$\newcommand{ \poly}{ _{ \operatorname{ poly}}^{ \bullet}}\newcommand{ \td}{ (\operatorname{ td}_{ L/A}^{ \nabla})^{ \frac{ 1}{ 2}}}\newcommand{ \cx}[1]{ \operatorname{ tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{ #1}\poly\big)}\newcommand{ \cy}[1]{ \mathbb{ H}^\bullet_{ \operatorname{ CE}}(A,\mathcal{ #1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ 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 $\mathfrak{ g}$-manifolds. The spaces $\cx{ T}$ and $\cx{ D}$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to $L_\infty$ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups $\cy{ T}$ and $\cy{ D}$ admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an $L_\infty$ quasi isomorphism from $\cx{ T}$ to $\cx{ D}$ whose first Taylor coefficient is equal to $\operatorname{ hkr}\circ\td$. Here $\td$ acts on $\cx{ T}$ 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 $\cy{ T}$ to $\cy{ D}$. As applications, we establish formality theorems and Kontsevich--Duflo type theorems for complex manifolds, foliations, and $\mathfrak{ g}$-manifolds. In the case of complex manifolds, we recover the Kontsevich--Duflo theorem of complex geometry.
Categories: math.QA math.DG