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arXiv:2201.06043 · Journal of High Energy Physics

Entanglement entropy of gravitational edge modes

Justin R. David, Jyotirmoy Mukherjee

Published Aug 4, 2022• 25 citations
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Abstract

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 labels 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 S2 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 $$ -\frac{16}{3} $$ − 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 Sd.

DOI
10.1007/jhep08(2022)065

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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 labels 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 S2 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 −163-\frac{16}{3}−316​ − 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 Sd.
We compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the l=0l=0l=0 and l=1l=1l=1 modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.
We consider an antisymmetric gauge field in the Minkowski space of ddd-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of ppp-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact ppp-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric ppp-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p−1)(p-1)(p−1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.
A bstract We consider an antisymmetric gauge field in the Minkowski space of d -dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p -form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p -form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p -form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact ( p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.
We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in d=3+1d=3+1d=3+1 dimensions. In spherical coordinates the problem decomposes into one-dimensional ones along the radial coordinate for each angular momentum. We show that the entanglement entropy of a Maxwell field is equivalent to one of two identical massless scalars from which the mode of l=0l=0l=0 has been removed. This shows the relation ${c}_{\mathrm{log}}^{M}=2({c}_{\mathrm{log}}^{S}\ensuremath{-}{c}_{\mathrm{log}}^{{S}_{l=0}})$ between the logarithmic coefficient in the entropy for a Maxwell field clogM{c}_{\mathrm{log}}^{M}clogM​, the one for a d=3+1d=3+1d=3+1 massless scalar clogS{c}_{\mathrm{log}}^{S}clogS​, and the logarithmic coefficient clogSl=0{c}_{\mathrm{log}}^{{S}_{l=0}}clogSl=0​​ for a d=1+1d=1+1d=1+1 scalar with a Dirichlet boundary condition at the origin. Using the accepted values for these coefficients ${c}_{\mathrm{log}}^{S}=\ensuremath{-}1/90$ and clogSl=0=1/6{c}_{\mathrm{log}}^{{S}_{l=0}}=1/6clogSl=0​​=1/6, we get ${c}_{\mathrm{log}}^{M}=\ensuremath{-}16/45$, which coincides with Dowker's calculation, but does not match the coefficient $\ensuremath{-}\frac{31}{45}$ in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers clogM{c}_{\mathrm{log}}^{M}clogM​, clogS{c}_{\mathrm{log}}^{S}clogS​ and clogSl=0{c}_{\mathrm{log}}^{{S}_{l=0}}clogSl=0​​, verifying the relation, as well as checked that they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.
In this paper, I study the entanglement entropy in Hartle-Hawking states of JT gravity set up by a Euclidean path integral with an operator inserted somewhere along the Euclidean boundary. I show that the entanglement entropy between the dual SYK models can be written to leading order in 1/GN1/G_N1/GN​ in a way that resembles a formula for entanglement entropy in a compact lattice gauge theory, but where the link that we appear to compute the entanglement across depends on where we put the operator along the Euclidean boundary. I then describe a toy model where we can write down an explicit family of density matrices whose von Neumann entropies exhibit such a phase transition as we tune an interpolating parameter. This behavior relies on a certain superselection sector structure and I comment on the possibility that the gravitational result comes from a similar emergent superselection sector structure in SYK/JT holography. The main implication is that each copy of the SYK model should then contain Hilbert space sectors labeled by multiple SL(2,R)SL(2,R)SL(2,R) representations whose entanglement gives rise to space.