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The Super Mumford Form and Sato Grassmannian

Katherine A. Maxwell

TL;DR

The paper constructs a supersymmetric analogue of the Kontsevich–Arbarello–De Concini–Kac–Procesi program by relating the moduli of super Riemann surfaces with the semi‑infinite (super) Grassmannian via the super Krichever map. It establishes a flat holomorphic connection on the Mumford line bundle $\lambda_{3/2}\otimes\lambda_{1/2}^{-5}$ over the moduli of NS‑punctured SRSs with formal coordinates, using a relative superconformal Noether normalization and the perfectness of a Lie superalgebroid of superconformal fields. The approach unifies central extensions of the super Witt/Neveu–Schwarz algebras with Berezinian line bundles on Gr$(H_j)$, providing a mechanism to transport determinant data from the Grassmannian to the moduli side and to derive differential equations for the Polyakov‑type measures in the supersymmetric setting. The results illuminate how integrating over supermoduli can be approached via the super Grassmannian, potentially informing superstring measure constructions and higher‑genus formulations. The construction hinges on coherent interactions among Lie superalgebroids, the super Krichever map, and the Bézoutian/Berezinian framework, yielding a robust flat connection that mirrors the classical Mumford setup in a supersymmetric context.

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 $λ_{3/2}\otimesλ_{1/2}^{-5}$ 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.

The Super Mumford Form and Sato Grassmannian

TL;DR

The paper constructs a supersymmetric analogue of the Kontsevich–Arbarello–De Concini–Kac–Procesi program by relating the moduli of super Riemann surfaces with the semi‑infinite (super) Grassmannian via the super Krichever map. It establishes a flat holomorphic connection on the Mumford line bundle over the moduli of NS‑punctured SRSs with formal coordinates, using a relative superconformal Noether normalization and the perfectness of a Lie superalgebroid of superconformal fields. The approach unifies central extensions of the super Witt/Neveu–Schwarz algebras with Berezinian line bundles on Gr, providing a mechanism to transport determinant data from the Grassmannian to the moduli side and to derive differential equations for the Polyakov‑type measures in the supersymmetric setting. The results illuminate how integrating over supermoduli can be approached via the super Grassmannian, potentially informing superstring measure constructions and higher‑genus formulations. The construction hinges on coherent interactions among Lie superalgebroids, the super Krichever map, and the Bézoutian/Berezinian framework, yielding a robust flat connection that mirrors the classical Mumford setup in a supersymmetric context.

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 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.

Paper Structure

This paper contains 30 sections, 47 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Lie superalgebroids
  3. Action Lie superalgebroids
  4. Atiyah Lie superalgebroids
  5. Pullbacks over a base change
  6. The supermoduli space of triples $\mathfrak{M}_{g,1^{\infty}_{\mathrm{NS}}}$
  7. Background material on super Riemann surfaces and their superconformal structure
  8. Definition
  9. Superconformal Noether normalization and a perfect Lie superalgebroid
  10. Action of the super Witt algebra
  11. The super Mumford isomorphism
  12. The semi-infinite super Grassmannian of $\mathbb{C}(\!(z)\!)[\zeta]\,[dz|d\zeta]^{\otimes j}$
  13. Background material on finite dimensional super Grassmannians
  14. Definition
  15. Action of the general linear Lie group and algebra
  16. ...and 15 more sections

Key Result

Proposition 2.5

Consider the action Lie superalgebroid $\mathcal{G}$ associated to a Lie superalgebra morphism $\varphi\colon\mathfrak{g}\to\Gamma(S,TS)$. Separately consider a line bundle $L\to S$. There is a bijection between

Theorems & Definitions (133)

  • Definition 2.1
  • Definition 2.2: cf. KosmannSchwarzbach-Mackenzie--2002
  • Definition 2.3: cf. KosmannSchwarzbach-Mackenzie--2002
  • Definition 2.4: cf. KosmannSchwarzbach-Mackenzie--2002
  • Proposition 2.5: cf. KosmannSchwarzbach-Mackenzie--2002
  • proof
  • Definition 2.6
  • Lemma 2.7: cf. Beilinson-Shechtman--1988
  • proof
  • Definition 2.8
  • ...and 123 more