Formality theorem for [formula omitted]-manifolds

Abstract

With any g-manifold M are associated two dglas tot(Λ•g∨⊗kTpoly•(M)) and tot(Λ•g∨⊗kDpoly•(M)), whose cohomologies HCE•(g,Tpoly•(M)→0Tpoly•+1(M)) and HCE•(g,Dpoly•(M)→dHDpoly•+1(M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L∞ quasi-isomorphism Φ:tot(Λ•g∨⊗kTpoly•(M))→tot(Λ•g∨⊗kDpoly•(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 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 g-manifold M is an isomorphism of Gerstenhaber algebras from HCE•(g,Tpoly•(M)→0Tpoly•+1(M)) to HCE•(g,Dpoly•(M)→dHDpoly•+1(M)).