Formality theorem for g-manifolds

With any $\mathfrak{g}$-manifold M are associated two dglas $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))$ and $\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$, whose cohomologies $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ and $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$ are Gerstenhaber algebras. We establish a formality theorem for $\mathfrak{g}$-manifolds: there exists an $L_{\infty }$ quasi-isomorphism $\mathrm{\Phi }:\mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{T}_{\mathrm{poly}}^{?}(M))\to \mathrm{tot}({\mathrm{\Lambda }}^{?}{\mathfrak{g}}^{\vee }{?}_{\mathbb{k}}{D}_{\mathrm{poly}}^{?}(M))$ 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 $H_{\mathrm{CE}}^{?}(\mathfrak{g},T_{\mathrm{poly}}^{?}(M)\stackrel{0}{\to }{T}_{\mathrm{poly}}^{?+1}(M))$ to $H_{\mathrm{CE}}^{?}(\mathfrak{g},D_{\mathrm{poly}}^{?}(M)\stackrel{d_H}{\to }{D}_{\mathrm{poly}}^{?+1}(M))$.