Formality and Kontsevich--Duflo type theorems for Lie pairs

Abstract

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.

55 pages, several typos corrected, some references added, some minor cosmetic changes in the presentation

References in corpus (8)

• Supergroupoids, double structures, and equivariant cohomology
• The Atiyah class of a dg-vector bundle
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Atiyah classes and dg-Lie algebroids for matched pairs
• Formality for g-manifolds
• Noncommutative calculus and operads
• L-infinity Formality check for the Hochschild Complex of Certain Universal Enveloping Algebras
• Polyvector fields and polydifferential operators associated with Lie pairs

Cited by in corpus (6)

• Hochschild cohomology of dg manifolds associated to integrable distributions
• Fedosov dg manifolds associated with Lie pairs
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Hopf algebras arising from dg manifolds
• Polyvector fields and polydifferential operators associated with Lie pairs
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair