Entanglement entropy of gravitational edge modes

TL;DR

This work computes the entanglement entropy contributed by edge modes in linearised gravity across a spherical entangling surface. By decomposing the graviton into tensor harmonics and fixing an appropriate gauge, the authors identify two radial-curvature components, R_{ hat{t} hat{r} hat{e} hat{m}} and R_{ hat{t} hat{r} hat{t} hat{r}}, that locally label superselection sectors via the Gauss-law constraints, and they evaluate the associated logarithmic term from the two-point functions on S^2. The resulting edge contribution is S_edge = −(16/3) log(R/ε), matching the logarithmic coefficient extracted from the edge Harish-Chandra character of the massless spin-2 field on S^4, thus linking gravitational edge entropy to the edge partition function. The analysis parallels the U(1) case across even dimensions and suggests extensions to higher spins and non-Abelian theories, with implications for the role of edge degrees of freedom in gravitational entanglement and holography.

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 $S^d$.