BFV extensions for mechanical systems with Lie-2 symmetry

TL;DR

This paper explores extending the BFV formalism to mechanical systems with Lie-2 symmetry, identifying obstructions and explicit conditions for Hamiltonian extensions, especially in systems with singular constraints like angular momentum.

Contribution

It introduces a method to construct BFV extensions for systems with Lie-2 algebroid symmetries and explicitly characterizes the obstructions to extending the Hamiltonian.

Findings

BFV extension of the Hamiltonian may be obstructed.

Explicit complex governing the second extension problem is identified.

Standard Hamiltonian lacks BFV extension at the origin in angular momentum example.

Abstract

We consider mechanical systems on $T^*M$ with possibly irregular and reducible first class contraints 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 $(\cal M,Q)$ covering them, where we restrict to the case of Lie-2 algebroids. We propose to consider $T^*\cal M$ as a potential 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 $T^*(\mathbb R^3 \backslash \{0 \})$, with a relatively involved ghost contribution singular at the origin.