On the relation of Lie algebroids to constrained systems and their BV/BFV formulation

Understand

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M.

• The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1].
• Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge.
• Adding a Hamiltonian to the system corresponds to a metric g on M.