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Shifted Contact Structures and Their Local Theory

Kadri İlker Berktav

TL;DR

The paper introduces k-shifted contact structures on derived (Artin) stacks, proving a Darboux-type local model for negatively shifted cases (k<0) and constructing a canonical symplectification that yields a k-shifted symplectic space. It builds on PTVV shifted symplectic geometry, using refined local models via minimal standard form cdgas to obtain explicit Darboux forms and to describe nondegeneracy conditions in the derived setting. The main contributions include a precise local model for shifted contact structures, a proof of a Darboux-type theorem, and a functorial construction of symplectifications in the derived category, connecting contact and symplectic geometries in DAG. The results pave the way for stacky generalizations to derived Artin stacks and suggest future directions toward Legendrians and broader shifted contact theory with potential applications in moduli problems and quantization within derived algebraic geometry.

Abstract

In this paper, we formally define the concept of shifted contact structures on derived (Artin) stacks and study their local properties in the context of derived algebraic geometry. In this regard, for negatively shifted contact derived $\mathbb{K}$-schemes, we develop a Darboux-like theorem and formulate the notion of symplectification.

Shifted Contact Structures and Their Local Theory

TL;DR

The paper introduces k-shifted contact structures on derived (Artin) stacks, proving a Darboux-type local model for negatively shifted cases (k<0) and constructing a canonical symplectification that yields a k-shifted symplectic space. It builds on PTVV shifted symplectic geometry, using refined local models via minimal standard form cdgas to obtain explicit Darboux forms and to describe nondegeneracy conditions in the derived setting. The main contributions include a precise local model for shifted contact structures, a proof of a Darboux-type theorem, and a functorial construction of symplectifications in the derived category, connecting contact and symplectic geometries in DAG. The results pave the way for stacky generalizations to derived Artin stacks and suggest future directions toward Legendrians and broader shifted contact theory with potential applications in moduli problems and quantization within derived algebraic geometry.

Abstract

In this paper, we formally define the concept of shifted contact structures on derived (Artin) stacks and study their local properties in the context of derived algebraic geometry. In this regard, for negatively shifted contact derived -schemes, we develop a Darboux-like theorem and formulate the notion of symplectification.
Paper Structure (28 sections, 11 theorems, 61 equations)

This paper contains 28 sections, 11 theorems, 61 equations.

Table of Contents

  1. Introduction and summary
  2. Results and the outline.
  3. Acknowledgments.
  4. Conventions.
  5. Shifted symplectic structures
  6. Some derived algebraic geometry
  7. Nice local models for derived ${\mathbb K}$-schemes.
  8. Nice local models for cotangent complexes of derived schemes.
  9. PTVV's shifted symplectic geometry on derived schemes
  10. Shifted symplectic Darboux models
  11. Prototype Darboux model for $k<0$ odd.
  12. Darboux forms for the other cases of $k.$
  13. Shifted contact structures and a Darboux-type theorem
  14. Basics of classical contact geometry
  15. Shifted contact structures and Darboux models
  16. ...and 13 more sections

Key Result

Theorem 1.1

Every $k$-shifted contact derived ${\mathbb K}$-scheme $\bf X$ is locally equivalent to $({\rm Spec} A, \alpha_0)$ for $A$ a minimal standard form cdga and $\alpha_0$ a contact Darboux form.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 38 more