Modular structures in the DSSYK partition function

Abstract

We study the low-temperature expansion of the disk partition function $Z({\hat{I}}^2)$ of the double-scaled SYK model (DSSYK) at fixed coupling $\hat{I}»=2p^2/N$, 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 $Z({\hat{I}}^2)$, 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 $\hat{I}»$) low-temperature expansion of $Z({\hat{I}}^2)$, splitting it into a perturbative tower and a non-perturbative sector controlled by $\tilde{q}=e^{-4\ddot{I}{€}^2/\hat{I}»}$. At each order in $\tilde{q}$, we determine the non-perturbative correction in closed form up to second order in $\hat{I}»$; the resulting series resums into a compact expression in the same Eisenstein series, extending previous semiclassical results beyond their strict ${\hat{I}}^2\to \infty$ 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 $Z({\hat{I}}^2)$ is exactly supported, to all orders in $\hat{I}»$, 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.

23 pages. v2: more references

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