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

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 Ramond case we take the number of punctures to be large compared to the genus. We consider for the case of Neveu-Schwarz punctures 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 $H^0(X, \omega^j)$ for $\omega$ the Berezinian line bundle of a family of super Riemann surfaces.