Formality theorem for differential graded manifolds

Abstract

We establish a formality theorem for smooth dg manifolds. More precisely, we prove that, for any finite-dimensional dg manifold (M,Q), there exists an L∞ quasi-isomorphism of dglas from (Tpoly•⊕(M),[Q,−],[−,−]) to (Dpoly•⊕(M),〚m+Q,−〛,〚−,−〛) whose first Taylor coefficient (1) is equal to the composition hkr∘(td(M,Q)∇)12:Tpoly•⊕(M)→Dpoly•⊕(M) of the action of (td(M,Q)∇)12∈∏k≥0(Ωk(M))k on Tpoly•⊕(M) (by contraction) with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we prove the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo-type theorem holds for all finite-dimensional smooth dg manifolds.