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

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.