Hochschild cohomology of dg manifolds associated to integrable distributions

Abstract

For the field K = R or C, and an integrable distribution F ⊂eq TM R K on a smooth manifold M, we study the Hochschild cohomology of the dg manifold (F[1],dF) and establish a canonical isomorphism with the Hochschild cohomology of the algebra of functions on leaf space in terms of transversal polydifferential operators of F. In particular, for the dg manifold (TX0,1[1],∂) associated with a complex manifold X, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomology HH(X) of the complex manifold X. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold (TX0,1[1],∂) implies the Duflo-Kontsevich theorem for complex manifolds.