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The geometric deformation of curved $L_\infty$ algebras and Lie algebroids

Xiaoyi Cui

TL;DR

This work establishes a precise correspondence between geometric deformations of curved $L_\infty$ algebras assembled from a vector bundle $V\to M$ and Lie algebroid structures on $V$, using a Fedosov-type recursive framework to solve defining and consistency equations. It clarifies how Atiyah and Atiyah-Chern classes emerge in this setting, showing the first Atiyah-Chern class is the $d^{DR}$-coboundary of the modular class and that action algebroids yield equivariant Chern characters at leading order. The results illuminate how geometric deformations encode both smooth and multiplicative structures on bundles and yield a unifying lens for BV theories, where deforming cotangent theories naturally produces Poisson sigma models. The approach provides computationally explicit bridges between $L_\infty$-algebroid data, Weil algebra representatives, and field-theoretic constructions, with potential applications to equivariant cohomology, modular invariants, and topological sigma-models.

Abstract

While $L_\infty$ algebras are fundamental structures in differential geometry and mathematical physics, the geometric information encoded in such structures is often implicit. We address the following question: What constitutes a geometrically meaningful deformation of an $L_\infty$ algebra arising from vector bundles, and how can such deformations classify new geometric invariants? Inspired by nonabelian extension theory of Lie algebras, we define geometric deformations of curved $L_\infty$ algebras constructed from a vector bundle $V\to M$, and demonstrate that such deformations uniquely correspond to Lie algebroid structures on $V$. Explicit computations reveal that the first Atiyah-Chern class, expressible via deformed $L_\infty$ brackets, transgresses to the de Rham coboundary of the modular class. In the case of action Lie algebroids, the leading-order Atiyah-Chern classes correspond to the equivariant Chern characters. Applications to BV theories show that the geometric deformations naturally generate Poisson sigma models. These results provide a coherent framework for deriving field theories from geometric deformations of $L_\infty$ algebras.

The geometric deformation of curved $L_\infty$ algebras and Lie algebroids

TL;DR

This work establishes a precise correspondence between geometric deformations of curved algebras assembled from a vector bundle and Lie algebroid structures on , using a Fedosov-type recursive framework to solve defining and consistency equations. It clarifies how Atiyah and Atiyah-Chern classes emerge in this setting, showing the first Atiyah-Chern class is the -coboundary of the modular class and that action algebroids yield equivariant Chern characters at leading order. The results illuminate how geometric deformations encode both smooth and multiplicative structures on bundles and yield a unifying lens for BV theories, where deforming cotangent theories naturally produces Poisson sigma models. The approach provides computationally explicit bridges between -algebroid data, Weil algebra representatives, and field-theoretic constructions, with potential applications to equivariant cohomology, modular invariants, and topological sigma-models.

Abstract

While algebras are fundamental structures in differential geometry and mathematical physics, the geometric information encoded in such structures is often implicit. We address the following question: What constitutes a geometrically meaningful deformation of an algebra arising from vector bundles, and how can such deformations classify new geometric invariants? Inspired by nonabelian extension theory of Lie algebras, we define geometric deformations of curved algebras constructed from a vector bundle , and demonstrate that such deformations uniquely correspond to Lie algebroid structures on . Explicit computations reveal that the first Atiyah-Chern class, expressible via deformed brackets, transgresses to the de Rham coboundary of the modular class. In the case of action Lie algebroids, the leading-order Atiyah-Chern classes correspond to the equivariant Chern characters. Applications to BV theories show that the geometric deformations naturally generate Poisson sigma models. These results provide a coherent framework for deriving field theories from geometric deformations of algebras.
Paper Structure (19 sections, 34 theorems, 211 equations)

This paper contains 19 sections, 34 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Curved $L_\infty$ structures from vector bundles and geometric deformation
  3. General framework
  4. Geometric realization from vector bundles and the deformation
  5. Step 1: Degree Analysis
  6. Step 2: Defining Equation
  7. Step 3: Consistency Equation
  8. Family version and homotopy equivalence
  9. Atiyah class and character
  10. Atiyah class of Lie algebroids
  11. The Weil algebra
  12. Atiyah-Chern class
  13. Modular class and the first Atiyah-Chern class
  14. Action algebroids
  15. $L_\infty$ algebras in field theories
  16. ...and 4 more sections

Key Result

Proposition 8

Let $\mathfrak g$ be a curved $L_\infty$ algebra over $(A, d_A)$ and let $\mathbb V$ be a finitely generated $\mathbb Z$-graded projective $A$-module. Suppose that operations $\{\tilde{l}_n\}_{n=0}^\infty$ define an $(A, d_A)$-linear curved $L_\infty$ structure on $\mathfrak g\oplus \mathbb V$ such Then $\mathbb V$ is an $L_\infty$-module of $\mathfrak g$ such that $\{\tilde{l}_n\}_{n=0}^\infty$

Theorems & Definitions (65)

  • proof : Definition 1
  • proof : Remark 2
  • proof : Definition 3
  • proof : Definition 4
  • proof : Definition 5
  • proof : Definition 6
  • proof : Definition 7
  • Proposition 8
  • proof
  • Proposition 9: GG_ahatGLL
  • ...and 55 more