Discrete analogue of the Weil-Petersson volume in double scaled SYK

TL;DR

This paper explores the connection between double scaled SYK model correlators and a discrete analogue of Weil-Petersson volumes, providing explicit computations and confirming their classical limit correspondence.

Contribution

It introduces a discrete analogue of Weil-Petersson volumes in the context of double scaled SYK and computes these volumes explicitly for low genus orders.

Findings

Discrete Weil-Petersson volume reduces to classical volume in semi-classical limit

Explicit calculations of the discrete volume for initial genus orders

Decomposition of correlators into trumpet and discrete volume components

Abstract

We show that the connected correlators of partition functions in double scaled SYK model can be decomposed into ``trumpet'' and the discrete analogue of the Weil-Petersson volume, which was defined by Norbury and Scott. We explicitly compute this discrete volume for the first few orders in the genus expansion and confirm that the discrete volume reduces to the Weil-Petersson volume in a certain semi-classical limit.