Moduli of stable supermaps

TL;DR

This paper studies the moduli space of stable supermaps from SUSY curves to a fixed target superscheme, establishing its structure as a Deligne-Mumford superstack and analyzing its properties and dimensions.

Contribution

It introduces the moduli superstack of stable supermaps, characterizes its bosonic reduction, and computes its virtual dimension using super Grothendieck-Riemann-Roch.

Findings

The moduli superstack is a Deligne-Mumford superstack with separated diagonal.

The bosonic reduction maps onto the moduli stack of stable maps from spin curves.

The virtual dimension matches known formulas for bosonic targets.

Abstract

We review the notion of stable supermap from SUSY curves to a fixed target superscheme, and prove that when the target is (super)projective, stable supermaps are parameterized by a Deligne-Mumford superstack with superschematic and separated diagonal. We characterize the bosonic reduction of this moduli superstack and see that it has a surjective morphism onto the moduli stack of stable maps from spin curves to the bosonic reduction of the target, whose fibers are linear schemes; for this reason, the moduli superstack of stable supermaps is not proper unless such linear schemes reduce to a point. Using Manin-Penkov-Voronov's super Grothendieck-Riemann-Roch theorem we also make a formal computation of the virtual dimension of the moduli superstack, which agrees with the characterization of the bosonic reduction just mentioned and with the dimension formula for the case of bosonic target existing in the literature.