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arXiv:2409.08198 · Physical review. D/Physical review. D.

Holographic T T ¯ deformation of the entanglement entropy in ( A ) dS 3 / CFT 2

Jing-Cheng Chang, Song He, Yu-Xiao Liu, Long Zhao

Published Jun 4, 2025• 7 citations
arXivHTMLPDF

Abstract

In recent years, the holographic duality between T T ¯ -deformed conformal field theory (CFT) and Anti-de Sitter (AdS) spacetime with finite radial cutoff has received significant attention. The study of T T ¯ deformation within the framework of de Sitter (dS)/CFT duality has also progressed. This paper shows that the trace flow equation in dS spacetime can be analytically extended from its AdS counterpart through double Wick rotations. Meanwhile, we generalize the replica method in both AdS and dS holography to derive a general expression for the entanglement entropy of arbitrary single spatial intervals within the T T ¯ -deformed framework. For both finite size and finite temperature systems, we obtain the analytical expression for the entanglement entropy after T T ¯ deformation. Finally, in dS/dS holography and half-dS holography, we find that the dual field theory exhibits nonlocality by analyzing the strong subadditivity and boosted strong subadditivity of entanglement entropy.

DOI
10.1103/21wr-76pq

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In recent years, the holographic duality between T T ¯ -deformed conformal field theory (CFT) and Anti-de Sitter (AdS) spacetime with finite radial cutoff has received significant attention. The study of T T ¯ deformation within the framework of de Sitter (dS)/CFT duality has also progressed. This paper shows that the trace flow equation in dS spacetime can be analytically extended from its AdS counterpart through double Wick rotations. Meanwhile, we generalize the replica method in both AdS and dS holography to derive a general expression for the entanglement entropy of arbitrary single spatial intervals within the T T ¯ -deformed framework. For both finite size and finite temperature systems, we obtain the analytical expression for the entanglement entropy after T T ¯ deformation. Finally, in dS/dS holography and half-dS holography, we find that the dual field theory exhibits nonlocality by analyzing the strong subadditivity and boosted strong subadditivity of entanglement entropy.
We compute entanglement entropy for free massive scalar fields in anti-de Sitter (AdS) space. The entangling surface is a minimal surface whose boundary is a sphere at the boundary of AdS. The entropy can be evaluated from the thermal free energy of the fields on a topological black hole by using the replica method. In odd-dimensional AdS, exact expressions of the Rényi entropy S n are obtained for arbitrary n. We also evaluate 1-loop corrections coming from the scalar fields to holographic entanglement entropy. Applying the results, we compute the leading difference of entanglement entropy between two holographic CFTs related by a renormalization group flow triggered by a double trace deformation. The difference is proportional to the shift of a central charge under the flow.
TT‾ T\overline{T} TT and EE, with implications for (A)dS subregion encodings
Abstract We initiate a study of subregion dualities, entropy, and redundant encoding of bulk points in holographic theories deformed by TT‾ T\overline{T} TT T T ¯ and its generalizations. This includes both cut off versions of Anti de Sitter spacetime, as well as the generalization to bulk de Sitter spacetime, for which we introduce two additional examples capturing different patches of the bulk and incorporating the second branch of the square root dressed energy formula. We provide new calculations of entanglement entropy (EE) for more general divisions of the system than the symmetric ones previously available. We find precise agreement between the gravity side and deformed-CFT side results to all orders in the deformation parameter at large central charge. An analysis of the fate of strong subadditivity for relatively boosted regions indicates nonlocality reminiscent of string theory. We introduce the structure of operator algebras in these systems. The causal and entanglement wedges generalize to appropriate deformed theories but exhibit qualitatively new behaviors, e.g. the causal wedge may exceed the entanglement wedge. This leads to subtleties which we express in terms of the Hamiltonian and modular Hamiltonian evolution. Finally, we exhibit redundant encoding of bulk points, including the cosmological case.
We study the entanglement entropy (EE) and the R\'enyi entropy (RE) of multiple intervals in two-dimensional TT‾T\overline{T}TT- deformed conformal field theory (CFT) at a finite temperature by field theoretic and holographic methods. First, by the replica method with the twist operators, we construct the general formula of the RE and EE up to the first order of a deformation parameter. By using our general formula, we show that the EE of multiple intervals for a holographic CFT is just a summation of the single interval case even with the small deformation. This is a nontrivial consequence from the field theory perspective, though it may be expected by the Ryu-Takayanagi formula in holography. However, the deformed RE of the two intervals is a summation of the single interval case only if the separations between the intervals are big enough. It can be understood by the tension of the cosmic branes dual to the RE. We also study the holographic EE for single and two intervals with an arbitrary cutoff radius (dual to the TT‾T\overline{T}TT deformation) at any temperature. We confirm our holographic results agree with the field theory results with a small deformation and high temperature limit, as expected. For two intervals, there are two configurations for EE: disconnected (sss-channel) and connected (ttt-channel) ones. We investigate the phase transition between them as we change parameters: as the deformation or temperature increases the phase transition is suppressed and the disconnected phase is more favored.
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.
We show that the quantum entanglement between the conformal field theories (CFTs) living on the future and past boundaries in the de Sitter/conformal field theory (dS/CFT) correspondence can be described by Wick rotating to a geometry with two timelike directions. We propose a new realization of the dS/CFT correspondence in which quantum gravity on this two-time geometry is holographically dual to a CFT defined on a Lorentzian torus. We show that this duality reproduces key features of dS holography, including the dS entropy, correlation functions, and pseudoentropy. Finally, by extending the framework of path-integral optimization, we explain how dS spacetime emerges from the CFT on the Lorentzian torus.
It was proposed by Ryu and Takayanagi that the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. We present all possible minimal surfaces spanned on one or two spherical boundaries at AdS infinity. We give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities. In the case of two disjoint boundaries the existence of two different phases of the entanglement entropy is confirmed. A trivial phase corresponds to two disconnected minimal surfaces, while the other one corresponds to a tube connecting the spherical boundaries. A transition between these phases is reminiscent of the finite temperature deconfinement transition in the CFT on the boundary. The exact analytical results are thus consistent with previous numerical and approximate computations. We also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries.