Formality theorem for g-manifolds Théorème de formalité pour les g-variétés

With any g-manifold M are associated two dglas tot(Λ ^• g ^∨ ⊗ _k T _poly ^• (M)) and tot(Λ ^• g ^∨ ⊗ _k D _poly ^• (M)), whose cohomologies H _CE ^• (g,T _poly ^• (M)→0T _poly ^•+1 (M)) and H _CE ^• (g,D _poly ^• (M)→d _H D _poly ^•+1 (M)) are Gerstenhaber algebras. We establish a formality theorem for g-manifolds: there exists an L _∞ quasi-isomorphism Φ:tot(Λ ^• g ^∨ ⊗ _k T _poly ^• (M))→tot(Λ ^• g ^∨ ⊗ _k D _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 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 H _CE ^• (g,T _poly ^• (M)→0T _poly ^•+1 (M)) to H _CE ^• (g,D _poly ^• (M)→d _H D _poly ^•+1 (M)).