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PhD Thesis: Shifted Contact Structures on Differentiable Stacks

Antonio Maglio

TL;DR

This work develops stacky contact geometry by defining 0- and +1-shifted contact structures on differentiable stacks, using line-bundle valued 1-forms and Morita-invariant kernels. The kernel is realized as a mapping cone (a representation up to homotopy, a RUTH) of the line-bundle 1-form, with the 0-shifted case yielding a cone in degrees $-1,0,1$ and the +1-shifted case in degrees $-1,0$, which can be promoted to a VB-groupoid; curvature is encoded as a RUTH morphism. The framework builds on a Symplectic-to-Contact dictionary, promoting stacky kernels and curvatures to VB-groupoids and line-bundle-groupoids, and it provides Morita-invariant invariants that ensure well-defined structures on differentiable stacks. Examples include 0-shifted contact structures on orbifolds and +1-shifted contact structures arising from prequantization of +1-shifted symplectic structures and the integration of Dirac-Jacobi structures, highlighting potential connections to mathematical physics. Collectively, the results extend Contact Geometry to a stacky setting, establish a robust toolbox (VB-groupoids, LBGs, RUTHs, and Atiyah structures), and pave the way for higher analogues and reductions (e.g., shifted Jacobi or Marsden–Weinstein-type reductions) in differentiable stacks.

Abstract

This thesis focuses on developing "stacky" versions of contact structures, extending the classical notion of contact structures on manifolds. A fruitful approach is to study contact structures using line bundle-valued $1$-forms. Specifically, we introduce the notions of $0$ and $+1$-shifted contact structures on Lie groupoids. To define the kernel of a line bundle-valued $1$-form $θ$ on a Lie groupoid, we draw inspiration from the concept of the homotopy kernel in Homological Algebra. That kernel is essentially given by a representation up to homotopy (RUTH). Similarly, the curvature is described by a specific RUTH morphism. Both the definitions are motivated by the Symplectic-to-Contact Dictionary, which establishes a relationship between Symplectic and Contact Geometry. Examples of $0$-shifted contact structures can be found in contact structures on orbifolds, while examples of $+1$-shifted contact structures include the prequantization of $+1$-shifted symplectic structures and the integration of Dirac-Jacobi structures.

PhD Thesis: Shifted Contact Structures on Differentiable Stacks

TL;DR

This work develops stacky contact geometry by defining 0- and +1-shifted contact structures on differentiable stacks, using line-bundle valued 1-forms and Morita-invariant kernels. The kernel is realized as a mapping cone (a representation up to homotopy, a RUTH) of the line-bundle 1-form, with the 0-shifted case yielding a cone in degrees and the +1-shifted case in degrees , which can be promoted to a VB-groupoid; curvature is encoded as a RUTH morphism. The framework builds on a Symplectic-to-Contact dictionary, promoting stacky kernels and curvatures to VB-groupoids and line-bundle-groupoids, and it provides Morita-invariant invariants that ensure well-defined structures on differentiable stacks. Examples include 0-shifted contact structures on orbifolds and +1-shifted contact structures arising from prequantization of +1-shifted symplectic structures and the integration of Dirac-Jacobi structures, highlighting potential connections to mathematical physics. Collectively, the results extend Contact Geometry to a stacky setting, establish a robust toolbox (VB-groupoids, LBGs, RUTHs, and Atiyah structures), and pave the way for higher analogues and reductions (e.g., shifted Jacobi or Marsden–Weinstein-type reductions) in differentiable stacks.

Abstract

This thesis focuses on developing "stacky" versions of contact structures, extending the classical notion of contact structures on manifolds. A fruitful approach is to study contact structures using line bundle-valued -forms. Specifically, we introduce the notions of and -shifted contact structures on Lie groupoids. To define the kernel of a line bundle-valued -form on a Lie groupoid, we draw inspiration from the concept of the homotopy kernel in Homological Algebra. That kernel is essentially given by a representation up to homotopy (RUTH). Similarly, the curvature is described by a specific RUTH morphism. Both the definitions are motivated by the Symplectic-to-Contact Dictionary, which establishes a relationship between Symplectic and Contact Geometry. Examples of -shifted contact structures can be found in contact structures on orbifolds, while examples of -shifted contact structures include the prequantization of -shifted symplectic structures and the integration of Dirac-Jacobi structures.

Paper Structure

This paper contains 39 sections, 84 theorems, 588 equations.

Table of Contents

  1. Lie Groupoids and Morita equivalence
  2. Lie groupoids
  3. Lie algebroids
  4. The Atiyah algebroid
  5. Morita equivalence and differentiable stacks
  6. VB-groupoids and VB-Morita equivalence
  7. VB-groupoids
  8. Trivial core VB-groupoids
  9. Twisted dual VB-groupoid
  10. The Atiyah VB-groupoid
  11. Representations up to homotopy
  12. VB-Morita equivalence and VB-stacks
  13. Linear natural isomorphisms
  14. Shifted symplectic structures
  15. Shifted $2$-forms
  16. ...and 24 more sections

Key Result

Proposition 1

Let $(F,f)\colon (E\to M)\to (E'\to M')$ be a regular VB morphism. The map is a VB isomorphism.

Theorems & Definitions (383)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Example 1: Lie groups
  • ...and 373 more