A discrete analogue of $\mathcal{N}=2$ super Weil-Petersson volumes

We introduce a discrete spectral curve from $\mathcal{N}=2$ double-scaled SYK whose topological recursion defines quasi-polynomial volumes in integer boundary lengths. The curve has two edge theories: a local $\mathcal{N}=2$ theory at the lower edge and a bosonic theory at the upper edge. Focusing on the local $\mathcal{N}=2$ theory, we compute its finite-$q$ volumes and independently check the first few using the string equation. In the $q\to 1$ continuum limit, these recover the super Weil-Petersson volumes of Turiaci and Witten. Finally, we give an all-genus intersection-theoretic interpretation of these local volumes on the moduli space of stable marked curves, in terms of a deformed spin class with a gap-dependent $\kappa$-class dressing.