We study the thermodynamic properties of warped AdS$_3$ black hole within the framework of thermodynamic information geometry. Our analysis focuses on finding the set of proper thermodynamic Riemannian metrics on the space of equilibrium states, together with the conditions for local and global thermodynamic stability. We use our findings to constrain the values of left and right central charges from the dual CFT theory and the parameters of the bulk gravitational theory.
We study the asymptotic symmetries of three-dimensional Warped Anti-de Sitter (WAdS) spaces in three-dimensional New Massive Gravity (NMG). For a specific choice of asymptotic boundary conditions, we find that the algebra of charges is infinite-dimensional and coincides with the semidirect sum of Virasoro algebra with non-vanishing central charge and an affine $\hat{u}(1)_k$ Kač-Moody algebra. We show that the WAdS black hole configurations organize in terms of two commuting Virasoro algebras. We identify the Virasoro generators that expand the associated representations in the dual Warped Con…
In a previous paper, it has been shown that the entropy of non-extremal black holes in Warped Anti-de Sitter (WAdS) spaces in massive gravity can be computed microscopically in terms of a dual conformal field theory. Here, we extend this computation to a set of asymptotic boundary conditions that, while still gathering the WAdS$_3$ black holes, also allow for new solutions that are not locally equivalent to WAdS$_3$ space, and therefore are associated to the local degrees of freedom of the theory (bulk massive gravitons). After presenting explicit examples of such geometries, we compute the as…
WAdS/WCFT correspondence is an interesting realization of non-AdS holography. It relates 3-dimensional Warped-Anti-de Sitter (WAdS$_3$) spaces to a special class of 2-dimensional quantum field theory with chiral scaling symmetry that acts only on right-moving modes. The latter are often called Warped Conformal Field Theories (WCFT$_2$), and their existence makes WAdS/WCFT particularly interesting as a tool to investigate a new type of 2-dimensional conformal structure. Besides, WAdS/WCFT is interesting because it enables to apply holographic techniques to the microstate counting problem of non…
We 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 WAdS$_3$ black hole geometries exhibit a Gregory-Laflamme-like instability, and that WAdS$_3$ throats do not admit a decoupling limit.
In arXiv:1601.02634 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 AdS$_3$ (WAdS$_3$) with Dirichlet boundary conditions. In this paper, we consider AdS$_3$ and WAdS$_3$ 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 AdS$_3$/WCFT and WAdS$…
We consider the timelike version of Warped Anti-de Sitter space (WAdS), which corresponds to the three-dimensional section of the Gödel solution of four-dimensional cosmological Einstein equations. This geometry presents closed timelike curves (CTCs), which are inherited from its four-dimensional embedding. In three dimensions, this type of solutions can be supported without matter provided the graviton acquires mass. Here, among the different ways to consistenly give mass to the graviton in three dimensions, we consider the parity-even model known as New Massive Gravity (NMG). In the bulk of …
Warped 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 no…
Besides black holes, the phase space of three-dimensional massive gravity about Warped-Anti de Sitter (WAdS) space contains solutions that decay exponentially in time. They describe evanescent graviton configurations that, while governed by a wave equation with non-vanishing effective mass, do not carry net gravitational energy. Explicit examples of such solutions have been found in the case of Topologically Massive Gravity; here, we generalize them to a much more general ghost-free massive deformation, with the difference being that the decay rate gets corrected due to the presence of higher-…
We study real-time correlators in the warped AdS/CFT correspondence. We apply the prescription used in the usual AdS/CFT correspondence and obtain the retarded Green's functions for the scalar and vector fields in the spacelike warped and the null warped black hole backgrounds. We find that the retarded Green's functions and the cross sections are well consistent with the predictions from dual CFT. Our results not only support strongly the conjectured warped AdS/CFT correspondence, but also show that the usual relativistic AdS/CFT prescription of obtaining the real-time correlators remain effe…
We analytically calculate the quasi-normal modes of various perturbations of spacelike stretched and null warped $AdS_3$ black holes. From AdS/CFT correspondence, these quasi-normal modes are expected to appear as the poles in momentum space of retarded Green functions of dual operators in CFT at finite temperature. We find that this is indeed the case, after taking into account of the subtle identification of quantum numbers. The subtlety comes from the fact that only after appropriate coordinate transformation the asymptotic geometries of warped black holes are the same as the ones of warped…
String theory contains solutions with SL(2,R)_R x U(1)_L-invariant warped AdS3 (WAdS3) factors arising as continuous deformations of ordinary AdS3 factors. We propose that some of these are holographically dual to the IR limits of nonlocal dipole-deformed 2D D-brane gauge theories, referred to as "dipole CFTs". Neither the bulk nor boundary theories are currently well-understood, and consequences of the proposed duality for both sides is investigated. The bulk entropy-area law suggests that dipole CFTs have (at large N) a high-energy density of states which does not depend on the deformation p…
We calculate the scalar quasinormal modes of warped $AdS_3$ black holes analytically. We find that in general they are not in agreement with the prediction of the usual AdS/CFT correspondence. Nevertheless, for the black hole not deviating far from BTZ black hole, the quasinormal frequencies are still well consistent with AdS/CFT correspondence.
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 information between two intervals at large distance in two-dimensional holographic warped conformal field theory, which is conjectured to be dual to the gravity on AdS3 or warped AdS3 spacetime under the Dirichlet-Newman boundary conditions. By using the operator product expansion of twist operators up to level 3, we read the leading oder and the next-toleading order RMI in the large central charge and small cross-ratio limits. The leading ord…
On the Stability of Warped Anti-de Sitter Spaces in Supergravity
Politecnico di Torino（都灵理工大学）；INFN, Sezione di Torino（意大利国家核物理研究所都灵分部）；CEICO, Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Universidade de Santiago de Compostela（圣地亚哥德孔波斯特拉大学）；Texas A&M University（德克萨斯农工大学）；ENS de Lyon（里昂高等师范学校）
In this talk we discuss two classes of examples of warped products of AdS spaces in the context of the AdS/CFT correspondence. The first class of examples appears in the construction of dual Type I' string descriptions to five dimensional supersymmetric fixed points with E_{N_f+1} global symmetry. The background is obtained as the near horizon geometry of the D4-D8 brane system in massive Type IIA supergravity. The second class of examples appears when considering the N=2 superconformal theories defined on a 3+1 dimensional hyperplane intersection of two sets of M5 branes. We use the dual stri…
We show that for a certain low frequency limit the wave equation of a generic massive scalar field in the background of the spacelike warped AdS_3 black hole can be written as the Casimir of an SL(2,R) symmetry. Two sets of SL(2,R) generators are found which uncover the hidden SL(2,R)\times SL(2,R) symmetry of the solution. This symmetry is only locally defined and is spontaneously broken to U(1)\times U(1) by a periodic identification of the ϕcoordinate. By using the generator of the identification we read the left and right temperatures (T_L,T_R) of the proposed dual conformal field theory w…
We investigate the absorption cross section for minimal-coupled scalars in the warped AdS_3 black hole. According to our calculation, the cross section reduces to the horizon area in the low energy limit as usually expected in contrast to what was previously found. We also calculate the greybody factor and find that the effective temperatures for the two chiral CFT's are consistent with that derived from the quasinormal modes. Observing the conjectured warped AdS/CFT correspondence, we suspect that a specific sector of the CFT operators with the desired conformal dimension could be responsible…
The AdS/CFT conjecture offers the possibility of a quantum description for a black hole in terms of a CFT. This has led to the study of general AdS^3 type black holes with a view to constructing an explicit toy quantum black hole model. Such a CFT description would be characterized by its central charge and the dimensions of its primary fields. Recently the expression for the central charges (C_L, C_R) of the CFT dual to the warped AdS^3 have been determined using asymptotic symmetry arguments. The central charges depend, as expected, on the warping factor. We show that topological arguments, …
We study a new class of solutions of three-dimensional topological massive gravity. These solutions can be taken as non-extremal black holes, with their extremal counterparts being discrete quotients of spacelike warped AdS$_3$ along the $U(1)_L$ isometry. We study the thermodynamics of these black holes and show that the first law is satisfied. We also show that for consistent boundary conditions, the asymptotic symmetry generators form only one copy of the Virasoro algebra with central charge $c_L = \frac{4ν\ell}{G(ν^2+3)}$, with which the Cardy formula reproduces the black hole entropy. We …
We study a toy model of the Kerr/CFT correspondence using string theory on AdS$_3 \times S^3$. We propose a single trace irrelevant deformation of the dual CFT generated by a vertex operator with spacetime dimensions (2,1). This operator shares the same quantum numbers as the integrable $T\bar{J}$ deformation of two-dimensional CFTs where $\bar{J}$ is a chiral $U(1)$ current. We show that the deformation is marginal on the worldsheet and that the target spacetime is deformed to null warped AdS$_3$ upon dimensional reduction. We also calculate the spectrum of the deformed theory on the cylinder…
We study some aspects of a class of non-AdS holography where the 3d bulk gravity is given by Generalized Minimal Massive Gravity (GMMG). We consider the spacelike warped $AdS_3$ ($WAdS_3$) black hole solution of this model where the 2d dual boundary theory is the warped conformal field theory (WFCT). The charge algebra of the isometries in the bulk and the charge algebra of the vacuum symmetries at the boundary are compatible and this is an evidence for the duality conjecture. Further evidence for this duality is the equality of entanglement entropy and modular Hamiltonian on both sides of the…
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 AdS$_3$ spacetime with Comp$\grave{\mathrm{e}}$re-Song-Strominger boundary conditions. The WCFT calculation amounts to the calculation of one-point functions on torus, whose high temperature limit can be app…
We give a direct computation of the mass of black holes in Warped Anti-de Sitter space (WAdS) in terms of the Brown-York stress-tensor at the boundary. This permits to explore to what extent the holographic renormalization techniques can be applied to such type of deformation of AdS. We show that, despite some components of the boundary stress-tensor diverge and resist to be regularized by the introduction of local counterterms, the precise combination that gives the quasilocal energy density yields a finite integral. The result turns out to be in agreement with previous computations of the bl…
We study real-time correlators in the Kerr/CFT correspondence. The near-horizon geometry of extreme and near-extremal Kerr black holes can be taken as warped spacetimes with the warping factors depending on the transverse radius and the angular variable. By considering the Kerr/CFT correspondence as a generalized warped AdS/CFT correspondence, we show how to compute the CFT real-time correlation functions. We find that the retarded Green's functions, the cross sections and the quasi-normal modes are all in perfect match with the predictions of the dual CFT. The same analysis is also generalize…
The general model of higher derivative (HD) gravity is considered. The search of brane-world cosmology in such theory is presented when bulk is d5 AdS and boundary is spherical, hyperbolic or flat (single) brane. It is found the wide range of theory parameters where such cosmology may be realized. Special attention is paid to the version of HD theory representing SG dual of ${\cal N}=2$ $Sp(N)$ SCFT (in next-to-leading order of large $N$ expansion). In particular, it is shown that inflationary brane Universe does not occur for SG dual while hyperbolic brane occurs (which was not possible in le…
We consider heterotic strings on a warped Eguchi-Hanson space with five-brane and line bundle gauge fluxes. The heterotic string admits an exact CFT description in terms of an asymmetrically gauged SU(2)xSL(2,R) WZW model, in a specific double scaling limit in which the blow-up radius and the string scale are sent to zero simultaneously. This allows us to compute the perturbative 6D spectra for these models in two independent fashions: i) Within the supergravity approximation we employ a representation dependent index; ii) In the double scaling limit we determine all marginal vertex operators …
We use the notion of double holography to study Hawking radiation emitted by the eternal BTZ black hole in equilibrium with a thermal bath, but in the form of warped CFT$_2$ degrees of freedom. In agreement with the literature, we find entanglement islands and a phase transition in the entanglement surface, but our results differ significantly from work in AdS/CFT in three major ways: (1) the late-time entropy decreases in time, (2) island degrees of freedom exist at all times, and not just at late times, with the phase transition changing whether or not these degrees of freedom include the bl…
We use group theoretic methods to calculate the spectrum of short multiplets around the extremum of N=8 gauged supergravity potential which possesses N=2 supersymmetry and SU(3) global symmetry. Upon uplifting to M-theory, it describes a warped product of AdS_4 and a certain squashed and stretched 7-sphere. We find quantum numbers in agreement with those of the gauge invariant operators in the N=2 superconformal Chern-Simons theory recently proposed to be the dual of this M-theory background. This theory is obtained from the U(N)xU(N) theory through deforming the superpotential by a term quadr…
In this paper we apply the thermodynamics method to investigate the holographic pictures for the BTZ black hole, the spacelike and the null warped black holes in three-dimensional topologically massive gravity (TMG) and new massive gravity (NMG). Even though there are higher derivative terms in these theories, the thermodynamics method is still effective. It gives consistent results with the ones obtained by using asymptotical symmetry group (ASG) analysis. In doing the ASG analysis we develop a brute-force realization of the Barnich-Brandt-Compere formalism with Mathematica code, which also a…
In this work we explore the complexity path integral optimization process for the case of warped $\text{AdS}_3$/warped $\text{CFT}_2$ correspondence. We first present the specific renormalization flow equations and analyze the differences with the case of CFT. We discuss how the "chiral Liouville action" could replace the Liouville action as the suitable cost function for this case. Starting from the other side of the story, we also show how the deformed Liouville actions could be derived from the spacelike, timelike and null warped metrics and how the behaviors of boundary topological terms c…
We consider a black hole solution whose spatial boundary is a squashed three sphere in Einstein gravity with negative cosmological constant. We solve the Einstein equations numerically and find a warped AdS black hole solution with arbitrary squashing parameter. This solution becomes the ordinary AdS-Schwarzschild solution when the squashing parameter is chosen appropriately. Motivated by this fact, we study N=4 super Yang-Mills theory with zero coupling constant on a squashed three sphere and show that the thermodynamical entropy of this theory roughly agrees with that of the warped AdS black…
We consider the various perturbations of self-dual warped AdS$_3$ black hole and obtain the exact expressions of quasinormal modes by imposing the vanishing Dirichlet boundary condition at asymptotic infinity. It is expected that the quasinormal modes agree with the poles of retarded Green's functions of the dual CFT. Our results provide a quantitative test of the warped AdS/CFT correspondence.
We consider asymptotically Warped AdS$_3$ black holes in Topologically Massive Gravity. We study their thermodynamic stability and show the existence of a Hawking-Page phase transition between the black hole and thermal background phases. At zero angular potential, the latter is shown to occur at the self-dual point of the dual Warped Conformal Field Theory partition function, in analogy with the phase transition for BTZ black holes in AdS$_3$/CFT$_2$. We also discuss how the central and vacuum charges can be obtained from inner horizon mechanics in the presence of a gravitational anomaly.
We obtain a family of heterotic supergravity backgrounds describing non-Kahler warped conifolds with three-form flux and an Abelian gauge bundle, preserving N=1 supersymmetry in four dimensions. At large distance from the singularity the usual Ricci-flat conifold is recovered. By performing a Z_2 orbifold of the T^{1,1} base, the conifold singularity can be blown-up to a four-cycle, leading to a completely smooth geometry. Remarkably, the throat regions of the solutions, which can be isolated from the asymptotic Ricci-flat geometry using a double-scaling limit, possess a worldsheet CFT descrip…
We extend the recently proposal of hidden conformal symmetry to the self-dual warped AdS$_3$ black holes in topological massive gravity. It is shown that the wave equation of massive scalar field with sufficient small angular momentum can be reproduced by the SL(2, R) Casimir quadratic operator. Due to the periodic identification in the $ϕ$ direction, it is found that only the left section of hidden conformal symmetry is broken to U(1), while the right section is unbroken, which only gives the left temperature of dual CFT. As a check of the dual CFT conjecture of self-warped AdS$_3$ black hole…
We study integrable deformations of AdS/CFT by focusing upon three kinds of warped AdS_3 geometries, 1) space-like warped AdS_3, 2) time-like warped AdS_3 and 3) null warped AdS_3. These geometries are embedded into type IIB supergravity solutions and are regarded as consistent string backgrounds. By restricting the classical motion of strings on the warped AdS_3xS^1 subspace, the Landau-Lifshitz sigma models are derived by taking the fast-moving limit. The first two warped AdS_3 spaces correspond to anisotropic deformations of the sl(2) spin chain and the last one to Jordanian deformations. A…
We reconsidered the quantum gravity description of the near horizon extremal Reissner-Nordstrøm black hole in the viewpoint of the AdS$_2$/CFT$_1$ correspondence. We found that, for pure electric case, the right moving central charge of dual 1D CFT is $6 Q^2$ which is different from the previous result $6 Q^3$ of left moving sector obtained by warped AdS$_3$/CFT$_2$ description. We discussed the discrepancy in these two approaches and examined novel properties of our result.
We compute the spin-2 Kaluza-Klein modes around a warped product of AdS_4 and a seven-ellipsoid. This background with global G_2 symmetry is related to a U(N) x U(N) N=1 superconformal Chern-Simons matter theory with sixth order superpotential. The mass-squared in AdS_4 is quadratic in G_2 quantum number and KK excitation number. We determine the dimensions of spin-2 operators using the AdS/CFT correspondence. The connection to N=2 theory preserving SU(3) x U(1)_R is also discussed.
We consider (warped) AdS string backgrounds which allow for a GKP spinning string/null cusp solution. Integrability implies that the worldsheet S-matrix should factorize, which in turn constrains the form of the warp factor as a function of the coordinates of the internal space. This constraint is argued to rule out integrability for all supersymmetric AdS7 and AdS6 backgrounds as well as AdS5 Gaiotto-Maldacena backgrounds and a few highly supersymmetric AdS4 and AdS3 backgrounds.
Three dimensional spacelike warped anti-de Sitter space is studied in the context of Einstein theories of gravity and string theory, where there is no gravitational Chern-Simons term in the action. We propose that it is holographically dual to a two-dimensional conformal field theory with equal left and right moving central charges. Various checks of the central charges are offered, based on the Bekenstein-Hawking entropy of the stretched warped black holes and warped self-dual solutions. The proposed central charges are applied to compute the Bekenstein-Hawking entropy of the Hopf T-dual of s…
We provide further evidence that the recently constructed warped $AdS_6$ solutions in Type IIB supergravity are dual to 5d SCFTs that correspond to $(p,q)$ 5-brane webs with large numbers of like-charged external 5-branes. We study a number of specific examples, including the $T_N$ theory, and identify the bulk states dual to a class of operators with ${\cal O}(N)$ scaling dimensions in terms of strings and string-webs embedded in the solutions.
In this letter, we provide an alternative method to study the area spectrum of certain classes of extremal black holes which have near-horizon geometry as warped AdS. We argue that previous methods which are based on the existence of quasinormal modes may not be applicable in the extremal limit. The topology difference of the near-horizon geometry between non-extremal and extremal black holes implies a separate treatment is needed to study the area discreteness in the extremal limit. To be specific, we will study area spectrum of supersymmetric BMPV black holes/black rings and Reissner-Norstro…
We study the holographic dual of asymmetrically warped space-times, which are asymptotically AdS. The self-tuning of the cosmological constant is reinterpreted as a cancellation of the visible sector stress-energy tensor by the contribution of a hidden CFT, charged under a spontaneously broken global symmetry. The apparent violation of 4D causality due to bulk geodesics is justified by considering that the CFT feels the background metric as smeared out over a length of the order of the AdS radius.
We examine supersymmetric properties of null-warped AdS$_3$, or alternatively Schrodinger geometries, dual to putative warped CFTs in two dimensions. We classify super Schrodinger subalgebras of the superalgebra psu(1, 1$|$2) $\oplus$ psu(1, 1$|$2), corresponding to the superconformal algebra of the AdS$_3 \times$ S$^3$ geometry. We comment on geometric realisations and provide a string theory description with enhanced supersymmetry in terms of intersecting D3-branes. For type IIB supergravity solutions based on T$^{1,1}$, we consider the relationship between five-dimensional Schrodinger solut…
String theory on NS-NS AdS_3 x S^3 admits an exactly marginal deformation which breaks the SL(2,R)_R x SL(2,R)_L isometry of AdS_3 down to SL(2,R)_R x U(1)_L. The holographic dual is an exotic and only partially understood type of two-dimensional CFT with a reduced unbroken global conformal symmetry group. In this paper we study the deformed theory on the string worldsheet. It is found to be related by a spectral flow which is nonlocal in spacetime to the undeformed worldsheet theory. An exact formula for the spectrum of massive strings is presented.
We study the applicability of the covariant holographic entanglement entropy proposal to asymptotically warped AdS$_3$ spacetimes with an SL(2,R) x U(1) isometry. We begin by applying the proposal to locally AdS$_3$ backgrounds which are written as a real-line fibration over AdS$_2$. We then perturb away from this geometry by considering a warping parameter $a=1+δ$ to get an asymptotically warped AdS$_3$ spacetime and compute the dual entanglement entropy perturbatively in $δ$. We find that for large separation in the fiber coordinate, the entanglement entropy can be computed to all orders in …
In this paper we show that warped AdS$_{3}$ black hole spacetime is a solution of the generalized minimal massive gravity (GMMG) and introduce suitable boundary conditions for asymptotically warped AdS$_{3}$ spacetimes. Then we find the Killing vector fields such that transformations generated by them preserve the considered boundary conditions. We calculate the conserved charges which correspond to the obtained Killing vector fields and show that the algebra of the asymptotic conserved charges is given as the semi direct product of the Virasoro algebra with $U(1)$ current algebra. We use a pa…
In this paper we study spin 2 fluctuations around a warped $AdS_3 \times S^2 \times T^4 \times \mathcal{I}_ρ$ background in type IIA supergravity with small $\mathcal{N} = (0,4)$ supersymmetry. We find a class of fluctuations, which will be called \textit{universal}, that is independent of the background data and corresponds to operators with scaling dimension $Δ= 2l +2$, being $l$ the angular-momentum-quantum-number on the $S^2$ which realises the $SU(2)_R$ symmetry. We compute the central charge for $\mathcal{N} = (0,4)$ two-dimensional superconformal theories from the action of the spin 2 f…
We study the breaking of supersymmetry in five-dimensional (5d) warped spaces, using the Randall-Sundrum model as a prototype. In particular, we present a supersymmetry-breaking mechanism which has a geometrical origin, and consists of imposing different boundary conditions between the fermions and bosons living in the 5d bulk. The scale of supersymmetry breaking is exponentially small due to the warp factor of the AdS metric. We apply this mechanism to a supersymmetric standard model where supersymmetry breaking is transmitted through the AdS bulk to matter fields confined on the Planck-brane…
