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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2307.07497 |
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| _version_ | 1866909917922721792 |
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| author | Cui, Xiaoyi |
| author_facet | Cui, Xiaoyi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07497 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The geometric deformation of curved $L_\infty$ algebras and Lie algebroids Cui, Xiaoyi Differential Geometry Mathematical Physics 53D17, 53Z05, 57R91 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. |
| title | The geometric deformation of curved $L_\infty$ algebras and Lie algebroids |
| topic | Differential Geometry Mathematical Physics 53D17, 53Z05, 57R91 |
| url | https://arxiv.org/abs/2307.07497 |