Quantum observables, Lie algebra homology and TQFT

Abstract

Let us consider a Lie (super)algebra $G$ spanned by $T_{\alpha}$ where $T_{\alpha}$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{\alpha_1...\alpha_k}$ that determines a homology class of the Lie algebra $G$ the expression $c^{\alpha_1...\alpha_k}T_{\alpha _1}...T_{\alpha_k}$ 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.