2023/07/14 by Xiaoyi Cui, Cui, Xiaoyi
Mathematics · Medicine · #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #Neurosurgical Procedures and Complications
paper · pdf · doi:10.48550/arxiv.2307.07497
While L_∞ 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_∞ 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_∞ algebras constructed from a vector bundle V→ 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_∞ 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_∞ algebras.