Formality and Kontsevich–Duflo type theorems for Lie pairs

Hsuan Yi Liao, Mathieu Stiénon, Ping Xu

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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(ΛA)⊗RTpoly ) and tot(Γ(ΛA)⊗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(ΛA)⊗RTpoly ) to tot(Γ(ΛA)⊗RDpoly ) whose first Taylor coefficient is equal to hkr∘(tdL/A ) [Formula presented]. Here the cocycle (tdL/A ) [Formula presented] acts on tot(Γ(ΛA)⊗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.

Original languageEnglish (US)
Pages (from-to)406-482
Number of pages77
JournalAdvances in Mathematics
StatePublished - Aug 20 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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