PhD Thesis: Shifted Contact Structures on 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 $\theta$ 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.