Recurrence Relations for Finite-Temperature Correlators via AdS$_{2}$/CFT$_{1}$

TL;DR

This paper introduces a Lie-algebraic method to derive recurrence relations for frequency-space two-point functions in CFT$_{1}$ dual to AdS$_{2}$ black holes, enabling exact solutions for correlators with charge and frequency dependence.

Contribution

It presents a novel algebraic approach using Lie algebra representations to compute and solve recurrence relations for CFT$_{1}$ correlators in AdS$_{2}$/CFT$_{1}$ correspondence.

Findings

Derived exact recurrence relations for two-point functions.

Obtained Lorentzian correlators consistent with known results.

Provided a new algebraic framework for finite-temperature correlators.

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 $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. We derive novel recurrence relations for Euclidean CFT$_{1}$ two-point functions, which are exactly solvable and completely determine the frequency- and 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.