### 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) |
---|---|

Pages (from-to) | 406-482 |

Number of pages | 77 |

Journal | Advances in Mathematics |

Volume | 352 |

DOIs | |

State | Published - Aug 20 2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

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

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*Advances in Mathematics*, vol. 352, pp. 406-482. https://doi.org/10.1016/j.aim.2019.04.047

**Formality and Kontsevich–Duflo type theorems for Lie pairs.** / Liao, Hsuan Yi; Stienon, Mathieu Philippe; Xu, Ping.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Formality and Kontsevich–Duflo type theorems for Lie pairs

AU - Liao, Hsuan Yi

AU - Stienon, Mathieu Philippe

AU - Xu, Ping

PY - 2019/8/20

Y1 - 2019/8/20

N2 - Kontsevich's formality theorem states that there exists an L∞ quasi-isomorphism from the dgla Tpoly •(M) of polyvector fields on a smooth manifold M to the dgla Dpoly •(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∨)⊗RTpoly •) and tot(Γ(Λ•A∨)⊗RDpoly •) 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 HCE •(A,Tpoly •) and HCE •(A,Dpoly •) 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∨)⊗RTpoly •) to tot(Γ(Λ•A∨)⊗RDpoly •) whose first Taylor coefficient is equal to hkr∘(tdL/A ∇) [Formula presented]. Here the cocycle (tdL/A ∇) [Formula presented] acts on tot(Γ(Λ•A∨)⊗RTpoly •) 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 HCE •(A,Tpoly •) to HCE •(A,Dpoly •). 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.

AB - Kontsevich's formality theorem states that there exists an L∞ quasi-isomorphism from the dgla Tpoly •(M) of polyvector fields on a smooth manifold M to the dgla Dpoly •(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∨)⊗RTpoly •) and tot(Γ(Λ•A∨)⊗RDpoly •) 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 HCE •(A,Tpoly •) and HCE •(A,Dpoly •) 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∨)⊗RTpoly •) to tot(Γ(Λ•A∨)⊗RDpoly •) whose first Taylor coefficient is equal to hkr∘(tdL/A ∇) [Formula presented]. Here the cocycle (tdL/A ∇) [Formula presented] acts on tot(Γ(Λ•A∨)⊗RTpoly •) 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 HCE •(A,Tpoly •) to HCE •(A,Dpoly •). 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.

UR - http://www.scopus.com/inward/record.url?scp=85067295665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067295665&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.04.047

DO - 10.1016/j.aim.2019.04.047

M3 - Article

VL - 352

SP - 406

EP - 482

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -