2025/03/31 by Maglio, Antonio · 1 citation
#18N50 #22A22 #53D10 #53D17 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2503.24238
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.