Support of dS/CFT correspondence from perturbations of three dimensional spacetime

We discuss the relation between bulk de Sitter three-dimensional spacetime and the corresponding conformal field theory at the boundary, in the framework of the exact quasinormal mode spectrum. We show that the quasinormal mode spectrum corresponds exactly to the spectrum of thermal excitations of Conformal Field Theory at the past boundary I, together with the spectrum of a Conformal Field Theory at the future boundary I. PACS number(s): 04.06.Kz, 04.70.Dy e-mail:[email protected] e-mail:[email protected] 1 The study of quasinormal modes has been an intriguing research activity for the last 30 years [1], leading to important contributions to the understanding of black holes [2-11]. Recently the case of Anti-de Sitter (AdS) space has been particularly focused due to its proposed relation to the conformal field theory (CFT) [12]. Qualitative correspondences between quasinormal modes in AdS spaces and the decay of perturbations in the due CFT have been obtained [5-10]. More encouraging, in the three-dimensional (3D) BTZ black hole model [13], a precise quantitative agreement between the quasinormal frequencies and the location of poles of the retarded correlation function of the corresponding perturbations in the dual CFT has been presented [11]. This gives a further evidence of the correspondence between gravity in AdS spacetime and quantum field theory at the boundary. There has been an increasing interest in gravity on de Sitter (dS) spacetimes in view of recent observational support for a positive cosmological constant. A holographic duality relating quantum gravity on D-dimensional dS space to CFT on (D-1)-sphere has been proposed [14]. It is of interest to extend the study in [11] to dS space by displaying the exact solution of the quasinormal mode problem in the dS bulk space and exploring its relation to the CFT theory at the boundary. This could serve as a quantitative test of the dS/CFT correspondence. This is the motivation of the present paper. We will concentrate on nontrivial 3D dS spacetimes. The mathematical simplicity in these models renders all computations analytical. As shown in [15] [16] 3D gravity is directly related to the twodimensional (2D) WZW theory at the border [17], where the whole theoretical apparatus of CFT can be fully used to obtain exact results. In such a case, Brown and Henneaux [18] already obtained one relation between 3D AdS gravity and conformal algebra with a (classical version of the) central charge, recently developed into the quantum theory by Maldacena [12]. The metric of the 3D rotating dS spacetime is given by[19] ds = −(M − r 2 l + J 4r )dt + (M − r 2 l + J 4r )dr + r(dφ− J 2r dt), (1) where J is associated to the angular momentum. The horizon of such spacetime can be obtained from M − r 2 l + J 4r = 0. (2)