Graded contact manifolds and contact Courant algebroids

Abstract

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally, n-linear) contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid, for which the corresponding cohomology operator is represented not by a vector field (de Rham derivative) but a first-order differential operator. It is shown that one can view Kirillov or Jacobi brackets as homological Hamiltonians on linear contact manifolds. Contact manifolds of degree 2 are studied, as well as contact analogs of Courant algebroids. We define lifting procedures that provide us with constructions of canonical examples of the structures in question.

39 pages, some changes of the terminology, references extended and updated

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• Vector Bundle Valued Differential Forms on $\mathbb{N}Q$-manifolds
• Reductions: precontact versus presymplectic
• Graded Bundles in the Category of Lie Groupoids
• A contact covariant approach to optimal control with applications to sub-Riemannian geometry
• VB-structures and generalizations
• Polarisation of Graded Bundles
• The Local Structure of Generalized Contact Bundles
• Contactifications: a Lagrangian description of compact Hamiltonian systems
• Contact structures and supersymmetric mechanics
• Contact structures on Lie algebroids
• Multiplicative Connections and Their Lie Theory
• Duality for graded manifolds
• Generalised contact geometry as reduced generalised complex geometry
• Connections Adapted to Non-Negatively Graded Structures
• Integrating Nijenhuis Structures
• On the concept of a filtered bundle
• Kirillov structures up to homotopy
• Jacobi algebroids and Jacobi sigma models
• Generalized symmetries as homotopy Lie algebras
• Jacobi Geometry and Hamiltonian Mechanics: the Unit-Free Approach
• Carrollian ${\mathbb{R}}^{\times }$-bundles: Connections and Beyond
• A Darboux classification of homogeneous Pfaffian forms on graded manifolds
• Contact Courant algebroids and $L_{\infty }$-algebras
• On the matrix realization of the Lie superalgebra of contact projective vector fields $\mathfrak{spo}(2l+2|n)$
• The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$
• Higher Lie and Leibniz algebras
• The Deformation $L_{\infty }$ algebra of a Dirac--Jacobi structure
• Heisenberg order of differential operators on the superspaces ${\mathbb{R}}^{2l+1|n}$