Formality Theorem with Coefficients in a Module

Abstract

In this paper, X will denote a $ {\user1{\mathcal{C}}}^{\infty } $ manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, $\Gamma \left( {X,^E D_{\text{poly}}^* } \right)$ and showed that it is formal. Denote by $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ the DGLA of E-polyvector fields. Considering M, a module over E, we define $\Gamma \left( {X,^E T_{\text{poly}}^* \left( M \right)} \right)$ the $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ -module of E-polyvector fields with values in M. Similarly, we define the $\Gamma \left( {X,^E D_{\text{poly}}^* } \right)$ -module of E-polydifferential operators with values in M, $\Gamma \left( {X,^E D_{\text{poly}}^* \left( M \right)} \right)$ . We show that there is a quasi-isomorphism of L ∞-modules over $\Gamma \left( {X,^E T_{\text{poly}}^* } \right)$ from $\Gamma \left( {X,^E T_{\text{poly}}^* \left( M \right)} \right)$ to $\Gamma \left( {X,^E D_{\text{poly}}^* \left( M \right)} \right)$ . Our result extends Calaque’s (and Kontsevich’s) result.