Discrete analogue of the Weil-Petersson volume in double scaled SYK
Kazumi Okuyama
TL;DR
This work formulates DSSYK as a hermitian one-matrix model and shows that the connected multi-boundary correlators of partition functions decompose into a discrete volume $N_{g,n}(b_1,\ldots,b_n)$ and trumpet factors, with $Z_{\text{trumpet}}(\beta,b)=I_b(\beta a)$. Using the Eynard–Orantin topological recursion on the DSSYK spectral curve, the authors compute $N_{g,n}$ for small $(g,n)$ and demonstrate that, in the semi-classical limit $\lambda\to0$ (with $L_i=\lambda b_i$), $N_{g,n}$ reduces to the Weil-Petersson volumes $V^{\mathrm{WP}}_{g,n}(L_1,\ldots,L_n)$. The trumpet–volume decomposition is shown to hold generically for one-cut matrix models, and the Gaussian case provides explicit checks that reproduce known counting results and Kontsevich-volume limits. The paper also outlines future directions, including non-perturbative corrections, two-matrix generalizations, and potential lattice interpretations of the discrete moduli data in DSSYK.
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.