Holographic thermal correlators from recursions

TL;DR

This paper develops a recurrence relation approach to compute holographic thermal correlators in AdS black hole backgrounds, revealing pole dynamics and connecting to conformal blocks in CFT.

Contribution

It introduces a recurrence relation method for holographic correlators using the Heun equation, linking it to Virasoro conformal blocks and analyzing different black hole solutions.

Findings

Derived recurrence relations for holographic correlators.

Analyzed pole movement and eigenvalue repulsions.

Connected the recurrence approach to conformal block techniques.

Abstract

We express holographic thermal correlators using a recurrence relation of $\{a_n\}$ at $n\to\infty$, building on recent advances in the connection formula for the Heun equation. We consider two gravitational solutions that correspond to distinct states in different subsectors of $\mathcal{N}=4$ super-Yang-Mills theory at finite temperature and density. The first is the Reissner-Nordstr\"{o}m-AdS$_5$ black hole, which has finite entropy at zero temperature, and the second is a charged dilatonic black hole in AdS$_5$, which has zero entropy at zero temperature. In both cases, we perturb the system with a charged scalar field and express the perturbation equation in terms of the Heun equation. We find interesting moving patterns of the poles of the correlators including eigenvalue repulsions. We discuss the relation between the recurrence relation and the Virasoro conformal block as two equivalent approaches to write the connection formula for the Heun equation.