Graded contact manifolds and contact Courant algebroids

TL;DR

This paper introduces a systematic framework for contact and Jacobi structures on graded supermanifolds, revealing new algebraic structures like Kirillov algebroids and contact analogs of Courant algebroids.

Contribution

It develops a unified approach to graded contact structures, introduces Kirillov algebroids, and explores contact analogs of Courant algebroids with new lifting procedures.

Findings

Linear contact structures correspond to first jets of line bundles.

Kirillov brackets are represented as homological Hamiltonians.

Contact analogs of Courant algebroids are constructed and studied.

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.