TY - JOUR

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

AU - Liao, Hsuan Yi

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information:
Research partially supported by NSF grants DMS-1707545, DMS-1406668, DMS-1101827, and NSA grant H98230-14-1-0153. Hsuan-Yi Liao was supported by the Ministry of Education of the Republic of China (Taiwan) through a Government Scholarship to Study Abroad.We would like to thank Ruggero Bandiera, Damien Broka, Martin Bordemann, Vasily Dolgushev, Olivier Elchinger, Camille Laurent-Gengoux, Kirill Mackenzie, Dominique Manchon, Marco Manetti, Rajan Mehta, Michael Pevzner, Boris Shoikhet, Jim Stasheff, Dmitry Tamarkin, and Thomas Willwacher for fruitful discussions and useful comments. Stiénon is grateful to Université Paris 7 for its hospitality during his sabbatical leave in 2015–2016. Liao would like to thank National Center for Theoretical Science in Taiwan for its hospitality in August–September 2018. Liao was supported by the Ministry of Education of the Republic of China (Taiwan) through a Government Scholarship to Study Abroad, 2016–2018.
Publisher Copyright:
© 2019 Elsevier Inc.

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.

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U2 - 10.1016/j.aim.2019.04.047

DO - 10.1016/j.aim.2019.04.047

M3 - Article

AN - SCOPUS:85067295665

VL - 352

SP - 406

EP - 482

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -