JT supergravity and Brezin-Gross-Witten tau-function

TL;DR

This paper explores the connection between JT supergravity and the Brezin-Gross-Witten tau-function, revealing a matrix model representation, genus expansion, and non-perturbative aspects, with implications for understanding super Riemann surfaces.

Contribution

It establishes that JT supergravity's matrix model is a special case of the BGW model with specific couplings, extending the relation between gravity and integrable hierarchies.

Findings

Matrix model of JT supergravity is a special case of BGW model.

Computed genus expansion and low-temperature behavior of one-point function.

Proposed a non-perturbative completion in the Bessel case.

Abstract

We study thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy.