study this phenomenon in the context of the electromagnetic duality of abelian $p$-forms. A careful calculation of the duality anomaly on an arbitrary $D$-dimensional manifold shows that the effective actions agree exactly in odd $D$, while in even $D$ they differ by a term proportional to the Euler number. Despite this anomaly, the trace of the stress tensor agrees between the dual theories. We also compute the change in the vacuum entanglement entropy under duality, relating this entanglement anomaly to the duality of an "edge mode" theory in two fewer dimensions. Previous work on this subject has led to conflicting results; we explain and resolve these discrepancies.