Geometric entropy and edge modes of the electromagnetic field

TL;DR

The paper resolves the long-standing puzzle about the negative Kabat contact term in Maxwell entanglement entropy by performing a Kaluza-Klein reduction on product manifolds and isolating an edge-mode contribution. By mapping Proca-like KK modes to dual scalars and carefully treating boundary conditions, the authors show that the geometric entropy equals the entanglement entropy when edge modes are included, with the contact term identified as the edge-mode entropy. In four dimensions, this edge contribution reconciles the entanglement entropy with the trace anomaly, providing a consistent statistical interpretation and insights for black hole thermodynamics and Newton’s constant renormalization. The work also clarifies regulator dependence and sets a framework for extending to nonabelian gauge theories and gravity.

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.