Enumerative geometry via the moduli space of super Riemann surfaces
Paul Norbury
TL;DR
The paper develops an algebraic-geometric framework to study volumes of moduli spaces of super Riemann surfaces by expressing them as intersection numbers on the moduli space of stable curves via the Theta-classes $\Theta_{g,n}$. It proves a recursion for the super volumes that matches the Stanford–Witten recursions and shows this recursion is equivalent to the statement that the generating function $Z^{\Theta}$ is a KdV tau-function, specifically tied to the Brézin–Gross–Witten system. A key technical achievement is extending the bundle $E_{g,n}$ to the compactification and constructing its Euler form so that volumes become intersection numbers, enabling a topological-recursion formulation with a spectral curve $x=\frac{1}{2}z^2$, $y=\frac{\cos(2\pi z)}{z}$. The work parallels Mirzakhani’s approach for the bosonic case, providing a rigorous algebro-geometric route to super-volume recursions and Virasoro constraints, and suggests directions toward a fully supergeometric proof and Ramond puncture generalizations.
Abstract
In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces $\overline{\cal M}_{g,n}$. This allows us to prove via algebraic geometry a recursion between the volumes of moduli spaces of super hyperbolic surfaces previously proven via super geometry techniques by Stanford and Witten. The recursion between the volumes of moduli spaces of super hyperbolic surfaces is proven to be equivalent to the fact that a generating function for the intersection numbers of a natural collection of cohomology classes $Θ_{g,n}$ with tautological classes on $\overline{\cal M}_{g,n}$ is a KdV tau function. This is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem regarding a generating function for the intersection numbers of tautological classes on $\overline{\cal M}_{g,n}$ using volumes of moduli spaces of hyperbolic surfaces.