Non-perturbative effects in JT gravity from KdV equations

TL;DR

This work constructs a systematic one-parameter transseries solution to the KdV equation governing the perturbative genus expansion of JT gravity, and shows that non-perturbative sectors are captured by instanton actions from a critical-point effective potential. Specializing to JT gravity, the authors derive explicit one- and multi-instanton contributions to Weil-Petersson volumes and the JT free energy, and verify a precise match with non-perturbative results obtained from topological recursion in random matrix models. They also connect the Baker–Akhiezer function and Christoffel–Darboux kernel to the topological-recursion wave function and the quantum curve, providing a unified framework for perturbative and non-perturbative JT gravity. Overall, the paper demonstrates non-perturbative completions of JT gravity via transseries and highlights deep links between topological gravity, matrix models, and TR/quantum curves.

Abstract

It is well-known that the partition function of the Jackiw-Teitelboim (JT) gravity is obtained by an integral transformation of volumes of moduli spaces for Riemann surfaces, also known as the Weil-Petersson volumes. This fact enables us to compute the perturbative genus expansion of the partition function by solving a KdV-type non-linear partial differential equation. In this work, we find that this KdV equation also admits transseries solutions. We give a systematic algorithm to explicitly construct a one-parameter transseries solution to the KdV equation. Our approach is based on general two-dimensional topological gravity, and the results for the JT gravity are easily obtained as a special case. The results in the leading non-perturbative sector perfectly agree with another independent calculation from topological recursions in random matrices.