Bulk entanglement entropy for photons and gravitons in AdS$_3$

Abstract

We study quantum corrections to holographic entanglement entropy in AdS_33/CFT_22; these are given by the bulk entanglement entropy across the Ryu-Takayanagi surface for all fields in the effective gravitational theory. We consider bulk U(1)U(1) gauge fields and gravitons, whose dynamics in AdS_33 are governed by Chern-Simons terms and are therefore topological. In this case the relevant Hilbert space is that of the edge excitations. A novelty of the holographic construction is that such modes live not only on the bulk entanglement cut but also on the AdS boundary. We describe the interplay of these excitations and provide an explicit map to the appropriate extended Hilbert space. We compute the bulk entanglement entropy for the CFT vacuum state and find that the effect of the bulk entanglement entropy is to renormalize the relation between the effective holographic central charge and Newton’s constant. We also consider excited states obtained by acting with the U(1)U(1) current on the vacuum, and compute the difference in bulk entanglement entropy between these states and the vacuum. We compute this UV-finite difference both in the bulk and in the CFT finding a perfect agreement.

Extended actions, dynamics of edge modes, and entanglement entropy

“…There have also been efforts to characterize, for local subsystems, the most general boundary symmetry algebras spanned by the edge modes [14][15][16][17][18][19][20][21], with potentially important consequences for quantum gravity [22,23]. Another important development at finite distance has been the realization that a proper treatment of the edge modes is crucial even when dealing with fictitious entangling interfaces, which has consequences in the computations of entanglement entropy [24][25][26][27][28][29][30][31][32][33][34]. At infinity on the other hand, a lot of work has been dedicated towards understanding the intricate infrared properties of theories with massless excitations, and there a central role is played by large gauge transformations and soft modes [35].…”

“…The computation of entanglement entropy from the surface symmetry follows from [34,43,44,48,48,49]. It relies on the extended Hilbert space construction, and on the factorization…”

Endpoint contributions to excited-state modular Hamiltonians

“…Related works have appeared in the literature, including studies of entanglement and bulk reconstruction for excited states [4][5][6][7][8] and the behavior of modular Hamiltonians and entanglement in excited states and under shape deformations [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In contrast to these previous works, we consider excited states produced by perturbing the vacuum by a generic operator integrated over a null plane.…”

Interfaces and the extended Hilbert space of Chern-Simons theory

“…Because this includes trivial interfaces, this gives a systematic extension of the extended Hilbert space construction to all (compact) Chern-Simons theories on closed manifolds. This is a significant addition to the small (but growing) handful of existing extended Hilbert space constructions in continuum field theory [1,[26][27][28][29][30][31].…”

“…This fact makes three-dimensional holography an ideal testing ground for exploring questions of bulk entanglement and bulk factorization (or more precisely, the lack thereof). Interesting work JHEP07(2020)009 has already appeared in this direction [29]. It would be interesting if our construction (with suitable generalization), can provide precise realizations of entanglement wedge reconstruction, quantum error correction, and the "area operator" in 3d holography.…”