208527744
Holographic entanglement entropy in AdS3/WCFT
Entanglement is the most fundamental phenomenon in quantum mechanics, which denotes the unusual interactions of many-body systems in quantum regime. The entanglement entropy, defined by von Neumann entropy, thus, provides us a significant tool to understand the physics in quantum field theory, such as correlation, phase transition and quantum information. The dual picture of entanglement entropy in gravitational spacetime introduces the quantum behaviors into gravity, which suggests its prodigious role in quantum gravity apparently. In this paper, we consider AdS$_3$ with Comp$\grave{\mathrm{e}}$re-Song-Strominger (CSS) boundary conditions, under which the dual field theory is warped conformal field theory (WCFT), featuring a Virasoro-Kac-Moody algebra. We provide a holographic dual picture of the single interval entanglement entropy in WCFT in the bulk AdS$_3$. We explicitly show that bulk object captures the entanglement entropy in WCFT is a massive spinning particle's wordline that anchored on the boundary of the interval. Given the mass and spin of the particle as functions of quantum numbers of the twist operator in WCFT, the on-shell wordline action will match the entanglement entropy in WCFT.
119441463
Holographic Derivation of Entanglement Entropy from AdS/CFT
A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdSd+2, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS3. We also compare the entropy computed in AdS5×S 5 with that of the free N = 4 super Yang-Mills.
14858887
Aspects of Holographic Entanglement Entropy
This is an extended version of our short report [1], where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.
4506295
Chiral scale and conformal invariance in 2D quantum field theory.
It is well known that a local, unitary Poincaré-invariant 2D quantum field theory with a global scaling symmetry and a discrete non-negative spectrum of scaling dimensions necessarily has both a left and a right local conformal symmetry. In this Letter, we consider a chiral situation beginning with only a left global scaling symmetry and do not assume Lorentz invariance. We find that a left conformal symmetry is still implied, while right translations are enhanced either to a right conformal symmetry or a left U(1) Kac-Moody symmetry.
51809783
Chiral Liouville gravity
A bstractClassical two-dimensional Liouville gravity is often considered in conformal gauge which has a residual left and right Virasoro symmetry algebra. We consider an alternate, chiral, gauge which has a residual right Virasoro Kac-Moody algebra, and no left Virasoro algebra. The Kac-Moody zero mode is the left-moving energy. Dirac brackets of the constrained Hamiltonian theory are derived, and the residual symmetries are shown to be generated by integrals of the conserved chiral currents. The central charge and Kac-Moody level are computed. The possible existence of a corresponding quantum theory is discussed.
27769925
Warped conformal field theory as lower spin gravity
Two dimensional Warped Conformal Field Theories (WCFTs) may represent the simplest examples of field theories without Lorentz invariance that can be described holographically. As such they constitute a natural window into holography in non-AdS space–times, including the near horizon geometry of generic extremal black holes. It is shown in this paper that WCFTs posses a type of boost symmetry. Using this insight, we discuss how to couple these theories to background geometry. This geometry is not Riemannian. We call it Warped Geometry and it turns out to be a variant of a Newton–Cartan structure with additional scaling symmetries. With this formalism the equivalent of Weyl invariance in these theories is presented and we write two explicit examples of WCFTs. These are free fermionic theories. Lastly we present a systematic description of the holographic duals of WCFTs. It is argued that the minimal setup is not Einstein gravity but an SL(2, R) × U( 1) Chern–Simons Theory, which we call Lower Spin Gravity. This point of view makes manifest the definition of boundary for these non-AdS geometries. This case represents the first step towards understanding a fully invariant formalism for WN field theories and their holographic duals. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
119244766
Locality and anomalies in warped conformal field theory
A bstractWe study various aspects of warped conformal field theories (WCFTs) in two dimensions. We find new Lagrangian WCFTs, and show that all known Lagrangian WCFTs are local only after an infinite tuning. We deduce the anomalies of WCFTs and relate them to central charges, as well as analyze WCFT hydrostatics, thereby re-deriving the warped analogue of the Cardy formula. Finally, we point out that holographic WCFTs are semi-local, that matter fields in spacelike WAdS3 black hole geometries exhibit a Gregory-Laflamme-like instability, and that WAdS3 throats do not admit a decoupling limit.
119226873
A note on the complex SYK model and warped CFTs
A bstractWe discuss the connections between the complex SYK model at the conformal limit and warped conformal field theories. Both theories have an SL(2, ℝ) × U(1) global symmetry. We present comparisons on symmetries, correlation functions, the effective action and the entropy formula. We also use modular covariance to reinterpret results in the complex SYK model.
119020858
Correlation functions of warped CFT
A bstractWarped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra. In this paper, we study correlation functions for primary operators in WCFT. Similar to conformal symmetry, warped conformal symmetry is very constraining. The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio. The warped conformal bootstrap equation are constructed by formulating the notion of crossing symmetry. In the large central charge limit, four point functions can be decomposed into global warped conformal blocks, which can be solved exactly. Furthermore, we revisit the scattering problem in warped AdS spacetime (WAdS), and give a prescription on how to match the bulk result to a WCFT retarded Green’s function. Our result is consistent with the conjectured holographic dualities between WCFT and WAdS.
17389870
Extreme Kerr throat geometry: A vacuum analog of AdS 2 × S 2
We study the near-horizon limit of a four-dimensional extreme rotating black hole. The limiting metric is a completely nonsingular vacuum solution, with an enhanced symmetry group $\mathrm{SL}(2,R)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1).$ We show that many of the properties of this solution are similar to the ${\mathrm{AdS}}_{2}{\ifmmode\times\else\texttimes\fi{}\mathrm{S}}^{2}$ geometry arising in the near-horizon limit of extreme charged black holes. In particular, the boundary at infinity is a timelike surface. This suggests the possibility of a dual quantum mechanical description. A five-dimensional generalization is also discussed.
983459
Microscopic realization of the Kerr/CFT correspondence
Supersymmetric M/string compactifications to five dimensions contain BPS black string solutions with magnetic graviphoton charge P and near-horizon geometries which are quotients of AdS3 × S2. The holographic duals are typically known 2D CFTs with central charges cL = cR = 6P3 for large P. These same 5D compactifications also contain non-BPS but extreme Kerr-Newman black hole solutions with SU(2)L spin JL and electric graviphoton charge Q obeying Q3 ≤ JL2. It is shown that in the maximally charged limit Q3 → JL2, the near-horizon geometry coincides precisely with the right-moving temperature TR = 0 limit of the black string with magnetic charge P = JL1/3. The known dual of the latter is identified as the cL = cR =6JL CFT predicted by the Kerr/CFT correspondence. Moreover, at linear order away from maximality, one finds a TR ≠ 0 quotient of the AdS3 factor of the black string solution and the associated thermal CFT entropy reproduces the linearly sub-maximal Kerr-Newman entropy. Beyond linear order, for general Q3 < JL2, one has a finite-temperature quotient of a warped deformation of the magnetic string geometry. The corresponding dual deformation of the magnetic string CFT potentially supplies, for the general case, the cL = cR =6JL CFT predicted by Kerr/CFT.
18278418
Warped AdS3 black holes
Three dimensional topologically massive gravity (TMG) with a negative cosmological constant ???2 and positive Newton constant G admits an AdS3 vacuum solution for any value of the graviton mass ?. These are all known to be perturbatively unstable except at the recently explored chiral point ?? = 1. However we show herein that for every value of ?? ? 3 there are two other (potentially stable) vacuum solutions given by SL(2,) ? U(1)-invariant warped AdS3 geometries, with a timelike or spacelike U(1) isometry. Critical behavior occurs at ?? = 3, where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null U(1) isometry. For 3$>?? > 3, there are known warped black hole solutions which are asymptotic to warped AdS3. We show that these black holes are discrete quotients of warped AdS3 just as BTZ black holes are discrete quotients of ordinary AdS3. Moreover new solutions of this type, relevant to any theory with warped AdS3 solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for 3$>?? > 3, the warped AdS3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges and .
118836818
New boundary conditions for AdS3
A bstractNew chiral boundary conditions are found for quantum gravity with matter on AdS3. The associated asymptotic symmetry group is generated by a single right-moving U(1) Kac-Moody-Virasoro algebra with $ {c_R}=\frac{{3\ell }}{2G } $. The Kac-Moody zero mode generates global left-moving translations and equals, for a BTZ black hole, the sum of the total mass and spin. The level is positive about the global vacuum and negative in the black hole sector, corresponding to ergosphere formation. Realizations arising in Chern-Simons gravity and string theory are analyzed. The new boundary conditions are shown to naturally arise for warped AdS3 in the limit that the warp parameter is taken to zero.
123487065
Semi-classical central charge in topologically massive gravity
It is shown that the warped black hole geometries discussed recently in arXiv:0807.3040 (Anninos et al 2008) admit an algebra of asymptotic symmetries isomorphic to the semi-direct product of a Virasoro algebra and an algebra of currents. The realization of this asymptotic symmetry by canonical charges allows us to find the central charge of the Virasoro algebra. The right-moving central charge is obtained when the Virasoro generators are normalized in order to have a positive zero-mode spectrum for the warped black holes. If one normalizes the Virasoro generators in order to have a positive central charge, the zero mode is then unbounded from below. The current algebra is also shown to be centrally extended.
11863169
Boundary conditions for spacelike and timelike warped AdS 3 spaces in topologically massive gravity
We propose a set of consistent boundary conditions containing the spacelike warped black holes solutions of Topologically Massive Gravity. We prove that the corresponding asymptotic charges whose algebra consists in a Virasoro algebra and a current algebra are finite, integrable and conserved. A similar analysis is performed for the timelike warped AdS3 spaces which contain a family of regular solitons. The energy of the boundary Virasoro excitations is positive while the current algebra leads to negative (for the spacelike warped case) and positive (for the timelike warped case) energy boundary excitations. We discuss the relationship with the Brown-Henneaux boundary conditions.
118675928
Two Virasoro symmetries in stringy warped AdS3
A bstractWe study three-dimensional consistent truncations of type IIB supergravity which admit warped AdS3 solutions. These theories contain subsectors that have no bulk dynamics. We show that the symplectic form for these theories, when restricted to the non-dynamical subsectors, equals the symplectic form for pure Einstein gravity in AdS3. Consequently, for each consistent choice of boundary conditions in AdS3, we can define a consistent phase space in warped AdS3 with identical conserved charges. This way, we easily obtain a Virasoro × Virasoro asymptotic symmetry algebra in warped AdS3; two different types of Virasoro × Kač-Moody symmetries are also consistent alternatives.Next, we study the phase space of these theories when propagating modes are included. We show that, as long as one can define a conserved symplectic form without introducing instabilities, the Virasoro × Virasoro asymptotic symmetries can be extended to the entire (linearised) phase space. This implies that, at least at semi-classical level, consistent theories of gravity in warped AdS3 are described by a two-dimensional conformal field theory, as long as stability is not an issue.
15010088
Kerr/CFT Correspondence
Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by J=GM{sup 2}) is considered. It is shown that consistent boundary conditions exist, for which the asymptotic symmetry generators form one copy of the Virasoro algebra with central charge c{sub L}=(12J/({Dirac_h}/2{pi})). This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature T{sub L}=(1/2{pi}) and conjugate energy given by the zero mode generator, L{sub 0}, of the Virasoro algebra. Assuming unitarity, the Cardy formula then gives a microscopic entropy S{sub micro}=(2{pi}J/({Dirac_h}/2{pi})) for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy S{sub macro}=(Area/4({Dirac_h}/2{pi})G). The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge c{sub L}=(12J/({Dirac_h}/2{pi})), and, in particular, that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with c{sub L}{approx}2x10{sup 79}.
118515727
Warped Conformal Field Theory
We study field theories in two spacetime dimensions invariant under a chiral scaling symmetry that acts only on right-movers. The local symmetries include one copy of the Virasoro algebra and a U(1) current algebra. This differs from the 2d conformal group, but in some respects is equally powerful in constraining the theory. In particular, the symmetries on a torus lead to modular covariance of the partition function, which is used to derive a universal formula for the asymptotic density of states. For an application we turn to the holographic description of black holes in quantum gravity, motivated by the fact that the symmetries in the near horizon geometry of any extremal black hole are identical to those of a 2d field theory with chiral scaling. We consider two examples: black holes in warped AdS_3 in topologically massive gravity, and in string theory. In both cases, the density of states in the 2d field theory reproduces the Bekenstein-Hawking entropy of black holes in the gravity theory.
55860369
Entanglement entropy in warped conformal field theories
A bstractWe present a detailed discussion of entanglement entropy in (1+1)-dimensional Warped Conformal Field Theories (WCFTs). We implement the Rindler method to evaluate entanglement and Renyi entropies for a single interval and along the way we interpret our results in terms of twist field correlation functions. Holographically a WCFT can be described in terms of Lower Spin Gravity, a SL (2, ℝ) × U (1) Chern-Simons theory in three dimensions. We show how to obtain the universal field theory results for entanglement in a WCFT via holography. For the geometrical description of the theory we introduce the concept of geodesic and massive point particles in the warped geometry associated to Lower Spin Gravity. In the Chern-Simons description we evaluate the appropriate Wilson line that captures the dynamics of a massive particle.
119505812
Structure constants from modularity in warped CFT
Abstract
 
 We derive a universal formula for the asymptotic growth of the mean value of three-point coefficient for Warped Conformal Field Theories (WCFTs), and provide a holographic calculation in BTZ black holes. WCFTs are two dimensional quantum field theories featuring a chiral Virasoro and U(1) Kac-Moody algebra, and are conjectured to be holographically dual to quantum gravity on asymptotically AdS3 spacetime with Compère- Song-Strominger boundary conditions. The WCFT calculation amounts to the calculation of one-point functions on torus, whose high temperature limit can be approximated by using modular covariance of WCFT, similar to the derivation of Cardy formula. The bulk process is given by a tadpole diagram, with a massive spinning particle propagates from the infinity to the horizon, and splits into particle and antiparticle which annihilate after going around the horizon of BTZ black holes. The agreement between the bulk and WCFT calculations indicates that the black hole geometries in asymptotically AdS3 spacetimes can emerge upon coarse-graining over microstates in WCFTs, similar to the results of Kraus and Maloney in the context of AdS/CFT [1].
119159068
Modifications to holographic entanglement entropy in warped CFT
A bstractIn [1] it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the Ryu-Takayanagi proposal must be modified for warped AdS3 (WAdS3) with Dirichlet boundary conditions. In this paper, we consider AdS3 and WAdS3 with Dirichlet-Neumann boundary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS3/WCFT and WAdS3/WCFT dualities. Our bulk results are consistent with the WCFT results derived by Castro-Hofman-Iqbal using the Rindler method. Comparing with [1], we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.
119099852
Entanglement and chaos in warped conformal field theories
A bstractVarious aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Rényi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Rényi entropy after a local quench. We find that the change in the Rényi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.
73616640
Spinning probes and helices in AdS3
We study extremal curves associated with a functional which is linear in the curve's torsion. The functional in question is known to capture the properties of entanglement entropy for two-dimensional conformal field theories with chiral anomalies and has potential applications in elucidating the equilibrium shape of elastic linear structures. We derive the equations that determine the shape of its extremal curves in general ambient spaces in terms of geometric quantities. We show that the solutions to these shape equations correspond to a three-dimensional version of Mathisson's helical motions for the centers of mass of spinning probes. Thereafter, we focus on the case of maximally symmetric spaces, where solutions correspond to cylindrical helices and find that the Lancret ratio of these equals the relative speed between the Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all possible helical motions in three-dimensional manifolds with constant negative curvature. In particular, we discover a rich space of helices in AdS$_3$ which we explore in detail.
119491144
Bootstrapping holographic warped CFTs or: how I learned to stop worrying and tolerate negative norms
A bstractWe use modular invariance to derive constraints on the spectrum of warped conformal field theories (WCFTs) — nonrelativistic quantum field theories described by a chiral Virasoro and U(1) Kac-Moody algebra. We focus on holographic WCFTs and interpret our results in the simplest holographic set up: three dimensional gravity with Compère-Song-Strominger boundary conditions. Holographic WCFTs feature a negative U(1) level that is responsible for negative norm descendant states. Despite the violation of unitarity we show that the modular bootstrap is still viable provided the (Virasoro-Kac-Moody) primaries carry positive norm. In particular, we show that holographic WCFTs must feature either primary states with negative norm or states with imaginary U(1) charge, the latter of which have a natural holographic interpretation. For large central charge and arbitrary level, we show that the first excited primary state in any WCFT satisfies the Hellerman bound. Moreover, when the level is positive we point out that known bounds for CFTs with internal U(1) symmetries readily apply to unitary WCFTs.
102491707
Rényi mutual information in holographic warped CFTs
Abstract
 
 The study of Rényi mutual information (RMI) sheds light on the AdS/CFT correspondence beyond classical order. In this article, we study the Rényi mutual in- formation between two intervals at large distance in two-dimensional holographic warped conformal field theory, which is conjectured to be dual to gravity on AdS3 or warped AdS3 spacetimes under Dirichlet-Neumann boundary conditions. By using the operator product expansion of twist operators up to level 3, we read the leading oder and the next-to-leading order RMI in the large central charge and small cross-ratio limits. The leading order result is furthermore confirmed using the conformal block expansion. Finally, we match the next-to-leading order result by a 1-loop calculation in the bulk.