### Abstract

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^{∨})⊗_{R}T_{poly} ^{•}) and tot(Γ(Λ^{•}A^{∨})⊗_{R}D_{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^{∨})⊗_{R}T_{poly} ^{•}) to tot(Γ(Λ^{•}A^{∨})⊗_{R}D_{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^{∨})⊗_{R}T_{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.

Original language | English (US) |
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Pages (from-to) | 406-482 |

Number of pages | 77 |

Journal | Advances in Mathematics |

Volume | 352 |

DOIs | |

State | Published - Aug 20 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*352*, 406-482. https://doi.org/10.1016/j.aim.2019.04.047