Enumerative geometry via the moduli space of super Riemann surfaces

Abstract

In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces ${\overline{\mathcal{M}}}_{g,n}$. 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 ${\Theta }_{g,n}$ with tautological classes on ${\overline{\mathcal{M}}}_{g,n}$ 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 ${\overline{\mathcal{M}}}_{g,n}$ using volumes of moduli spaces of hyperbolic surfaces.

Proved properties of a natural Euler form on a bundle over the moduli space used in the definition of the super volume; 74 pages

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