Dynamical Edge Modes and Entanglement in Maxwell Theory

TL;DR

This paper investigates boundary degrees of freedom in Maxwell theory related to black hole horizons, showing they are Goldstone bosons and deriving their contributions to entropy and partition functions using covariant phase space formalism.

Contribution

It identifies boundary conditions that produce edge modes as gauge Goldstone bosons and analyzes their role in entropy and partition functions in Maxwell and Proca theories.

Findings

Edge modes are Goldstone bosons of boundary gauge transformations.

The thermal edge partition function is a ghost scalar on the horizon.

The entanglement entropy matches the conformal anomaly in 4D.

Abstract

Previous work on black hole partition functions and entanglement entropy suggests the existence of "edge" degrees of freedom living on the (stretched) horizon. We identify a local and "shrinkable" boundary condition on the stretched horizon that gives rise to such degrees of freedom. They can be interpreted as the Goldstone bosons of gauge transformations supported on the boundary, with the electric field component normal to the boundary as their symplectic conjugate. Applying the covariant phase space formalism for manifolds with boundary, we show that both the symplectic form and Hamiltonian exhibit a bulk-edge split. We then show that the thermal edge partition function is that of a codimension-two ghost compact scalar living on the horizon. In the context of a de Sitter static patch, this agrees with the edge partition functions found by Anninos et al. in arbitrary dimensions. It also yields a 4D entanglement entropy consistent with the conformal anomaly. Generalizing to Proca theory, we find that the prescription of Donnelly and Wall reproduces existing results for its edge partition function, while its classical phase space does not exhibit a bulk-edge split.