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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.20517 |
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| _version_ | 1866910061911080960 |
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| author | François, Quentin García-Zelada, David Lévy, Thierry Tarrago, Pierre |
| author_facet | François, Quentin García-Zelada, David Lévy, Thierry Tarrago, Pierre |
| contents | We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the $\mathrm{U}(n)$-valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20517 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces François, Quentin García-Zelada, David Lévy, Thierry Tarrago, Pierre Probability Mathematical Physics Symplectic Geometry We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the $\mathrm{U}(n)$-valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process. |
| title | A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces |
| topic | Probability Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2603.20517 |