Moduli stacks of maps for supermanifolds

TL;DR

This paper develops the algebraic geometry of supermanifolds and superstacks to construct and analyze moduli spaces of stable maps into supermanifolds, with applications in twistor-string theory and Gromov-Witten invariants.

Contribution

It introduces a framework for moduli problems of stable maps into supermanifolds, establishing the existence of Deligne-Mumford superstacks in this context.

Findings

Construction of moduli superstacks as Deligne-Mumford superstacks

Analysis of properties of these moduli superstacks

Remarks on applications in physics and Gromov-Witten theory

Abstract

We consider the moduli problem of stable maps from a Riemann surface into a supermanifold; in twistor-string theory, this is the instanton moduli space. By developing the algebraic geometry of supermanifolds to include a treatment of superstacks we prove that such moduli problems, under suitable conditions, give rise to Deligne-Mumford superstacks (where all of these objects have natural definitions in terms of supergeometry). We make some observations about the properties of these moduli superstacks, as well as some remarks about their application in physics and their associated Gromov-Witten theory.