Exact low-temperature Green's functions in AdS/CFT: From Heun to confluent Heun

TL;DR

This paper derives exact low-temperature Green's functions in AdS/CFT using Heun functions, providing analytic insights into correlation functions, quasinormal modes, and critical phenomena in holographic superconductors.

Contribution

It introduces a novel analytic method employing Heun and confluent Heun functions to compute Green's functions at low temperature in holographic models.

Findings

Exact expressions for charged scalar correlators at low T and finite density.

Analytic determination of the critical temperature for holographic superconductors.

Precise agreement between analytic results and numerical simulations.

Abstract

We obtain exact expressions for correlation functions of charged scalar operators at finite density and low temperature in CFT$_4$ dual to the RN-AdS$_5$ black brane. We use recent developments in the Heun connection problem in black hole perturbation theory arising from Liouville CFT and the AGT correspondence. The connection problem is solved perturbatively in an instanton counting parameter, which is controlled in a double-scaling limit where $\omega, T \to 0$ holding $\omega/T$ fixed. This provides analytic control over the emergence of the zero temperature branch cut as a confluent limit of the Heun equation. From the Green's function we extract analytic results for the critical temperature of the holographic superconductor, as well as dispersion relations for both gapped and gapless low temperature quasinormal modes. We demonstrate precise agreement with numerics.