Entanglement in BF theory II: Edge-modes

TL;DR

This paper investigates entanglement entropy in Abelian p-form topological theories, revealing universal topological corrections, a novel chiral mixed Maxwell edge theory, and an infinite-dimensional current algebra structure that links edge modes to entanglement spectrum.

Contribution

It introduces a new chiral mixed Maxwell theory for edge modes and establishes a precise correspondence between the edge theory's partition function and the entanglement spectrum.

Findings

Universal area laws with topological corrections proportional to Betti numbers.

Explicit evaluation of the thermal partition function of the chiral mixed Maxwell theory.

Identification of an infinite-dimensional current algebra organizing the edge modes.

Abstract

We consider the entanglement entropy arising from edge-modes in Abelian $p$-form topological field theories in $d$ dimensions on arbitrary spatial topology and across arbitrary entangling surfaces. We find a series of descending area laws plus universal corrections proportional to the Betti numbers of the entangling surface, which can be taken as a higher-dimensional version of the "topological entanglement entropy." Our calculation comes in two flavors: firstly, through an induced edge-mode theory appearing on the regulated entangling surface in a replica path integral and secondly through a more rigorous definition of the entanglement entropy through an extended Hilbert space. Along the way we establish several key results that are of their own merit. We explain how the edge-mode theory is a novel combination of $(p-1)$-form and $(d-p-2)$-form Maxwell theories linked by a chirality condition, in what we coin a "chiral mixed Maxwell theory." We explicitly evaluate the thermal partition function of this theory. Additionally we show that the extended Hilbert space is completely organized into representations of an infinite-dimensional, centrally extended current algebra which naturally generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We construct the Verma modules and the representation characters of this algebra. Lastly, we connect the two approaches, showing that the thermal partition function of the chiral mixed Maxwell theory is precisely an extended representation character of our current algebra, establishing an exact correspondence of the edge-mode theory and the entanglement spectrum.