Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

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).

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s00220-021-04265-x

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Work Title Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras
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Open Access
Creators
  1. Seokbong Seol
  2. Mathieu Stiénon
  3. Ping Xu
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Communications in Mathematical Physics
Publication Date February 24, 2022
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  1. https://doi.org/10.1007/s00220-021-04265-x
Deposited January 12, 2024

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  • Added Creator Seokbong Seol
  • Added Creator Mathieu P Stienon
  • Added Creator Ping Xu
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    • This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov L<sub>∞</sub>[1] algebras associated with dg manifolds in the C<sup>∞</sup> 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<sub>∞</sub>[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<sub>∞</sub>[1] algebra structure is quasi-isomorphic to the standard L<sub>∞</sub>[1] algebra structure on the Dolbeault complex Ω0,∙(TX1,0).
    • 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).
  • Renamed Creator Mathieu Stiénon Show Changes
    • Mathieu P Stienon
    • Mathieu Stiénon
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