On the perturbative expansion of exact bi-local correlators in JT gravity

TL;DR

This paper analyzes the perturbative series of bi-local correlators in JT gravity, reproducing the Schwarzian expansion, deriving formulas with Apostol-Bernoulli polynomials, and exploring asymptotic and exact integral representations.

Contribution

It provides a detailed perturbative analysis of bi-local correlators in JT gravity, including exact formulas, asymptotic behavior, and connections to special functions.

Findings

Reproduces Schwarzian semiclassical expansion beyond leading order.

Derives formulas involving Apostol-Bernoulli polynomials for arbitrary temperature.

Expresses exact results as combinations of Mordell integrals.

Abstract

We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight $\lambda$ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for $\lambda \in \mathbb{N}/2$) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.