1112.0759 (Janusz Grabowski)

Graded contact manifolds and principal Courant algebroids    [PDF]

Janusz Grabowski

It is developed 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 contact structure. Linear contact structures are proven to be exactly
the canonical contact structures on first jets of line bundles. They give rise
to linear Kirillov (or Jacobi) brackets and the concept of a principal Lie
algebroid, a contact analog of a Lie algebroid. The corresponding cohomology
operator is represented not by a vector field (a 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 also studied as well as contact analogs of
Courant algebroids. Lifting procedures to tangent and cotangent bundles are
described and provide constructions of canonical examples of the structures in
question.

View original: http://arxiv.org/abs/1112.0759