In the presence of a scalar hair perturbation, the Cauchy horizon of a Reissner-Nordström black hole disappears and is replaced by the rapid collapse of the Einstein-Rosen bridge, which leads to a Kasner singularity [1,2]. We study the time-dependence of holographic complexity, both for the volume and for the action proposals, in a class of models with hairy black holes. Volume complexity can only probe a portion of the black hole interior that remains far away from the Kasner singularity. We provide numerical evidence that the Lloyd bound is satisfied by the volume complexity rate in all the parameter space that we explored. Action complexity can instead probe a portion of the spacetime closer to the singularity. In particular, the complexity rate diverges at the critical time $t_c$ for which the Wheeler-DeWitt patch touches the singularity. After the critical time the action complexity rate approaches a constant. We find that the Kasner exponent does not directly affect the details of the divergence of the complexity rate at $t=t_c$ and the late-time behaviour of the complexity. The Lloyd bound is violated by action complexity at finite time, because the complexity rate diverges at $t=t_c$. We find that the Lloyd bound is satisfied by the asymptotic action complexity rate in all the parameter space that we investigated.
We present a large class of holographic models where the boundary ${\mathbb R}^{2,1}$ dimensional conformal field theory has a thermal phase with a spontaneously broken global symmetry. The dual black branes in a Poincare patch of asymptotically $AdS_4$ violate the no-hair theorem.
We study the mixed state entanglement of an asymptotic AdS black hole system with scalar hair. Through numerical calculations, we find that the holographic entanglement entropy (HEE) presents a non-monotonic behavior with the system parameter, depending on the size of the subregion. In addition, the mutual information (MI) also shows non-monotonic behavior in certain ranges of system parameters. However, the entanglement wedge minimum cross-section (EWCS), which is a mixed state entanglement measure, increases monotonically with the AdS radius; meanwhile, shows non-monotonic with the temperatu…
In this paper, we considered new solutions for four-dimensional asymptotically AdS black holes with scalar hair and discuss about Hawking temperature in the context of dark energy by using the tunneling method. We obtain modification of the Hawking temperature due to presence of the dark energy.
We show that the Abelian Higgs field equations in the background of the four dimensional AdS-Schwarzschild black hole have a vortex line solution. This solution, which has axial symmetry, is a generalization of the AdS spacetime Nielsen-Olesen string. By a numerical study of the field equations, we show that black hole could support the Abelian Higgs field as its Abelian hair. Also, we conside the self gravity of the Abelian Higgs field both in the pure AdS spacetime and AdS-Schwarzschild black hole background and show that the effect of string as a black hole hair is to induce a deficit angle…
We present a family of $AdS_{2+1}$ asymptotically hairy black holes, within the context of Einstein's minimally coupled scalar field theory. We study the boundary conditions and construct the thermal superpotential. In the Euclidean section, we calculate the free energy and in the Lorentzian section, the Brown-York tensor, both regularized by two methods. Finally, we calculate the relevant thermodynamic quantities and analyze the different phases.
We investigate the phase diagram of a general class of $4$-dimensional exact regular hairy planar black holes. For some particular values of the parameters in the moduli potential, these solutions can be embedded in $ω$-deformed $\mathcal{N}=8$ gauged supergravity. We construct the hairy soliton that is the ground state of the theory and show that there exist first order phase transitions.
We provide an exact hairy black hole solution to an $n+1$ dimensional complex scalar field model coupled with gravity. The model is characterized by a potential with one parameter. Depending on the magnitude of this parameter, the effective mass square of the scalar field can be both positive and negative. No matter what case it is, there are temperature ranges in which the hairy black hole is thermodynamically stable against decaying into its no-hair counterpart. We use this solution as an AdS/CFT superconductor model and study its conductivity-frequency relation and find typical features obs…
Probing the Singularity of Scalar-Haired Black Holes with Holographic Complexity
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，ENS，PSL大学，CNRS，索邦大学，巴黎大学）；Departement Physik, ETH Zürich（苏黎世联邦理工学院物理系）；Département de Physique, École Polytechnique, IP Paris（巴黎综合理工学院物理系）
Recently a class of static spherical black hole solutions with scalar hair was found in four and five dimensional gauged supergravity with modified, but AdS invariant boundary conditions. These black holes are fully specified by a single conserved charge, namely their mass, which acquires a contribution from the scalar field. Here we report on a more detailed study of some of the properties of these solutions. A thermodynamic analysis shows that in the canonical ensemble the standard Schwarzschild-AdS black hole is stable against decay into a hairy black hole. We also study the stability of th…
In this work, we study the real-time dynamics of the charged hairy black hole with the time-dependent source of scalar field in asymptotically anti-de Sitter (AdS) spacetime. The numerical results reveal a novel descalarization mechanism. In order to obtain the hairy black hole as the initial data for the quench process, we first analyze the quasi-normal modes of the massive complex scalar field on the Reissner-Nordström anti-de Sitter (RN-AdS) black hole background. We find the dominant unstable modes for large and small RN-AdS black holes come from the zero-damped modes and AdS modes, respec…
We use a mix of analytic and numerical methods to exhaustively study a class of asymptotically global AdS solitons and hairy black hole solutions in negative cosmological constant Einstein Maxwell gravity coupled to a charged massless scalar field. Our results depend sensitively on the charge 'e' of the scalar field. The solitonic branch of solutions we study hit the Chandrashekhar limit at finite mass at small 'e', but extends to arbitrarily large mass at larger 'e'. At low values of 'e' no hairy black holes exist. At intermediate values of 'e' hairy black holes exist above a critical charge.…
We investigate three-dimensional (3D) Anti-de Sitter (AdS) gravity coupled to a complex scalar field ϕwith self-interaction potential V(|ϕ|). We show that the mass of static, rotationally symmetric, AdS black hole with scalar hairs is determined algebraically by the scalar charges. We recast the field equations as a linear system of first order differential equations. Exact solutions, describing 3D AdS black holes with real spherical scalar hairs and vortex-black hole solutions are derived in closed form for the case of a scalar field saturating the Breitenlohner-Freedman (BF) bound and for a …
We determine the thermodynamic properties of a class of spherically symmetric and static black holes in AdS4 with magnetic charges and scalar hair. These black holes are solutions in 4d N=2 gauged supergravity that can arise from eleven-dimensional supergravity compactified on S^7. At zero temperature, they preserve supersymmetry and hence are stable. At non-zero temperatures, we explore in detail the canonical ensemble and stability of solutions and find a first order phase transition between small and large hairy black holes. The transition emerges as a liquid-gas phase transition in the dua…
Superconductors in a cylindrical geometry respond periodically to a cylinder-threading magnetic flux, with the period changing from hc/2e to hc/e depending on whether the Aharonov-Bohm effects are suppressed or not. We show that Holographic Superconductors present a similar phenomenon, and that the different periodicities follow from classical no-hair theorems. We also give the Ginzburg-Landau description of the period-doubling phenomenon.
D-branes with a U-shaped geometry, like the D8 flavor branes in the Sakai-Sugimoto model of QCD, are encountered frequently in holographic backgrounds. We argue that the commonly used DBI action is inadequate as an effective field theory description of these branes, because it misses a crucial component of the low-energy dynamics: a light complex scalar mode. Following an idea of Erkal, Kutasov and Lunin we elaborate on an effective description based on the abelian tachyon-DBI action which incorporates naturally the non-local physics of the complex scalar mode. We demonstrate its power in a co…
The study of new BPS objects in AdS_5 has led to a deeper understanding of AdS/CFT. To help complete this picture, and to fully explore the consequences of the supersymmetry algebra, it is also important to obtain new solutions with bulk fermions turned on. In this paper we construct superpartners of the 1/2 BPS black hole in AdS_5 using a natural set of fermion zero modes. We demonstrate that these superpartners, carrying fermionic hair, have conserved charges differing from the original bosonic counterpart. To do so, we find the R-charge and dipole moment of the new system, as well as the ma…
We study the charged black hole of hyperbolic horizon with scalar hair (charged Martinez-Troncoso-Zanelli: CMTZ black hole) as a model of analytic hairy black hole for holographic superconductor. For this purpose, we investigate the second order phase transition between CMTZ and hyperbolic Reissner-Nordström-AdS (HRNAdS) black holes. However, this transition unlikely occur. As an analytic treatment for holographic superconductor, we develop superconductor in the bulk and superfluidity on the boundary using the CMTZ black hole below the critical temperature. The presence of charge destroys the …
We discuss some aspects related to holography of Anti-de Sitter (AdS) dyonic hairy black holes, which break the conformal symetry of the boundary. We use counterterms for the scalar field that satisfies mixed boundary conditions to compute the Euclidean action and dual stress tensor. We apply these results to show that the Cardy-Verlinde formula is not satisfied. However, when the magnetic (or electric) charge vanishes, the conformal symmetry is preserved and the entropy of the black hole can be put in the Cardy-Verlinde form. In our analysis, there is no need of adding extra finite counterter…
We investigate the properties of spherically symmetric black hole solutions in the generalized Einstein-Skyrme model theory in four-dimensional asymptotically anti-de Sitter space-time. The dependencies of the Skyrmion fields on the cosmological constant and on the strength of the effective gravitational coupling are examined. We show that the increase of the absolute value of the cosmological constant qualitatively yields the same effect as increasing of the effective gravitational coupling. We confirm that, similar to the model in the asymptotically flat space-time, a necessary condition for…
We study a holographic p-wave superconductor model in a four dimensional Einstein-Maxwell-complex vector field theory with a negative cosmological constant. The complex vector field is charged under the Maxwell field. We solve the full coupled equations of motion of the system and find black hole solutions with the vector hair. The vector hairy black hole solutions are dual to a thermal state with the U(1) symmetry as well as the spatial rotational symmetry breaking spontaneously. Depending on two parameters, the mass and charge of the vector field, we find a rich phase structure: zeroth order…
By using the scaling symmetry in the reduced action formalism, we derive the novel Smarr relation which holds even for the hairy rotating AdS_3 black holes. And then, by using the Smarr relation we argue that the hairy rotating AdS_3 black holes are stable thermodynamically, compared to the non-hairy ones.
We consider Gauss-Bonnet (GB) gravity in general dimensions, which is non-minimally coupled to a scalar field. By choosing the scalar potential of the type $V(ϕ)=2Λ_0+\fft 12m^2ϕ^2+γ_4ϕ^4$, we first obtain large classes of scalar hairy black holes with spherical/hyperbolic/planar topologies that are asymptotic to locally anti-de Sitter (AdS) space-times. We derive the first law of black hole thermodynamics using Wald formalism. In particular, for one class of the solutions, the scalar hair forms a thermodynamic conjugate with the graviton and nontrivially contributes to the thermodynamical fir…
We investigate Yang-Mills theory on a spatial torus at finite temperature in the presence of discrete electric and magnetic fluxes using the AdS/CFT correspondence. We calculate the leading dependence of the partition function on the fluxes using the dual supergravity theory and comment upon the interpretation of these fluxes as discrete quantum hair for black holes in AdS spacetime.
By considering AdS charged black hole in the context of extended thermodynamic as the working substance we use it as a heat engine. We investigate the effect of hairy charge on the evolution of efficiency and Carnot efficiency along with electric charge. Because of interesting thermodynamic behavior of hairy black holes it would be natural to know their effects when we use black hole as a heat engine. We show that the hairy charge increases the efficiency, and so maximum temperature would be happened for bigger Maxwell charge when this hairy charge grows. For the fixed electric charges, the ef…
We present a new family of topological charged hairy black hole solutions in asymptotically AdS space in $D$-dimensions. We solve the coupled Einstein-Maxwell-Scalar gravity system and obtain exact charged hairy black hole solutions with planar, spherical and hyperbolic horizon topologies. The scalar field is regular everywhere. We discuss the thermodynamics of the hairy black hole and find drastic changes in its thermodynamic structure due to the scalar field. For the case of planar and spherical horizons, we find charged hairy/RN-AdS black hole phase transition, with thermodynamically prefer…
We investigate the thermodynamics of hairy black holes in asymptotically Anti-de Sitter (AdS) space, including backreaction. Resorting to the Euclidean path integral approach, we show that matter conformally coupled to Einstein gravity in five dimensions may exhibit a phase transition whose endpoint turns out to be a hairy black hole in AdS5 space. The scalar field configuration happens to be regular everywhere outside and on the horizon, and behaves asymptotically in such a way that respects the AdS boundary conditions that are relevant for AdS/CFT. The theory presents other peculiar features…
We construct exact hairy planar black holes in D-dimensional AdS gravity. These solutions are regular except at the singularity and have stress-energy that satisfies the null energy condition. We present a detailed analysis of their thermodynamical properties and show that the first law is satisfied. We also discuss these solutions in the context of AdS/CFT duality and construct the associated c-function.
We study small hairy black holes in a consistent truncation of ${\cal N}=8$ gauged supergravity that consists of a single charged scalar field interacting with the metric and a U(1) gauge field. Small very near extremal RNAdS black holes in this system are unstable to decay by superradiant emission. The end point of this instability is a small hairy black hole that we construct analytically in a perturbative expansion in its charge. Unlike their RNAdS counterparts, hairy black hole solutions exist all the way down to the BPS bound, demonstrating that ${\cal N}=4$ Yang Mills theory has an ${\ca…
We study holographic conformal order compactified on $S^3$. The corresponding boundary CFT$ _4$ has a thermal phase with a nonzero expectation value of a certain operator. The gravitational dual to the ordered phase is represented by a black hole in asymptotically $AdS_5$ that violates the no-hair theorem. While the compactification does not destroy the ordered phase, it does not cure its perturbative instability: we identify the scalar channel QNM of the hairy black hole with Im$[w]>0$. On the contrary, we argue that the disordered thermal phase of the boundary CFT is perturbatively stable in…
We study the asymptotically flat quasi-local black hole/hairy black hole model with nonzero mass of the scalar filed. We disclose effects of the scalar mass on transitions in a grand canonical ensemble with condensation behaviors of a parameter $ψ_{2}$, which is similar to approaches in holographic theories. We find that more negative scalar mass makes the phase transition easier to happen. We also obtain an analytical relation $ψ_{2}\varpropto(T_{c}-T)^{1/2}$ around the critical phase transition points implying a second order phase transition. Besides the parameter $ψ_{2}$, we show that metri…
We argue that the electron star and the AdS Dirac hair solution are two limits of the free charged Fermi gas in AdS. Spectral functions of holographic duals to probe fermions in the background of electron stars have a free parameter that quantifies the number of constituent fermions that make up the charge and energy density characterizing the electron star solution. The strict electron star limit takes this number to be infinite. The Dirac hair solution is the limit where this number is unity. This is evident in the behavior of the distribution of holographically dual Fermi surfaces. As we de…
We construct hairy black hole solutions that merge with the anti-de Sitter (AdS$_4$) Reissner-Nordström black hole at the onset of superradiance. These hairy black holes have, for a given mass and charge, higher entropy than the corresponding AdS$_4$-Reissner-Nordström black hole. Therefore, they are natural candidates for the endpoint of the charged superradiant instability. On the other hand, hairy black holes never dominate the canonical and grand-canonical ensembles. The zero-horizon radius of the hairy black holes is a soliton (i.e. a boson star under a gauge transformation). We construct…
We use numerical methods to exhaustively study a novel family of hairy black hole solutions in AdS$_5$. These solutions can be uplifted to solutions of type IIB supergravity with AdS$_5\times S^5$ asymptotics and are thus expected to play an important role in our understanding of AdS/CFT. We find an intricate phase diagram, with the aforementioned family of hairy black hole solutions branching from the Reissner-Nordström black hole at the onset of the superradiance instability. We analyse black holes with spherical and planar horizon topology and explain how they connect in the phase diagram. …
Planar AdS black holes with axionic charge have finite DC conductivity due to momentum relaxation. We obtain a new family of exact asymptotically AdS$_4$ black branes with scalar hair, carrying magnetic and axion charge, and we study the thermodynamics and dynamic stability of these, as well as of a number of previously known electric and dyonic solutions with axion charge and scalar hair. The scalar hair for all solutions satisfy mixed boundary conditions, which lead to modified holographic Ward identities, conserved charges and free energy, relative to those following from the more standard …
We present a new family of asymptotically AdS four-dimensional black hole solutions with scalar hair of a gravitating system consisting of a scalar field minimally coupled to gravity with a self-interacting potential. For a certain profile of the scalar field we solve the Einstein equations and we determine the scalar potential. Thermodynamically we show that there is a critical temperature below which there is a phase transition of a black hole with hyperbolic horizon to the new hairy black hole configuration.
We study black hole solutions in asymptotically AdS_4 spacetime with scalar hair. Following AdS/CFT dictionary these black holes can be interpreted as thermal states of 2+1 dimensional conformal gauge theory plasma, deformed by a relevant operator. We discover a rich phase structure of the solutions. Surprisingly, we find thermodynamically stable phases with spontaneously broken global symmetries that exist only at high temperatures. These phases are metastable, and join the stable symmetric phase via a mean-field second-order phase transition.
We study the phenomena of superradiance for $F(R)$-Maxwell black holes in an AdS space-time. The AdS boundary plays the role of a mirror and provides a natural confining system that makes the superradiant waves bouncing back and forth between the region near the horizon and the reflective boundary, causing a possible superradiant instability. We obtain numerical solutions for static hairy black holes in this scenario and investigate their instability and explicitly address the stability of such solutions for spherical perturbations under specific conditions for the scalar charge and AdS radius…
Motivated by black hole solutions with matter fields outside their horizon, we study the effect of these matter fields in the motion of massless and massive particles. We consider as background a four-dimensional asymptotically AdS black hole with scalar hair. The geodesics are studied numerically and we discuss about the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit, that is, Schwarzschild AdS black holes. Mainly, we found that there are bounded orbits like planetary orbits in this background. Howeve…
Holographic Renormalization Group (RG) in nine dimensions is considered. The d8 holographic conformal anomaly is found. It should correspond to d8 CFT in AdS_9/CFT_8 correspondence. The comparison of holographic and QFT anomalies in d8 de Sitter space is done. It may give the indication for rigorous AdS_9/CFT_8 correspondence proposal.
Is it possible the no-hair paradigm is violated when a black hole undergoes an instability? Employing massless charged scalar perturbations, we address this question within the context of conformally invariant Einstein-Maxwell theory for some allowed $d$-dimensional ($d=4n+4$ with conformal parameter $n=0,1,2,...$ ) topological small AdS black-holes. We provide numerical analyses that show the non trivial scalar hairy black hole solutions to include planar and spherical horizon topologies in higher dimensions with an even conformal parameter $n$. It is also shown that the solutions presented h…
We compute the holographic stress tensor of colored Lifshitz spacetimes following the proposal by Ross-Saremi for gravity duals of non-relativistic theories. For a well-defined variational principle, we first construct a finite on-shell action for the Einstein-Yang-Mills model in four dimensions with Lifshitz spacetime as a solution. We then solve the linearised equations of motion and identify the modes that preserve the asymptotically Lifshitz condition. Employing these modes, we also show that the stress tensor is finite, obeying the scaling and the diffeomorphism Ward identities, i.e., con…
We construct a general class of exact, regular black hole solutions with toroidal horizon topology in 5-dimensional AdS gravity with a self-interacting scalar field. Due to the non-trivial backreaction of the scalar field, the no-hair theorems can be evaded so that an event horizon can be formed. The scalar field is regular everywhere outside the curvature singularity and it vanishes at the boundary where the potential is finite. We study the properties of these black holes in the context of AdS/CFT duality and comment on the dual operators, which saturate the unitarity bound. We present exact…
Motivated by the study of holographic superconductors, we generalize no-hair theorems for minimally coupled scalar fields charged under an Abelian gauge field, in arbitrary dimensions and with arbitrary horizon topology. We first present a straightforward generalization of no-hair theorems for neutral scalar hair. We then consider the existence of extremal black holes with scalar hair, and in the case of horizons with zero or positive curvature, provide a bound on the mass and charge of the scalar field that are necessary for the scalar hair to develop.
Asymptotic symmetries of AdS$_4$ quantum gravity and gauge theory are derived by coupling the dual CFT$_3$ to Chern-Simons gauge theory and 3D gravity in a "probe" large-level limit. The infinite-dimensional symmetries are shown to arise when one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS$_4$ quantum gravity. An AdS$_4$ analog of Minkowski "super-rotation" asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT$_2$ structure…
We consider Einstein gravity minimally coupled to a scalar field with a given potential in general dimensions. We obtain large classes of static hairy planar black holes which are asymptotic to AdS space-times. In particular, for a special case $μ=(n-2)/2$, we obtain new classes of exact dynamical solutions describing black holes formation. We find there are two classes of collapse solutions. The first class solutions describe the evolution start from AdS space-time with a naked singularity at the origin. The space-time is linearly unstable and evolves into stationary black hole states even un…
We propose the modified version of the canonical energy which was introduced originally by Hollands and Wald. Our construction depends only on the Euler-Lagrange expression of the system and thus is independent of the ambiguity in the Lagrangian. After some comments on our construction, we briefly mention on the relevance of our construction to the boundary information metric in the context of the AdS/CFT correspondence. We also study the stability of three-dimensional hairy extremal black holes by using our construction.
We show that our Universe lives in a topological and non-perturbative vacuum state full of a large amount of hidden quantum hairs, the hairons. We will discuss and elaborate on theoretical evidences that the quantum hairs are related to the gravitational topological winding number in vacuo. Thus, hairons are originated from topological degrees of freedom, holographically stored in the de Sitter area. The hierarchy of the Planck scale over the Cosmological Constant (CC) is understood as an effect of a Topological Memory intrinsically stored in the space-time geometry. Any UV quantum destabiliza…
Holography relates the quasinormal modes frequencies of AdS black holes to the pole structure of the dual field theory propagator. These modes thus provide the timescale for the approach to thermal equilibrium in the CFT. Here, we study how such pole structure and, in particular, the time to equilibrium can get modified in the presence of a black hole hair. More precisely, we consider in AdS a set of relaxed boundary conditions that allow for a low decaying graviton mode near the boundary, which triggers an additional degree of freedom. We solve the scalar field response on such background ana…
The superradiant instability of Kerr-AdS black holes is studied by numerically solving the full 3+1 dimensional Einstein equations. We find that the superradiant instability results in a two stage process with distinct initial and secondary instabilities. At the end of the secondary instability the geometry oscillates at several distinct fundamental frequencies -- a multi-oscillating black hole. The multi-oscillating black hole is remarkably close to a black resonator, albeit with a bit of gravitational hair. During the hairy black resonator epoch, the evolution of the horizon area is consiste…
We perturbatively construct a three-dimensional rotating AdS black hole with a real scalar hair. We choose the mass of a scalar field slightly above the Breitenlohner-Freedman bound and impose a more general boundary condition for the bulk scalar field at AdS infinity. We first show that rotating BTZ black holes are unstable against superradiant modes under our more general boundary condition. Next we construct a rotating hairy black hole perturbatively with respect to a small amplitude $ε$ of the scalar field, up to $O(ε^4)$. The lumps of non-linearly perturbed geometry admit only one Killing…
An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy-momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a discrete parameter which has neither topological nor Noether charge associated with it and therefore represents a hair. A $U(1)$ gauge field interacting with Einstein gravity can also be incl…
We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in $(1+1)$-dimensional dual conformal field theories through the $AdS/CFT$ correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature ($1+1$)-dimensional holographic conformal field the…
We derive the four-point correlators of scalar operators of dimension one in the supergravity limit of the D1D5 CFT holographically dual to string theory on AdS$_3\times S^3\times \mathcal{M}$, with $\mathcal{M}$ either $T^4$ or $K3$. We avoid the use of Witten diagrams but deduce our result from a limit of the heavy-heavy-light-light correlators computed in arXiv:1705.09250, together with several consistency requirements of the OPE in the various channels. This result represents the first holographic correlators of single-trace operators computed in AdS$_3$.
We look into the $AdS$ black holes from two-dimensional gravity perspective. In this work, we extend the previous results of holographic renormalization group flows to dimensions two. By introducing a superpotential, we derive the flow equations in two-dimensional dilaton gravity. We find a quantity which monotonically decreases along flows and give some comments on holographic $c$-theorem. As examples, we show that recently studied supersymmetric $AdS$ black hole solutions generically dimensionally reduce to two-dimensional dilaton gravity, and obtain the flow equations for black hole solutio…
