Formality theorem with coefficients in a module

Abstract

In this article, $X$ will denote a ${\cal C}^{\infty}$ manifold. In a very famous article, Kontsevich showed that the differential graded Lie algebra (DGLA) of polydifferential operators on $X$ is formal. Calaque extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid $E$ over $X$, he defined the DGLA of $E$-polydifferential operators, $\Gamma (X, ^{E}D^{*}_{poly})$, and showed that it is formal. Denote by $\Gamma (X, ^{E}T^{*}_{poly})$ the DGLA of $E$-polyvector fields. Considering $M$, a module over $E$, we define $\Gamma (X, ^{E}T_{poly}^{*}(M))$ the $\Gamma (X, ^{E}T^{*}_{poly})$-module of $E$-polyvector fields with values in $M$. Similarly, we define the $\Gamma (X, ^{E}D^{*}_{poly})$-module of $E$-polydifferential operators with values in $M$, $\Gamma (X, ^{E}D^{*}_{poly}(M))$. We show that there is a quasi-isomorphism of $L_{\infty}$-modules over $\Gamma (X, ^{E}T^{*}_{poly})$ from $\Gamma (X, ^{E}T^{*}_{poly}(M))$ to $\Gamma (X, ^{E}D^{*}_{poly}(M))$. Our result extends Calaque 's (and Kontsevich's) result.