Geometric entropy and edge modes of the electromagnetic field

TL;DR

This paper computes the vacuum entanglement entropy of Maxwell theory in curved spacetimes, revealing the role of edge modes and a contact term, and clarifies the statistical interpretation of the contact term including edge contributions.

Contribution

It provides a detailed calculation of electromagnetic entanglement entropy, including edge modes and contact terms, resolving a longstanding puzzle about their statistical meaning.

Findings

The geometric entropy matches the statistical entanglement entropy with edge modes.

The contact term is a negative scalar degree of freedom on the entangling surface.

Implications for black hole thermodynamics and Newton's constant renormalization.

Abstract

We calculate the vacuum entanglement entropy of Maxwell theory in a class of curved spacetimes by Kaluza-Klein reduction of the theory onto a two-dimensional base manifold. Using two-dimensional duality, we express the geometric entropy of the electromagnetic field as the entropy of a tower of scalar fields, constant electric and magnetic fluxes, and a contact term, whose leading order divergence was discovered by Kabat. The complete contact term takes the form of one negative scalar degree of freedom confined to the entangling surface. We show that the geometric entropy agrees with a statistical definition of entanglement entropy that includes edge modes: classical solutions determined by their boundary values on the entangling surface. This resolves a longstanding puzzle about the statistical interpretation of the contact term in the entanglement entropy. We discuss the implications of this negative term for black hole thermodynamics and the renormalization of Newton's constant.