The Neveu-Schwarz group and Schwarz's extended super Mumford form

TL;DR

This paper extends Schwarz's super Mumford form to the super Sato Grassmannian, demonstrating its invariance under the super Heisenberg-Neveu-Schwarz action, with implications for superstring theory.

Contribution

It computes the Neveu-Schwarz action on super tau functions and proves the invariance of Schwarz's extended Mumford form under this action, introducing related formal groups.

Findings

Schwarz's extended Mumford form is invariant under super Heisenberg-Neveu-Schwarz action

Constructed Neveu-Schwarz, super Witt, and super Heisenberg formal groups

Supports the idea of a universal moduli space within the Grassmannian

Abstract

In 1987, Albert Schwarz suggested a formula which extends the super Mumford form from the moduli space of super Riemann surfaces into the super Sato Grassmannian. His formula is a remarkably simple combination of super tau functions. We compute the Neveu-Schwarz action on super tau functions, and show that Schwarz's extended Mumford form is invariant under the the super Heisenberg-Neveu-Schwarz action, which strengthens Schwarz's proposal that a locus within the Grassmannian can serve as a universal moduli space with applications to superstring theory. Along the way, we construct the Neveu-Schwarz, super Witt, and super Heisenberg formal groups.