Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Seokbong Seol, Mathieu Stiénon, Ping Xu

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5 Scopus citations

Abstract

This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L[1] algebras associated with dg manifolds in the C context. We prove that, for a dg manifold, a ‘formal exponential map’ exists if and only if the Atiyah class vanishes. Inspired by Kapranov’s construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an L[1] algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative with respect to the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold (TX0,1[1],∂¯) arising from a complex manifold X, we prove that this L[1] algebra structure is quasi-isomorphic to the standard L[1] algebra structure on the Dolbeault complex Ω0,∙(TX1,0).

Original languageEnglish (US)
Pages (from-to)33-76
Number of pages44
JournalCommunications In Mathematical Physics
Volume391
Issue number1
DOIs
StatePublished - Apr 2022

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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