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arXiv:2202.11111 · Journal of High Energy Physics

Charged moments in W3 higher spin holography

Suting Zhao, Christian Northe, Konstantin Weisenberger, René Meyer

Published May 25, 2022• 46 citations
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Abstract

Abstract We consider the charged moments in SL(3, ℝ) higher spin holography, as well as in the dual two-dimensional conformal field theory with W3 symmetry. For the vacuum state and a single entangling interval, we show that the W3 algebra of the conformal field theory induces an entanglement W3 algebra acting on the quantum state in the entangling interval. The algebra contains a spin 3 modular charge which commutes with the modular Hamiltonian. The reduced density matrix is characterized by the modular energy and modular charge, hence our definition of the charged moments is also with respect to these conserved quantities. We evaluate the logarithm of the charged moments perturbatively in the spin 3 modular chemical potential, by computing the corresponding connected correlation functions of the modular charge operator up to quartic order in the chemical potential. This method provides access to the charged moments without using charged twist fields. Our result matches known results for the charged moment obtained from the charged topological black hole picture in SL(3, ℝ) higher spin gravity. Since our charged moments are not Gaussian in the chemical potential any longer, we conclude that the dual W3 conformal field theories must feature breakdown of equipartition of entanglement to leading order in the large c expansion.

DOI
10.1007/jhep05(2022)166

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Abstract We consider the charged moments in SL(3, ℝ) higher spin holography, as well as in the dual two-dimensional conformal field theory with W3 symmetry. For the vacuum state and a single entangling interval, we show that the W3 algebra of the conformal field theory induces an entanglement W3 algebra acting on the quantum state in the entangling interval. The algebra contains a spin 3 modular charge which commutes with the modular Hamiltonian. The reduced density matrix is characterized by the modular energy and modular charge, hence our definition of the charged moments is also with respect to these conserved quantities. We evaluate the logarithm of the charged moments perturbatively in the spin 3 modular chemical potential, by computing the corresponding connected correlation functions of the modular charge operator up to quartic order in the chemical potential. This method provides access to the charged moments without using charged twist fields. Our result matches known results for the charged moment obtained from the charged topological black hole picture in SL(3, ℝ) higher spin gravity. Since our charged moments are not Gaussian in the chemical potential any longer, we conclude that the dual W3 conformal field theories must feature breakdown of equipartition of entanglement to leading order in the large c expansion.
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
It is generally believed that the semiclassical AdS3 higher spin gravity could be described by a two dimensional conformal field theory with W \mathcal{W} W -algebra symmetry in the large central charge limit. In this paper, we study the single interval entanglement entropy on the torus in the CFT with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ deformation. More generally we develop the monodromy analysis to compute the two-point function of the light operators under a thermal density matrix with a W3 {\mathcal{W}}_{\mathcal{3}} W3​ chemical potential to the leading order. Holographically we compute the probe action of the Wilson line in the background of the spin-3 black hole with a chemical potential. We find exact agreement.
Abstract We test the proposal of [1] for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in U(1) Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS3. In the conformal field theory, these states are generated by the insertion of charged and uncharged heavy operators. We employ the monodromy method to calculate the ensuing four-point function between the heavy operators and the twist fields. For the two-interval case, we derive our results on the AdS and the conformal field theory side, respectively, from the generating function method of [1], as well as the vertex operator algebra. In all cases considered, we find equipartition of entanglement between the different charge sectors. We also clarify an aspect of conformal field theories with a large central charge and u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ u ̂ 1 k Kac-Moody symmetry used in our calculations, namely the factorization of the Hilbert space into a gravitational Virasoro sector with large central charge, and a u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ u ̂ 1 k Kac-Moody sector.
Higher spin entanglement and W N WN {\mathcal{W}}_{\mathrm{N}} WN​ conformal blocks
Two-dimensional conformal field theories with extended W \mathcal{W} W -symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS3 at large central charge. Observables that can be computed and compared in the two descriptions include Rényi and entanglement entropies, and correlation functions of local operators. We develop techniques for computing these, in a manner that sheds light on when and why one can expect agreement between such quantities on each side of the duality. We set up the computation of excited state Rényi entropies in the bulk in terms of Chern-Simons connections, and show how this directly parallels the CFT computation of correlation functions. More generally, we consider the vacuum conformal block for general operators with Δ ∼ c. When two of the operators obey Δc≪1 \frac{\varDelta }{c}\ll 1 cΔ​≪1 , we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS3 background with higher spin fields excited, the latter emerging as the effective bulk description of the excited state produced by the heavy operators. Among other things, this puts a previous proposal for computing higher spin entanglement entropy via Wilson lines on firmer footing, and clarifies its relation to CFT. We also study the corresponding computation in Toda theory and find that this provides yet another independent way to arrive at the same result.
We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential mu for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in mu in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) x SL(3, R) Chern-Simons theory. Our result suggests that the order mu(2) correction to the entanglement entropy may be universal for W-algebra CFTs with spin-three chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS(3).
We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential μ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in μ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, ℝ) × SL(3, ℝ) Chern-Simons theory. Our result suggests that the order μ 2 correction to the entanglement entropy may be universal for $ \mathcal{W} $ -algebra CFTs with spinthree chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS3.