Recurrence relations for finite-temperature correlators via AdS2/CFT1

Abstract

This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT 1 dual to AdS 2 black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS 2 black hole and projective unitary representations of the Lie algebra sl(2, ) ⊕ sl(2, ). We derive novel recurrence relations for Euclidean CFT 1 two-point functions, which are exactly solvable and completely determine the frequencyand charge-dependences of two-point functions. Wick-rotating back to Lorentzian signature, we obtain retarded and advanced CFT 1 two-point functions that are consistent with the known results. *

A Simple Derivation of Finite-Temperature CFT Correlators From the BTZ Black Hole

“…We will see that our method correctly reproduces the known results [5,8,9]. 1 This is true for non-extremal black hole with positive mass. The time-and angle-translation generators generate other one-parameter subgroups for the zero-mass limit of black hole (or black hole vacuum), the extremal black hole and the negative mass black hole (or black hole with naked singularity).…”

“…In this short paper we continue our investigation [1] and present a novel Lie-algebraic approach to momentum-space two-point functions of conformal field theory at finite temperature by using the AdS/CFT correspondence. The AdS/CFT correspondence relates strongly-coupled conformal field theory to classical gravity in one-higher spatial dimension.…”

“…where ∆ = 1 + m 2 + 1 is one of the solutions to the quadratic equation ∆(∆ − 2) = m 2 . Notice that near the AdS 3 boundary x = 0 the differential operator in the left-hand side of (2.13) behaves as (1). Hence the general solution has the following asymptotic near-boundary behavior:…”

Intertwining operator in thermal CFTd

“…which implies that x 0 (θ ) and x (θ ) can be put into the following alternative equivalent forms: Let us next move on to construct a coordinate patch for the diamond D where the temporal coordinate is identifies with the SO(1, 1) group parameter. 16 Such coordinate system is turned out to be of the form:…”

“…4 For the case of two-dimensional thermal CFT, the momentum-space two-and three-point Wightman functions for a scalar primary operator were calculated in [12][13][14] and [15]. See also our previous works on holographic approach to momentum-space thermal two-point functions of one-and two-dimensional CFT [16,17]. 5 A nice exposition of Sewell's theorem can be found in chapter 5 of [22].…”