BFV extensions for mechanical systems with Lie-2 symmetry

Abstract

We consider mechanical systems on ${T}^{*}M$ with possibly irregular and reducible first class constraints linear in the momenta, which thus correspond to singular foliations on $M$. According to a recent result, the latter ones have a Lie-infinity algebroid $(\mathcal{M},Q)$ covering them, where we restrict to the case of Lie-2 algebroids. We propose to consider ${T}^{*}\mathcal{M}$ as a potential Batalin-Fradkin-Vilkovisky (BFV) extended phase space of the constrained system, such that the canonical lift of the nilpotent vector field $Q$ yields automatically a solution to the BFV master equation. We show that in this case, the BFV extension of the Hamiltonian, providing a second corner stone of the BFV formalism, may be obstructed. We identify the corresponding complex governing this second extension problem explicitly (the first extension problem was circumvented by means of the lift of the Lie-2 algebroid structure). We repeatedly come back to the example of angular momenta on ${T}^{*}{\mathbb{R}}^{3}$: in this procedure, the standard free Hamiltonian does not have a BFV extension---while it does so on ${T}^{*}({\mathbb{R}}^{3}\{0})$, with a relatively involved ghost contribution singular at the origin.