Hochschild cohomology of dg manifolds associated to integrable distributions

Abstract

For the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and an integrable distribution $F\subseteq T_M{\otimes }_{\mathbb{R}}\mathbb{K}$ on a smooth manifold $M$, we study the Hochschild cohomology of the dg manifold $(F[1],d_F)$ 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 $(T_X^{0,1}[1],\bar{\partial })$ associated with a complex manifold $X$, we prove that its Hochschild cohomology is canonically isomorphic to the Hochschild cohomology $H{H}^{\bullet }(X)$ of the complex manifold $X$. As an application, we show that the Duflo-Kontsevich type theorem for the dg manifold $(T_X^{0,1}[1],\bar{\partial })$ implies the Duflo-Kontsevich theorem for complex manifolds.

arXiv comment32 pages, final version; references added; To appear in Comm.Math.Phys

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• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Keller admissible triples and Duflo theorem
• $A_{\infty }$-Algebras from Lie Pairs
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair