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Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity

Konstantin Eder, John Huerta, Simone Noja

Abstract

We study relative differential and integral forms on families of supermanifolds and investigate their cohomology. In particular, we establish a relative version of Poincaré-Verdier duality, relating the cohomology of differential and integral forms, and provide a concrete interpretation via Berezin fiber integration, which we introduce. To complement Poincaré duality, we prove compactly supported Poincaré lemmas for both differential and integral forms, filling a gap in the literature. We then apply our results to the mathematical foundations of supergravity. Specifically, we rigorously define picture-changing operators via relative Poincaré duality and use them to formulate a general action principle for geometric supergravity in a mathematically rigorous manner. As an example, we explicitly describe three-dimensional supergravity via higher Cartan structures, which are defined by certain classes of connections valued in $L_\infty$-superalgebras. Our construction provides a unified framework interpolating between two equivalent formulations of supergravity in the physics literature: the superspace approach and the group manifold approach.

Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity

Abstract

We study relative differential and integral forms on families of supermanifolds and investigate their cohomology. In particular, we establish a relative version of Poincaré-Verdier duality, relating the cohomology of differential and integral forms, and provide a concrete interpretation via Berezin fiber integration, which we introduce. To complement Poincaré duality, we prove compactly supported Poincaré lemmas for both differential and integral forms, filling a gap in the literature. We then apply our results to the mathematical foundations of supergravity. Specifically, we rigorously define picture-changing operators via relative Poincaré duality and use them to formulate a general action principle for geometric supergravity in a mathematically rigorous manner. As an example, we explicitly describe three-dimensional supergravity via higher Cartan structures, which are defined by certain classes of connections valued in -superalgebras. Our construction provides a unified framework interpolating between two equivalent formulations of supergravity in the physics literature: the superspace approach and the group manifold approach.
Paper Structure (21 sections, 30 theorems, 213 equations)

This paper contains 21 sections, 30 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Main Definitions and Families of Supermanifolds
  3. Differential and Integral Forms on Families of Supermanifolds
  4. Cohomology of Differential and Integral Forms
  5. Cohomology and Poincaré Lemmas
  6. Compactly Supported Cohomology and Poincaré Lemmas
  7. Differential Forms
  8. Integral Forms
  9. Families of Supermanifolds and Cohomology
  10. The Dualizing Complex $\omega^\bullet_{\mathpzc{X} / \mathpzc{S}}$
  11. Relative Poincaré Duality
  12. Berezin Fiber Integration
  13. Interlude on Berezin Integration.
  14. Trivial Families and Even Submanifolds
  15. Families of Ordinary Manifolds: Cohomology and Poincaré Lemma
  16. ...and 6 more sections

Key Result

Theorem 2.12

Let $\mathpzc{X}^\mathpzc{ev}_{/\mathpzc{S}}$ and $\mathpzc{Y}^{\mathpzc{ev}}_{/\mathpzc{S}}$ be families of supermanifolds with relative odd dimension $m|0$ and let $\zeta: \mathpzc{X}^\mathpzc{ev}_{/\mathpzc{S}, \mathpzc{red}} \to \mathpzc{Y}^\mathpzc{ev}_{/\mathpzc{S}, \mathpzc{red}}$ be a diffeo

Theorems & Definitions (93)

  • Definition 2.1: Superspaces and Morphisms
  • Definition 2.2: Nilpotent Sheaf
  • Definition 2.3: Smooth / Complex / Algebraic Supermanifold and Morphisms
  • Remark 2.4
  • Definition 2.5: Family of Supermanifolds
  • Definition 2.6: Morphisms of Families of Supermanifolds
  • Definition 2.7: Category of Relative Supermanifolds
  • Remark 2.8: Base Change and Families of Supermanifolds
  • Remark 2.9: Reduced Space and Relative Supermanifolds
  • Definition 2.10: Underlying Even Manifolds
  • ...and 83 more