Higher Riemann-Hilbert correspondence for foliations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912269998227456 |
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| author | Zeng, Qingyun |
| author_facet | Zeng, Qingyun |
| contents | This paper explores foliated differential graded algebras (dga) and their role in extending fundamental theorems of differential geometry to foliations. We establish an $A_{\infty}$ de Rham theorem for foliations, demonstrating that the classical quasi-isomorphism between singular cochains and de Rham forms lifts to an $A_{\infty}$ quasi-isomorphism in the foliated setting. Furthermore, we investigate the Riemann-Hilbert correspondence for foliations, building upon the established higher Riemann-Hilbert correspondence for manifolds. By constructing an integration functor, we prove a higher Riemann-Hilbert correspondence for foliations, revealing an equivalence between $\infty$-representations of $L_{\infty}$-algebroids and $\infty$-representations of Lie $\infty$-groupoids within the context of foliations. This work generalizes the classical Riemann-Hilbert correspondence to foliations, providing a deeper understanding of the relationship between representations of Lie algebroids and Lie groupoids in this framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08457 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Riemann-Hilbert correspondence for foliations Zeng, Qingyun Differential Geometry Algebraic Topology This paper explores foliated differential graded algebras (dga) and their role in extending fundamental theorems of differential geometry to foliations. We establish an $A_{\infty}$ de Rham theorem for foliations, demonstrating that the classical quasi-isomorphism between singular cochains and de Rham forms lifts to an $A_{\infty}$ quasi-isomorphism in the foliated setting. Furthermore, we investigate the Riemann-Hilbert correspondence for foliations, building upon the established higher Riemann-Hilbert correspondence for manifolds. By constructing an integration functor, we prove a higher Riemann-Hilbert correspondence for foliations, revealing an equivalence between $\infty$-representations of $L_{\infty}$-algebroids and $\infty$-representations of Lie $\infty$-groupoids within the context of foliations. This work generalizes the classical Riemann-Hilbert correspondence to foliations, providing a deeper understanding of the relationship between representations of Lie algebroids and Lie groupoids in this framework. |
| title | Higher Riemann-Hilbert correspondence for foliations |
| topic | Differential Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2503.08457 |