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)