The Super Mumford Form and Sato Grassmannian

TL;DR

This paper extends classical geometric constructions to supersymmetric settings, establishing a flat connection on a line bundle over the moduli space of super Riemann surfaces, linking algebraic geometry with superconformal field theory.

Contribution

It introduces a supersymmetric generalization of the Sato Grassmannian construction and proves the existence of a flat holomorphic connection on a specific line bundle over the super moduli space.

Findings

Existence of a flat holomorphic connection on the line bundle on the super moduli space.

Development of a superconformal Noether normalization lemma.

Extension of classical geometric methods to supersymmetric contexts.

Abstract

We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our main result is the existence of a flat holomorphic connection on the line bundle $\lambda_{3/2}\otimes\lambda_{1/2}^{-5}$ on the moduli space of triples: a super Riemann surface, a Neveu-Schwarz puncture, and a formal coordinate system. We also prove a superconformal Noether normalization lemma for families of super Riemann surfaces.