On the Super Mumford Form in the Presence of Ramond and Neveu-Schwarz Punctures

TL;DR

This paper extends the understanding of the super Mumford form on moduli spaces of super Riemann surfaces with Ramond and Neveu-Schwarz punctures, providing explicit expressions useful for superstring scattering amplitudes.

Contribution

It generalizes Voronov's result to include Ramond and Neveu-Schwarz punctures and large numbers of punctures, offering new formulas for the super Mumford form in these cases.

Findings

Derived explicit formulas for the super Mumford form with punctures

Connected the super Mumford form to superstring scattering amplitudes

Analyzed the form over moduli space components with odd spin structures

Abstract

We generalize the result of Voronov (1988) to give an expression for the super Mumford form $\mu$ on the moduli spaces of super Riemann surfaces with Ramond and Neveu-Schwarz punctures in the limit where the number of punctures is large compared to the genus. In the case of Neveu-Schwarz punctures we consider the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form $\mu$ can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express $\mu$ in terms of local bases of $H^0(\Sigma, \omega^j)$ for $\omega$ the Berezinian line bundle of a family of super Riemann surfaces.