Formality for g-manifolds

Abstract

To any $\mathfrak{g}$-manifold $M$ are associated two dglas $\operatorname{tot}\big(\Lambda^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big)$ and $\operatorname{tot} \big(\Lambda^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$, whose cohomologies $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ and $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_\infty$ quasi-isomorphism $\Phi: \operatorname{tot}\big(\Lambda^{\bullet} \mathfrak{g}^\vee \otimes_{\Bbbk} T_{\operatorname{poly}}^{\bullet} \big) \to \operatorname{tot} \big(\Lambda^{\bullet} \mathfrak{g}^\vee\otimes_{\Bbbk} D_{\operatorname{poly}}^{\bullet} \big)$ whose first `Taylor coefficient' (1) is equal to the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd cocycle of the $\mathfrak{g}$-manifold $M$ and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild-Kostant-Rosenberg map twisted by the square root of the Todd class of the $\mathfrak{g}$-manifold $M$ is an isomorphism of Gerstenhaber algebras from $H_{\operatorname{CE}}(\mathfrak{g}, T_{\operatorname{poly}}^{\bullet} \xrightarrow{0} T_{\operatorname{poly}}^{\bullet+1})$ to $H_{\operatorname{CE}}(\mathfrak{g}, D_{\operatorname{poly}}^{\bullet} \xrightarrow{0} D_{\operatorname{poly}}^{\bullet+1})$.