Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs

Liao, Hsuan-Yi

doi:10.1007/s00220-023-04854-y

Commun. Math. Phys.

2023

DOI: 10.1007/s00220-023-04854-y

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Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs

Hsuan-Yi Liao

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2024

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Cited by 3 publications

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“…The Atiyah class of a dg manifold (M, Q) is an obstruction to the existence of an affine connection compatible with the homological vector field Q. In particular, it was shown by Chen-Xiang-Xu [8] (see also [12]) that the Atiyah class of the dg manifold arising from a complex manifold is the Atiyah class of the complex manifold, introduced by Atiyah [2].…”

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confidence: 99%

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Seol

Stiénon

2022

Commun. Math. Phys.

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A. A dg manifold of amplitude +1 in C ∞ -context can be thought of as a derived intersection of a section s and the zero section of a vector bundle E. In this paper, we compute the Atiyah class of dg manifolds of amplitude +1. In particular, we prove that the Atiyah class of a dg manifold of amplitude +1 vanishes if and only if the intersection of s and the zero section is a clean intersection.As an example, given two submanifolds X and Y of W , we investigate the Atiyah class of a dg manifold which can be thought of as a model for the derived intersection of X and Y in the sense of Behrend-Liao-Xu. C 1. Introduction Notations and conventions 2. Backgrounds on dg manifolds 3. The Atiyah class and clean intersection 3.1. Clean intersection 3.2. Locality of the Atiyah class in positive amplitude 3.3. The Atiyah class of dg manifolds of amplitude +1 4. Proof of Theorem 3.2 4.1. Proof of (L1) ⇒ (L2): sample cases 4.2. Proof of (L1) ⇒ (L2): general case 5. The case of derived intersections Acknowledgement References

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confidence: 99%

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Seol

Stiénon

2022

Commun. Math. Phys.

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Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

Chang,

Liao

2024

Journal of Geometry and Physics

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Differential Graded Manifolds of Finite Positive Amplitude

Behrend,

Liao,

2024

International Mathematics Research Notices

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We prove that dg manifolds of finite positive amplitude, that is, bundles of positively graded curved $L_{\infty }[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_{\infty }[1]$-algebras. As an application, we study the derived intersections of manifolds.

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Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs

Cited by 3 publications

References 32 publications

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

Differential Graded Manifolds of Finite Positive Amplitude

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