Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants
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| Format: | Preprint |
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2021
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| _version_ | 1866929704158625792 |
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| author | Kinjo, Tasuki Koseki, Naoki |
| author_facet | Kinjo, Tasuki Koseki, Naoki |
| contents | The aim of this paper is two-fold: Firstly, we prove Toda's $χ$-independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank $r$ and Euler characteristic $χ$ which are not necessary coprime, and show that it does not depend on $χ$. This result extends the Hausel--Thaddeus conjecture on the $χ$-independence of E-polynomials proved by Mellit, Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption.
The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_10053 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants Kinjo, Tasuki Koseki, Naoki Algebraic Geometry High Energy Physics - Theory The aim of this paper is two-fold: Firstly, we prove Toda's $χ$-independence conjecture for Gopakumar--Vafa invariants of arbitrary local curves. Secondly, following Davison's work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank $r$ and Euler characteristic $χ$ which are not necessary coprime, and show that it does not depend on $χ$. This result extends the Hausel--Thaddeus conjecture on the $χ$-independence of E-polynomials proved by Mellit, Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda. |
| title | Cohomological $χ$-independence for Higgs bundles and Gopakumar-Vafa invariants |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2112.10053 |