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Moduli of stable supermaps

Ugo Bruzzo, Daniel Hernández Ruipérez

TL;DR

This work develops the moduli theory for stable supermaps from SUSY curves into a fixed projective target. It proves that the moduli form a Deligne–Mumford superstack with superschematic diagonal, and analyzes its bosonic reduction as a fibration over the spin-map stack with linear fibers, explaining non-properness in general. A super Grothendieck–Riemann–Roch framework is established to formally compute the virtual dimension, aligning with known bosonic results in the appropriate limit. The results provide a rigorous foundation for deformation and obstruction theories of stable supermaps and set the stage for a subsequent construction of a perfect obstruction theory. The study integrates super K-theory, characteristic classes, and GRR to connect the supergeometry with classical moduli theory of stable maps and spin maps.

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.

Moduli of stable supermaps

TL;DR

This work develops the moduli theory for stable supermaps from SUSY curves into a fixed projective target. It proves that the moduli form a Deligne–Mumford superstack with superschematic diagonal, and analyzes its bosonic reduction as a fibration over the spin-map stack with linear fibers, explaining non-properness in general. A super Grothendieck–Riemann–Roch framework is established to formally compute the virtual dimension, aligning with known bosonic results in the appropriate limit. The results provide a rigorous foundation for deformation and obstruction theories of stable supermaps and set the stage for a subsequent construction of a perfect obstruction theory. The study integrates super K-theory, characteristic classes, and GRR to connect the supergeometry with classical moduli theory of stable maps and spin maps.

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.

Paper Structure

This paper contains 17 sections, 22 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. The superstack of stable supermaps
  3. Stable and prestable SUSY curves
  4. Supercycles
  5. Stable supermaps
  6. The superstack of stable supermaps is algebraic
  7. The superstack of stable supermaps is Deligne-Mumford
  8. Bosonic reduction
  9. The stack of spin maps
  10. The bosonic reduction of the superstack ${\mathfrak S}{\mathfrak M}({\mathscr Y},\beta)$
  11. Characteristic classes and super Grothendieck-Riemann-Roch
  12. Super K-theory
  13. Characteristic classes
  14. Splitting principle
  15. The Todd character
  16. ...and 2 more sections

Key Result

Proposition 2.4

FKPpubl The category fibered in groupoids $p\colon {\mathfrak M}_{g,{\mathfrak n}_{NS},{\mathfrak n}_{RR}}\to {\mathfrak S}$ of stable SUSY curves with ${\mathfrak n}_{NS}$ NS punctures and ${\mathfrak n}_{RR}$ RR punctures is a proper and smooth Deligne-Mumford superstack. ∎

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 48 more