Super Weil-Petersson measures on the moduli space of curves

Abstract

The super Weil-Petersson metric defined over the moduli space of smooth super curves produces a natural measure over the moduli space of smooth curves. The construction of the measure uses the extra data of a spin structure on each smooth curve. When we allow marked points, the construction produces a collection of measures indexed by the behaviour of the spin structure at marked points -- Neveu-Schwarz or Ramond. In this paper we define these measures, and prove that they are finite. Each total measure gives the super volume of the moduli space of super curves with marked points. The Neveu-Schwarz volumes are polynomials that satisfy a recursion relation discovered by Stanford and Witten, analogous to Mirzakhani's recursion relations between Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. We prove here that the Ramond boundary behaviour produces deformations of the Neveu-Schwarz volume polynomials, satisfying a variant of the Stanford-Witten recursion relations.