The ETH matrix model for DSSYK: non-perturbative corrections and intersection theory
At leading order in the genus expansion the ETH matrix model for DSSYK reproduces its correlators by construction, while its higher-genus corrections are conjectured to capture higher-topology contributions in the dual sine-dilaton gravity -- a correspondence established so far only for the disk and the wormhole. At fixed genus the correlators are built from discrete volumes $N_{g,n}$, polynomial in $q$-deformed zeta values $ζ_q(2k)$ with $q=e^{-λ}$, $λ$ being the DSSYK coupling. These lie in the ring of quasimodular forms generated by the Eisenstein series $E_2,E_4,E_6$, whose $S$-duality yields an exact closed form for the leading non-perturbative correction as $λ\to0$, controlled by $\widetilde q=e^{-4π^2/λ}$. Known at disk level, this scale is shown here to govern the fixed-genus, higher-boundary amplitudes as well. We show that the term linear in $\widetilde q$, at leading order in $λ$, is captured entirely by the $q$-deformed Weil--Petersson volumes, and reduces to a finite sum of intersection numbers of $κ$-classes on the moduli space $\overline{\mathcal M}_{g,n}$ of stable curves, computable without repeating the topological recursion that produced the $N_{g,n}$. We tabulate it for every $(g,n)$ whose $q$-deformed volume is known in closed form, and extend it to $(3,1),(3,2),(4,1)$, where none is available. The construction is not restricted to leading order: we work out $O(\widetilde q^2)$ for the same cases.