Entanglement entropy of gravitational edge modes

TL;DR

This paper calculates the entanglement entropy contributions from gravitational edge modes in 4D Minkowski space, revealing a universal coefficient linked to the edge partition function and Harish-Chandra characters for spin-2 fields.

Contribution

It introduces a method to compute the entanglement entropy of gravitational edge modes using Riemann tensor components and relates these to edge partition functions and Harish-Chandra characters.

Findings

The logarithmic coefficient for gravitational edge modes is -16/3.

The coefficient matches the edge partition function of the massless spin-2 field on the 4-sphere.

The method extends to U(1) gauge fields, matching known results for spin-1 fields.

Abstract

We consider the linearised graviton in $4d$ Minkowski space and decompose it into tensor spherical harmonics and fix the gauge. The Gauss law of gravity implies that certain radial components of the Riemann tensor of the graviton on the sphere label the superselection sectors for the graviton. We show that among these 6 normal components of the Riemann tensor, 2 are related locally to the algebra of gauge-invariant operators in the sphere. From the two-point function of these components of the Riemann tensor on $S^2$ we compute the logarithmic coefficient of the entanglement entropy of these superselection sectors across a spherical entangling surface. For sectors labelled by each of the two components of the Riemann tensor these coefficients are equal and their total contribution is given by $-\frac{16}{3}$. We observe that this coefficient coincides with that extracted from the edge partition function of the massless spin-2 field on the 4-sphere when written in terms of its Harish-Chandra character. As a preliminary step, we also evaluate the logarithmic coefficient of the entanglement entropy from the superselection sectors labelled by the radial component of the electric field of the $U(1)$ theory in even $d$ dimensions. We show that this agrees with the corresponding coefficient of the edge Harish-Chandra character of the massless spin-1 field on $S^d$.