Modular structures in the DSSYK partition function

Abstract

We study the low-temperature expansion of the disk partition function $Z(\beta)$ of the double-scaled SYK model (DSSYK) at fixed coupling $\lambda=2p^{2}/N$, where $N$ is the number of Majorana fermions and $p$ is the number of fermions in each interaction term, both taken to infinity. We show that the exact Bessel-function representation of $Z(\beta)$, expanded at large argument (corresponding to low temperature), can be organized in terms of the classical ring of quasi-modular Eisenstein series $E_{2},E_{4},E_{6}$ and their differential identities. Exploiting the modular $S$-duality properties of this ring, we derive the semiclassical (small $\lambda$) low-temperature expansion of $Z(\beta)$, splitting it into a perturbative tower and a non-perturbative sector controlled by $\widetilde q=e^{-4\pi^{2}/\lambda}$. At each order in $\widetilde q$, we determine the non-perturbative correction in closed form up to second order in $\lambda$; the resulting series resums into a compact expression in the same Eisenstein series, extending previous semiclassical results beyond their strict $\beta\to\infty$ limit. We further show that this entire structure follows from a single, exact differential equation coupling a modular derivative to derivatives with respect to temperature. Finally, we prove that the non-perturbative sector of $Z(\beta)$ is exactly supported, to all orders in $\lambda$, on the same exponents as the on-shell actions of known bilocal-Liouville saddles of the DSSYK Schwarzian limit, pointing to a well-defined bulk origin for these non-perturbative corrections.