RN/CFT Correspondence From Thermodynamics

Recent studies suggest that in the Kerr/CFT correspondence, much universal information of the dual CFT, including the central charges and the temperatures, is fully encoded in the thermodynamics of the outer and inner horizons of the Kerr(-Newman) black holes. In this paper, we study holographic descriptions of Reissner-Nordstr\"om (RN) black holes in arbitrary dimensions by using the thermodynamics method.We refine the thermodynamics method proposed in arXiv:1206.2015 by imposing the "quantization" condition so that we can fix the ambiguity in determining the central charges of the dual CFT of RN black holes. Using the refined thermodynamics method, we find the holographic CFT duals for the RN black holes, and confirm these pictures by using conventional analysis of asymptotic symmetry group and the hidden conformal symmetry in the low-frequency scattering. In particular, we revisit the four-dimensional dyonic RN black hole and find a novel magnetic picture, besides the known electric CFT dual picture. We show how to generate a class of dual dyonic pictures by $SL(2,Z)$ transformations.Comment: 32 pages, references added, published versio

On short interval expansion of R\'enyi entropy

R\'enyi entanglement entropy provides a new window to study the AdS/CFT correspondence. In this paper we consider the short interval expansion of R\'enyi entanglement entropy in two-dimensional conformal field theory. This amounts to do the operator product expansion of the twist operators. We focus on the vacuum Verma module and consider the quasiprimary operators constructed from the stress tensors. After obtaining the expansion coefficients of the twist operators to level 6 in vacuum Verma module, we compute the leading contributions to the R\'enyi entropy, to order 6 in the short interval expansion. In the case of one short interval on cylinder, we reproduce the first several leading contributions to the R\'enyi entropy. In the case of two short disjoint intervals with a small cross ratio $x$, we obtain not only the classical and 1-loop quantum contributions to the R\'enyi entropy to order $x^6$, both of which are in perfect match with the ones found in gravity, but also the leading $1/c$ contributions, which corresponds to 2-loop corrections in the bulk.Comment: V1, 23 pages, 5 figures; V2, published version, typos corrected, references adde

Thermodynamics in Black-hole/CFT Correspondence

The area law of Bekenstein-Hawking entropy of the black hole suggests that the black hole should have a lower-dimensional holographic description. It has been found recently that a large class of rotating and charged black holes could be holographically described a two-dimensional (2D) conformal field theory (CFT). We show that the universal information of the dual CFT, including the central charges and the temperatures, is fully encoded in the thermodynamics laws of both outer and inner horizons. These laws, characterizing how the black hole responds under the perturbation, allows us to read different dual pictures with respect to different kinds of perturbations. The remarkable effectiveness of this thermodynamics method suggest that the inner horizon could play a key role in the study of holographic description of the black hole.Comment: 8 pages. Essay awarded honourable mention in the Gravity Research Foundation 2013 Awards for Essays on Gravitatio

Three-Scale Singular Limits of Evolutionary PDEs

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero

Convergence Rate Estimates for the Low Mach and Alfv\'en Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfv\'en numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors' previous paper, is much more complicated than for the classical two-scale singular limits

Note on Thermodynamics Method of Black Hole/CFT Correspondence

In the paper we further refine the thermodynamics method of black hole/CFT correspondence. We show that one can derive the central charges of different holographic pictures directly from the entropy product $S_+S_-$ if it is mass-independent, for a black hole in the Einstein gravity or the gravity without diffeomorphism anomaly. For a general black hole in the Einstein gravity that admits holographic descriptions, we show that the thermodynamics method and asymptotic symmetry group (ASG) analysis can always give consistent results in the extreme limit. Furthermore, we discuss the relation between black hole thermodynamics and the hidden conformal symmetry. We show that the condition $T_+A_+=T_-A_-$, with $A_\pm$ being the outer and inner horizon areas, is the necessary, but not sufficient, condition for a black hole to have the hidden conformal symmetry. In particular, for the Einstein(-Maxwell) gravity $T_+A_+=T_-A_-$ is just the condition $T_+S_+=T_-S_-$, with $S_\pm$ being the outer and inner horizon entropies, which is the condition for the entropy product $S_+S_-$ being mass-dependent. When there exists the hidden conformal symmetry in the low-frequency scattering off the generic non-extremal black hole, it always leads to the same temperatures of dual CFT as the ones got from the thermodynamics method.Comment: 31 pages, references added, published versio

Corrections to holographic entanglement plateau

We investigate the robustness of the Araki-Lieb inequality in a two-dimensional (2D) conformal field theory (CFT) on torus. The inequality requires that $\Delta S=S(L)-|S(L-\ell)-S(\ell)|$ is nonnegative, where $S(L)$ is the thermal entropy and $S(L-\ell)$, $S(\ell)$ are the entanglement entropies. Holographically there is an entanglement plateau in the BTZ black hole background, which means that there exists a critical length such that when $\ell \leq \ell_c$ the inequality saturates $\Delta S=0$. In thermal AdS background, the holographic entanglement entropy leads to $\Delta S=0$ for arbitrary $\ell$. We compute the next-to-leading order contributions to $\Delta S$ in the large central charge CFT at both high and low temperatures. In both cases we show that $\Delta S$ is strictly positive except for $\ell = 0$ or $\ell = L$. This turns out to be true for any 2D CFT. In calculating the single interval entanglement entropy in a thermal state, we develop new techniques to simplify the computation. At a high temperature, we ignore the finite size correction such that the problem is related to the entanglement entropy of double intervals on a complex plane. As a result, we show that the leading contribution from a primary module takes a universal form. At a low temperature, we show that the leading thermal correction to the entanglement entropy from a primary module does not take a universal form, depending on the details of the theory.Comment: 32 pages, 8 figures; V2, typos corrected, published versio

Holographic R\'enyi entropy for CFT with WW symmetry

In this paper we investigate the holographic R\'enyi entropy of two disjoint intervals on complex plane with small cross ratio $x$ for conformal field theory with $W$ symmetry in the ground state, which could be dual to a higher spin AdS$_3$ gravity. We focus on the cases of $W_3$ and $W_4$ symmetries. In order to see the nontrivial contributions from the $W$ fields, we calculate the R\'enyi entropy in the expansion of $x$ to order $x^8$ in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from $W$ fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the R\'enyi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of R\'enyi entanglement entropy, including the quantum correction, in both the AdS$_3$ gravity and the higher spin AdS$_3$ gravity.Comment: 32 pages, published versio