On the Relation of Lie Algebroids to Constrained Systems and their BV/BFV Formulation
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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, Q) of the Lie algebroid to its cotangent bundle \(T^*E\). 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 \(T^*E\) 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.
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T.S. wants to thank Anton Alekseev for a long-lasting and multiply inspiring friendship. N.I. thanks Anton Alekseev and the university of Geneva for the permission of his staying as a visiting scientist and their hospitality. We gratefully acknowledge the interest and critical and important feedback of Albin Grataloup and Sylvain Lavau on earlier versions of this paper. We also thank Camille Laurent–Gengoux for remarks on the manuscript and Maxim Grigoriev for drawing our attention to the references [18, 19] and . This work was supported by the project MODFLAT of the European Research Council (ERC) and the NCCR SwissMAP of the Swiss National Science Foundation.
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