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arXiv:1412.7520 · Journal of High Energy Physics

Higher spin entanglement and W N $$ {\mathcal{W}}_{\mathrm{N}} $$ conformal blocks

Jan De Boer A, Alejandra Castro, Eliot Hijano, Juan I. Jottar, Per Kraus

Published Jul 1, 2015• 87 citations
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Abstract

Two-dimensional conformal field theories with extended $$ \mathcal{W} $$ -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 $$ \frac{\varDelta }{c}\ll 1 $$ , 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.

DOI
10.1007/jhep07(2015)168

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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 1cΔ​≪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.
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.
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.
We extend and refine recent results on Rényi entropy in two-dimensional con-formal field theories at large central charge. To do so, we examine the effects of higher spin symmetry and of allowing unequal left and right central charges, at leading and sub-leading order in large total central charge. The result is a straightforward generalization of previously derived formulae, supported by both gravity and CFT arguments. The preceding statements pertain to CFTs in the ground state, or on a circle at unequal left- and right-moving temperatures. For the case of two short intervals in a CFT ground state, we derive certain universal contributions to Rényi and entanglement entropy from Virasoro primaries of arbitrary scaling weights, to leading and next-to-leading order in the interval size; this result applies to any CFT. When these primaries are higher spin currents, such terms are placed in one-to-one correspondence with terms in the bulk 1-loop determinants for higher spin gauge fields propagating on handlebody geometries.
In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS3 gravity and the higher spin AdS3 gravity.
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 consider symmetry-resolved entanglement entropy in AdS3/CFT2coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincaré patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons levelk, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ û1k Kac-Moody type. Employing the u^(1)k \hat{\mathfrak{u}}{(1)}_k u^(1)k​ û1k Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.