On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation

Abstract

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. Evolution invariance of the constraint surface introduces a connection $\nabla$ on E and one reobtains the compatibility of g with $(E,\rho ,\nabla )$ found previously in the literature. The covariantization of the Hamiltonian to a function on $T^*E[1]$ serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning $(E,\rho ,[ \cdot , \cdot ],\nabla )$ into a Cartan–Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.