Smoothness theorem for differential BV algebras

Abstract

Given a differential Batalin–Vilkovisky algebra (V, Q, Δ.) the associated odd differential graded Lie algebra (V, Q, + Δ, [,].) is always smooth formal. The quantum differential graded Lie algebra L ℏ : = ( V [ [ ℏ ] ] , Q + ℏ Δ , [ , ] ) is not always smooth formal, but when it is — for example, when a Q-Δ version of the ∂-∂ Lemma holds — there is a weak Frobenius manifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, we prove that Lħ is smooth formal if and only if the spectral sequence associated to the filtration F p : = ℏ p V [ [ ℏ ] ] on the complex ( V [ [ ℏ ] ] , Q + ℏ Δ ) degenerates at E1. A priori, this degeneration means that a collection of first-order obstructions vanish and we prove that it follows that all obstructions vanish. For those differential BV algebras that arise from the Hochschild complex of a Calabi–Yau category, the degeneration of this spectral sequence is an expression of the noncommutative Hodge to deRham degeneration, conjectured by Kontsevich and Soibelman and proved to hold in certain cases by Kaledin. The results in this paper imply that the noncommutative Hodge to de Rham degeneration conjecture is equivalent to the existence of a versal solution to the quantum master equation. At the end of the paper, some physical considerations are mentioned.