Formality and Kontsevich–Duflo type theorems for Lie pairs

Kontsevich's formality theorem states that there exists an L_∞ quasi-isomorphism from the dgla T_poly ^•(M) of polyvector fields on a smooth manifold M to the dgla D_poly ^•(M) of polydifferential operators on M, which extends the classical Hochschild–Kostant–Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds (that is, manifolds endowed with an action of a Lie algebra g). The spaces tot(Γ(Λ^•A^∨)⊗_RT_poly ^•) and tot(Γ(Λ^•A^∨)⊗_RD_poly ^•) associated with a Lie pair (L,A) each carry an L_∞ algebra structure canonical up to L_∞ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups H_CE ^•(A,T_poly ^•) and H_CE ^•(A,D_poly ^•) admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an L_∞ quasi-isomorphism from tot(Γ(Λ^•A^∨)⊗_RT_poly ^•) to tot(Γ(Λ^•A^∨)⊗_RD_poly ^•) whose first Taylor coefficient is equal to hkr∘(td_L/A ^∇) ^{[Formula presented]}. Here the cocycle (td_L/A ^∇) ^{[Formula presented]} acts on tot(Γ(Λ^•A^∨)⊗_RT_poly ^•) by contraction. Furthermore, we prove a Kontsevich–Duflo type theorem for Lie pairs: the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the Lie pair (L,A) is an isomorphism of Gerstenhaber algebras from H_CE ^•(A,T_poly ^•) to H_CE ^•(A,D_poly ^•). As applications, we establish formality and Kontsevich–Duflo type theorems for complex manifolds, foliations, and g-manifolds. In the case of complex manifolds, we recover the Kontsevich–Duflo theorem of complex geometry.