Thermofield duality for higher spin Rindler Gravity

Journal of High Energy Physics, Feb 2016

We study the Thermo-field realization of the duality between the Rindler-AdS higher spin theory and O(N) vector theory. The CFT represents a decoupled pair of free O(N) vector field theories. It is shown how this decoupled domain CFT is capable of generating the connected Rindler-AdS background with the full set of Higher Spin fields.

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Thermofield duality for higher spin Rindler Gravity

HJE eld duality for higher spin Rindler Gravity Antal Jevicki 0 1 Kenta Suzuki 0 1 E-mail: Antal 0 1 Kenta 0 1 0 dence , 1/N Expansion 1 Department of Physics, Brown University We study the Thermo- eld realization of the duality between the Rindler-AdS higher spin theory and O(N ) vector theory. The CFT represents a decoupled pair of free O(N ) vector eld theories. It is shown how this decoupled domain CFT is capable of generating the connected Rindler-AdS background with the full set of Higher Spin elds. Higher Spin Gravity; Gauge-gravity correspondence; AdS-CFT Correspon- - Thermo 1 Introduction 2 3 4 4.1 4.2 4.3 O(N ) vector model on hyperbolic space Transformation from Poincare to Rindler-AdS coordinates SO(2; 3) generators Bi-local map for the Rindler-AdS Hamiltonian of Rindler vector model Spin [24{27] dualities [28, 29] in AdS4 which have been intensively studied during the last few years [30{43]. They involve O(N ) vector models at their critical points of which the free theory is a particularly simple and tractable example. For pure AdS, it allows for a complete reconstruction of the bulk theory with higher spin elds and to all orders in 1=N . A direct map between CFT observables and bulk AdS elds was shown to be given in terms of a bi-local composite eld of the O(N ) CFT [31, 38]. A study of this bi-local construction { 1 { at nite temperature in the Thermo- eld formulation was given recently in [43] with results showing the characteristics of bulk black hole spacetimes. In this paper, we sharpen this correspondence by giving a construction of Rindler-AdS Higher Spin Gravity. In this case one has a pair of CFT's on hyperbolic spaces that are to be quantized in the Thermo- eld scheme. In addition to presenting a more complete example of \domain duality" [23] and the ability of reconstructing a connected spacetime, we clarify the stability of this system. The instability of the quotient hyperbolic space case was studied in [44]. The content of this paper is as follows: in section 2, we review the appearance of asymptotic Rindler-AdS spacetime as the massless limit of particular black holes (Topological Black Holes). We also give some relevant properties of this spacetime such as the presence of \evanescent" modes [45] whose CFT reconstruction will be of particular interest. In section 3, we discuss the quantization of the O(N ) vector model on hyperbolic space and its hamiltonian Thermo- eld version. In section 4, we give details of our bi-local map to Rindler-AdS bulk. Demonstration of \evanescent" modes given by this construction is one application. Sect 5 is reserved for Conclusions regarding the more general questions involving reconstruction of spacetime behind the horizon of BH [46{48]. 2 Massless limit of AdS BH and Rindler-AdS In this section, we review the static black hole solutions in asymptotically AdS spacetime and their massless limit, mainly following the discussion of [8]. Asymptotically AdS space admits three kinds of the static back hole solutions. These three solutions are characterized by the curvature of the horizons (transverse spacial coordinates). For AdSd+1, the metric is explicitly given by = { 2 { where ds2 = fk(r) dt2 + dr2 fk(r) + r 2 L2 d 2k;d 1 ; fk(r) = r 2 L2 rd 2 + k : Here L denotes the AdS radius and is a deformation parameter, which is proportional to the black hole mass. d 2k;d 1 represents the metric of the (d given by the unit metric of (Sd 1, R d 1, Hd 1) for k = (+1; 0; 1) case respectively. 1)-dimensional horizon The positive horizon curvature solution (k = +1) is called \AdS-Schwarzschild" black hole, the zero curvature solution (k = 0) is \AdS-Planar" black hole, and the negative curvature (k = 1) is \AdS-Hyperbolic" black hole. The horizon location r+ is determined by the larger solution of fk(r+) = 0. In the massless limit ! 0, each black hole solution is reduced to the global, Poincare, and Rindler coordinates of AdS spacetime, respectively. The temperature of these black holes is determined as black hole solution is dual to CFT's on R Hereafter, we set L = 1. The boundary metric is given by taking r ! 1. Therefore, one expects that each AdS Sd 1, R1;d 1, and R Hd 1, respectively [8]. (2.1) (2.2) (2.3) In this paper, we are mainly interested in the four-dimensional Rindler-AdS coordinates (this is sometimes called massless topological black hole), which is given as the k = 1 and = 0 case of (2.1). Note that for this case, the Rindler horizon location is given by r+ = 1 and = 2 , which agrees with the Unruh temperature. For our purpose, it's more convenient to use a Poincare-like radial coordinate ,1 de ned by = r 1, rather than the global-like coordinate r. Then, our Rindler-AdS metric is de ned by ds2 = 1 2 (1 modes exist in the AdS-Schwarzschild black hole background and in its in nite volume limit, AdS-Planar black hole background. The existence of the evanescent modes in RindlerAdS background was suggested in the discuss (...truncated)


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Antal Jevicki, Kenta Suzuki. Thermofield duality for higher spin Rindler Gravity, Journal of High Energy Physics, 2016, pp. 94, Volume 2016, Issue 2, DOI: 10.1007/JHEP02(2016)094