The absorption cross section is studied in the low-frequency region for a propagating scalar field under the warped AdS$_3$ black hole background in the topologically massive gravity. It can be shown that the absorption cross section is significantly deformed by the gravitational Chern-Simons term, which is proportional to the scattering area of black hole with an additional contribution depending on the left-moving and right-moving temperatures. It means that the cross section is larger than the area in spite of the s-wave limit. Finally, we discuss the left-right quasinormal modes for the scalar perturbation in this black hole.
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 study the Complexity=Volume conjecture for Warped AdS$_3$ black holes. We compute the spatial volume of the Einstein-Rosen bridge and we find that its growth rate is proportional to the Hawking temperature times the Bekenstein-Hawking entropy. This is consistent with expectations about computational complexity in the boundary theory.
We investigate stationary, rotationally symmetric solutions of a recently proposed three-dimensional theory of massive gravity. Along with BTZ black holes, we also obtain warped AdS_3 black holes, and (for a critical value of the cosmological constant) AdS_2 x S^1 as solutions. The entropy, mass and angular momentum of these black holes are computed.
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 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 AdS_3 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 bound…
We equip three-dimensional spin-3 gravity in the principal embedding with a new set of boundary conditions that we call "asymptotically null warped AdS". We find a chiral copy of the Polyakov-Bershadsky algebra as asymptotic symmetry algebra, reminiscent of the situation in topologically massive gravity with strict null warped AdS boundary conditions. We prove the invertibility of the map between zuvielbein and metric variables and construct a global gauge transformation to half of AdS spin-3 gravity in the diagonal embedding. This explains why the theory is chiral and why the Polyakov-Bershad…
We demonstrate that the warped AdS3 black hole solutions of Topologically Massive Gravity are classically stable against massive scalar field perturbations by analysing the quasinormal and bound state modes of the scalar field. In particular, it is found that although classical superradiance is present it does not give rise to superradiant instabilities. The stability is shown to persist even when the black hole is enclosed by a stationary mirror with Dirichlet boundary conditions. This is a surprising result in view of the similarity between the causal structure of the warped AdS3 black hole …
We first review spacelike stretched warped AdS$_3$ and we describe its black hole quotients by using accelerating and Poincaré coordinates. We then describe the maximal analytic extension of the black holes and present their causal diagrams. Finally, we calculate spacetime limits of the black hole phase space $(T_R,T_L)$. This is done by requiring that the identification vector $\partial_θ$ has a finite non-zero limit. The limits we obtain are the self-dual solution in accelerating or Poincaré coordinates, depending respectively on whether the limiting spacetimes are non-extremal or extremal, …
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…
The statistical entropy of a scalar field on the warped AdS$_3$ black hole in the cosmological topologically massive gravity is calculated based on the brick-wall method, which is different from the Wald's entropy formula giving the modified area law due to the higher-derivative corrections in that the entropy still satisfies the area law. It means that the entropy for scalar excitations on this background is independent of higher-order derivative terms or the conventional brick wall method has some limitations to take into account the higher-derivative terms.
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 investigate the asymptotic structure of the three dimensional warped anti de sitter black holes in the Bergshoeff, Hohm and Townsend massive gravity using the canonical Hamiltonian formalism. We define the canonical asymptotic guage generators, which produce the conserved charges and the asymptotic symmetry group for the warped anti de sitter black holes. The attained symmetry group is described by a semi direct sum of a Virasoro and a KacMoody algebra. Using the Sugawara construction, we obtain a direct sum of two Virasoro algebras. We show that not only the asymptotic conserved charges sa…
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 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.
We investigate the stability of the horizon in a warped anti-de Sitter black hole based on the three-dimensional new massive gravity by particle absorption. If a particle moving towards the black hole enters its outer horizon, the black hole changes because it absorbs conserved quantities of the particle. This variation is constrained by the equations of motion for the particle. We prove both the irreducibility of the entropy and stability of the horizon for the black hole through this process. However, the instability of the horizon is found in a near-extremal black hole by considering its se…
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…
We construct a stretched (aka Warped) Anti-de Sitter black hole in 3 dimensions supported by a real scalar field configuration. The latter is regular everywhere outside and on the horizon. No hair theorems in 3 dimensions demand the matter to be coupled to the curvature in a non-minimal way; however, this coupling can still be of the Horndeski type, i.e. yielding second order field equations similar to those appearing in the context of Galileon theories. These Warped-Anti-de Sitter black holes exhibit interesting thermodynamical properties, such as finite Hawking temperature and entropy. We co…
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…
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…
We identify the fractions of supersymmetry preserved by the most general warped flux AdS and flat backgrounds in both massive and standard IIA supergravities. We find that $AdS_n\times_w M^{10-n}$ preserve $2^{[{n\over2}]} k$ for $n\leq 4$ and $2^{[{n\over2}]+1} k$ for $4<n\leq 7$ supersymmetries, $k\in \bN_{>0}$. In addition we show that, for suitably restricted fields and $M^{10-n}$, the killing spinors of AdS backgrounds are given in terms of the zero modes of Dirac like operators on $M^{10-n}$. This generalizes the Lichnerowicz theorem for connections whose holonomy is included in a genera…
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 …
We show that there exists a precise kinematical notion of holography for the Euclidean warped $AdS_3$ black hole. This follows from the fact that the Euclidean warped $AdS_3$ black hole spacetime is a geometrically finite hyperbolic manifold. For such manifolds a theorem of Sullivan provides a one-to-one correspondence between the hyperbolic structure in the bulk and the conformal structure of its boundary. Using this theorem we obtain the holographic quasinormal modes for the warped $AdS_3$ black hole.
Warped AdS$_3$ black hole thermodynamics and the charged Cardy formula
Sofia University（索非亚大学）；INRNE, Bulgarian Academy of Sciences（保加利亚科学院核物理与核能研究所）；Università di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；INFN Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）
We have studied the quasinormal modes of a massive scalar field coupling to Einstein's tensor in the spacelike stretched $AdS_3$ black hole spacetime. We find that both the right-moving and left-moving quasinormal frequencies depend not only on the warped parameter $v$ of black hole, but also on the coupling between the scalar field and Einstein's tensor. Moreover, we also discuss the warped $AdS/CFT$ correspondence from the quasinormal modes and probe the effects of the coupling on the left and right conformal weights $h_L$ and $h_R$ of the operators dual to the scalar field in the boundary.
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…
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.
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…
Three dimensional topologically massive gravity (TMG) with a negative cosmological constant -\ell^{-2} and positive Newton constant G admits an AdS_3 vacuum solution for any value of the graviton mass μ. These are all known to be perturbatively unstable except at the recently explored chiral point μ\ell=1. However we show herein that for every value of μ\ell< 3 there are two other (potentially stable) vacuum solutions given by SL(2,R)x U(1)-invariant warped AdS_3 geometries, with a timelike or spacelike U(1) isometry. Critical behavior occurs at μ\ell=3, where the warping transitions from a st…
In this paper we consider the warped AdS$_{3}$ black hole solution of topologically massive gravity with a negative cosmological constant, and we investigate the possibility that it acts as a particle accelerator by analyzing the energy in the center of mass (CM) frame of two colliding particles in the vicinity of its horizon, which is known as the Bañados, Silk and West (BSW) process. Mainly, we show that the critical angular momentum $(L_c)$ of the particle decreases when the parameter that controls the stretching deformation ($ν$) increases. Also, we show that despite the particle with $L_c…
Asymptotically warped AdS spacetimes in topologically massive gravity with negative cosmological constant are considered in the case of spacelike stretched warping, where black holes have been shown to exist. We provide a set of asymptotic conditions that accommodate solutions in which the local degree of freedom (the "massive graviton") is switched on. An exact solution with this property is explicitly exhibited and possesses a slower fall-off than the warped AdS black hole. The boundary conditions are invariant under the semidirect product of the Virasoro algebra with a u(1) current algebra.…
We study a Hanany-Witten set-up relevant to N = 2 superconformal field theories. We find the exact near-horizon solution for this 11-dimensional system which involves intersecting M5-branes. The metric describes a warped product of AdS_5 with a manifold with SU(2) x U(1) isometry.
We study three-dimensional consistent truncations of type IIB supergravity which admit warped AdS$_3$ 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 AdS$_3$. Consequently, for each consistent choice of boundary conditions in AdS$_3$, we can define a consistent phase space in warped AdS$_3$ with identical conserved charges. This way, we easily obtain a Virasoro $\times$ Virasoro asymptotic symmetry algebra in warp…
We study the normal modes of a scalar field in the background of a warped AdS_3 black hole which arises in topologically massive gravity. We discuss the normal mode spectrum using the brick wall boundary condition. In addition, we investigate the possibility of a more general boundary condition for the scalar field.
In the context of warped conformal field theories (WCFT), the derivation of the warped Cardy formula relies on the zero mode spectrum being bounded from below. Generically, this is not true for holographic WCFTs in "canonical" ensemble, whereas this condition is satisfied in the "quadratic" ensemble, making it more natural in holographic setups. In this work, we find a new set of boundary conditions in three-dimensional Topologically Massive Gravity (TMG) such that the putative dual theory is a WCFT in quadratic ensemble. Surprisingly, imposing the equations of motion, we obtain a closed form …
Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3 dimensional self-dual space-like warped $AdS_3$ black hole background. We argue that in this background, symmetric conditions and gauge constraint can not be satisfied simultaneously. Using new suitable constraint, we find the exact solutions of equation of motion. Then, we obtain the quasi-normal modes by imposing appropriate boundary condition at horizon and infinity.
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…
We present the general form of all timelike supersymmetric solutions to 3D U(1)^3 gauged supergravity, a known consistent truncation of string theory. We uncover a rich vacuum structure, including an infinite class of new timelike-warped AdS_3 (Goedel) and timelike-warped dS_3 critical points. We outline the construction of supersymmetric flows, driven by irrelevant scalar operators in the SCFT, which interpolate between critical points. For flows from AdS_3 to Goedel, the natural candidate for the central charge decreases along the flow. Flows to timelike-warped dS_3 exhibit topology change.
The Complexity=Action conjecture is studied for black holes in Warped AdS$_3$ space, realized as solutions of Einstein gravity plus matter. The time dependence of the action of the Wheeler-DeWitt patch is investigated, both for the non-rotating and the rotating case. The asymptotic growth rate is found to be equal to the Hawking temperature times the Bekenstein-Hawking entropy; this is in agreement with a previous calculation done using the Complexity=Volume conjecture.
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 …
In this paper we consider pulsating strings in warped $AdS_6\times S^4$ background, which is a vacuum solution of massive type {\bf IIA} superstring. The case of rotating strings in this background was considered in hep-th/0402202 and it was found that the results significantly differs from those considered in $AdS_5\times S^5$. Motivated by this results we study pulsating strings in the warped spherical part of the type {\bf IIA} geometry and compare the results with those obtained in hep-th/0209047, hep-th/0310188 and hep-th/0404012. We conclude with comments on our solutions and the obtaine…
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 present all the symmetry superalgebras $\mathfrak{g}$ of all warped AdS$_k\times_w M^{d-k}$, $k>2$, flux backgrounds in $d=10, 11$ dimensions preserving any number of supersymmetries. First we give the conditions for $\mathfrak{g}$ to decompose into a direct sum of the isometry algebra of AdS$_k$ and that of the internal space $M^{d-k}$. Assuming this decomposition, we identify all symmetry superalgebras of AdS$_3$ backgrounds by showing that the isometry groups of internal spaces act transitively on spheres. We demonstrate that in type II and $d=11$ theories the AdS$_3$ symmetry superalgeb…
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.
Three-dimensional warped anti-de Sitter space in topologically massive gravity with a negative cosmological constant has been proposed to be holographically dual to a two-dimensional conformal field theory. We extend this proposal to both positive and vanishing values of the cosmological constant where stretched warped anti-de Sitter space is found to be a solution. For positive cosmological constant, another class of warped solutions is obtained by a spacelike (timelike) line fibration over Lorentzian (Euclidean) two-dimensional de Sitter space. These solutions exhibit a cosmological horizon …
We consider warped AdS_3 black holes in generic higher derivatives gravity theories in 2+1 dimensions. The asymptotic symmetry group of the phase space containing these black holes is the semi-direct product of a centrally extended Virasoro algebra and an affine u(1) Kac-Moody algebra. Previous works have shown that in some specific theories, the entropy of these black holes agrees with a Cardy-like entropy formula derived for warped conformal field theories. In this paper, we show that this entropy matching continues to hold for the most general higher derivative theories of gravity. We also …
We compute the ultraviolet divergences of holographic subregion complexity for the left and right factors of the thermofield double state in warped AdS$_3$ black holes, both for the action and the volume conjectures. Besides the linear divergences, which are also present in the BTZ black hole, additional logarithmic divergences appear. For the action conjecture, these log divergences are not affected by the arbitrarity in the length scale associated with the counterterm needed to ensure reparameterization invariance. We find that the subregion action complexity obeys the superadditivity proper…
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 study the thermodynamic stability of warped black holes in three-dimensional topologically massive gravity. The spacelike stretched black hole is parametrized by its mass and angular momentum. We determine the local and global stability properties in the canonical and grand canonical ensembles. The presence of a Hawking-Page type transition is established, and the critical temperature is determined. The thermodynamic metric of Ruppeiner is computed, and the curvature is shown to diverge in the extremal limit. The consequences of these results for the classical stability properties of warped…
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 generalize the swampland criterion forbidding stable non-supersymmetric AdS vacua and propose a new swampland conjecture forbidding stable non-supersymmetric "locally AdS" warped throats. The conjecture is motivated by the properties of systems of fractional D3-branes at singularities, and can be used to rule out large classes of warped throats with supersymmetry breaking ingredients, and their possible application to de Sitter uplift. In particular, this allows to reinterpret the runaway instabilities of the gravity dual of fractional branes in the dP$_1$ theory, and to rule out warped thr…
We consider null warped AdS(3) solutions of three-dimensional gravity coupled to a massive vector field. We isolate a certain set of non-propagating solutions to the equations of motion, which we argue are the ones relevant for understanding the stress tensor sector of the dual field theory. We construct a map from these modes to solutions of three-dimensional Einstein gravity with a negative cosmological constant, and thus show that they admit a Fefferman-Graham-like asymptotic expansion. We also compute the renormalized on-shell action for these modes at full non-linear level and propose tha…
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…
We determine the most general form of off-shell N=(1,1) supergravity field configurations in three dimensions by requiring that at least one off-shell Killing spinor exists. We then impose the field equations of the topologically massive off-shell supergravity and find a class of solutions whose properties crucially depend on the norm of the auxiliary vector field. These are spacelike-squashed and timelike-stretched AdS_3 for the spacelike and timelike norms, respectively. At the transition point where the norm vanishes, the solution is null warped AdS_3. This occurs when the coefficient of th…
