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Quasi-coherent sheaves and D-modules in Derived Differential Supergeometry

David Carchedi

TL;DR

This work develops a comprehensive framework for derived differential geometry with a focus on differential supergeometry, extending DDG to supermanifolds via the 2-sorted theory ${f SC}^{ ty}$ and establishing a robust correspondence between derived spaces and algebraic models. Central achievements include the universal and algebro-geometric descriptions of derived manifolds, the spectrum-based construction of derived ${f SC}^{ ty}$-schemes, and the demonstration that spaces of sections and differential operators can be treated coherently within an $ ext{∞}$-topos framework. A key technical result is the canonical equivalence QC$( ext{M}_{dR}) \, ilde{=}\, ext{D-Modules}( ext{M})$ for supermanifolds, which mirrors classical $ ext{D}$-module theory within derived differential geometry and enables a unified approach to jets, de Rham stacks, and formal neighborhoods. The paper also develops the homotopical $C^ ty$-algebra theory, including Koszul complexes and localization, to underpin a robust derived calculus for differential operators and modules, with implications for constructing derived spaces of solutions to Euler–Lagrange equations in gauge theories and for extending QC-sheaf theory to derived stacks and orbifolds. Overall, the framework provides foundational tools for rigorously integrating derived geometry with differential, infinite-dimensional, and supergeometric contexts, facilitating future work on derived spaces of solutions in field theories.

Abstract

Derived geometry provides powerful tools to handle non-transverse intersections and singular moduli problems arising in geometry and theoretical physics. While derived algebraic geometry has been extensively developed, classical field theories -- formulated as variational problems involving sections of smooth fiber bundles over manifolds -- naturally require the language of differential geometry, infinite-dimensional analysis (e.g., Fréchet manifolds), and additional geometric structures on spacetime, such as smooth metrics. Moreover, field theories incorporating fermionic matter fields necessitate extending the framework to include supermanifolds. This article is the first in a sequence aimed at rigorously modeling the derived space of solutions to the field equations of Lagrangian gauge theories as derived $\Ci$-stacks. While this article does not explicitly discuss physics or field theory, it develops foundational aspects of derived differential geometry which are useful in their own right and contribute to the further development of the field. Moreover, these results provide essential groundwork for subsequent papers rigorously constructing derived spaces of solutions to Euler-Lagrange equations. We establish foundational results extending existing work on derived manifolds into supergeometric and infinite-dimensional contexts, and explicitly relate these constructions to differential operators and PDE theory. This paper an excerpt from a larger manuscript currently in preparation and is made available now to disseminate key foundational developments.

Quasi-coherent sheaves and D-modules in Derived Differential Supergeometry

TL;DR

This work develops a comprehensive framework for derived differential geometry with a focus on differential supergeometry, extending DDG to supermanifolds via the 2-sorted theory and establishing a robust correspondence between derived spaces and algebraic models. Central achievements include the universal and algebro-geometric descriptions of derived manifolds, the spectrum-based construction of derived -schemes, and the demonstration that spaces of sections and differential operators can be treated coherently within an -topos framework. A key technical result is the canonical equivalence QC for supermanifolds, which mirrors classical -module theory within derived differential geometry and enables a unified approach to jets, de Rham stacks, and formal neighborhoods. The paper also develops the homotopical -algebra theory, including Koszul complexes and localization, to underpin a robust derived calculus for differential operators and modules, with implications for constructing derived spaces of solutions to Euler–Lagrange equations in gauge theories and for extending QC-sheaf theory to derived stacks and orbifolds. Overall, the framework provides foundational tools for rigorously integrating derived geometry with differential, infinite-dimensional, and supergeometric contexts, facilitating future work on derived spaces of solutions in field theories.

Abstract

Derived geometry provides powerful tools to handle non-transverse intersections and singular moduli problems arising in geometry and theoretical physics. While derived algebraic geometry has been extensively developed, classical field theories -- formulated as variational problems involving sections of smooth fiber bundles over manifolds -- naturally require the language of differential geometry, infinite-dimensional analysis (e.g., Fréchet manifolds), and additional geometric structures on spacetime, such as smooth metrics. Moreover, field theories incorporating fermionic matter fields necessitate extending the framework to include supermanifolds. This article is the first in a sequence aimed at rigorously modeling the derived space of solutions to the field equations of Lagrangian gauge theories as derived -stacks. While this article does not explicitly discuss physics or field theory, it develops foundational aspects of derived differential geometry which are useful in their own right and contribute to the further development of the field. Moreover, these results provide essential groundwork for subsequent papers rigorously constructing derived spaces of solutions to Euler-Lagrange equations. We establish foundational results extending existing work on derived manifolds into supergeometric and infinite-dimensional contexts, and explicitly relate these constructions to differential operators and PDE theory. This paper an excerpt from a larger manuscript currently in preparation and is made available now to disseminate key foundational developments.

Paper Structure

This paper contains 46 sections, 170 theorems, 759 equations.

Table of Contents

  1. Introduction
  2. Organization and Main Results:
  3. Algebraic theories
  4. Categorical properties of algebraic theories
  5. Finitely presented algebras
  6. Homotopical algebras
  7. $\mathbf{C}^\infty$-algebras
  8. $\mathbf{C}^\infty$-completion
  9. Localizations
  10. Reduced $\mathbf{C}^\infty$-algebras
  11. Derived manifolds
  12. A universal property
  13. An algebro-geometric description
  14. A geometric description
  15. Homotopical $\mathbf{C}^\infty$-algebras
  16. ...and 31 more sections

Key Result

Corollary 1.2

(Corollary cor:supdman) There is a canonical equivalence of $\infty$-categories between the $\infty$-category of derived supermanifolds and the opposite of the $\infty$-category of finitely presented ${\mathbf{SC}^{\infty}}$-algebras in $\mathcal{S} \operatorname{pc}.$

Theorems & Definitions (435)

  • Definition 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 1.9
  • Corollary 1.10
  • ...and 425 more