Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916824385323008 |
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| author | Maneval, Elsa |
| author_facet | Maneval, Elsa |
| contents | The Hausel-Thaddeus conjectures concern topological mirror symmetry between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles on a curve. A non-archimedean approach was introduced by Groechenig, Wyss and Ziegler, proving the conjecture for coprime rank and degree. This article is concerned with its generalisation to the non-coprime case. We treat both the classical ($D=K$) and meromorphic ($D>K$) settings. We prove an equality of $p$-adic volumes twisted by gerbes between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles of arbitrary degree. In the meromorphic case, building on results of Maulik and Shen, we show that these twisted $p$-adic volumes are related to intersection cohomology. We also conjecture a connection between these $p$-adic volumes and $BPS$ cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles Maneval, Elsa Algebraic Geometry The Hausel-Thaddeus conjectures concern topological mirror symmetry between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles on a curve. A non-archimedean approach was introduced by Groechenig, Wyss and Ziegler, proving the conjecture for coprime rank and degree. This article is concerned with its generalisation to the non-coprime case. We treat both the classical ($D=K$) and meromorphic ($D>K$) settings. We prove an equality of $p$-adic volumes twisted by gerbes between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles of arbitrary degree. In the meromorphic case, building on results of Maulik and Shen, we show that these twisted $p$-adic volumes are related to intersection cohomology. We also conjecture a connection between these $p$-adic volumes and $BPS$ cohomology. |
| title | Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2507.02588 |