Quantum observables, Lie algebra homology and TQFT

Abstract

Let us consider a Lie (super)algebra $G$ spanned by $T_α$ where $T_α$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{α_1...α_k}$ that determines a homology class of the Lie algebra $G$ the expression $c^{α_1...α_k}T_{α_1}...T_{α_k}$ is again a quantum observables. This theorem is used to construct quantum observables in BV sigma-model. We apply this construction to explain Kontsevich's results about the relation between homology of the Lie algebra of Hamiltonian vector fields and topological invariants of manifolds.