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As an important application of the spectral networks we have so far introduced, we present in this chapter the construction of non-abelianization and abelianization maps. These maps describe the equivalence between flat connections on a rank n bundle over a Riemann surface S and flat connections on a line bundle over the spectral curve \(\Sigma \) , an n-sheeted ramified covering of S. The monodromies of the abelianized local system are cluster \(\mathcal {X}\) -coordinates. The process of (non-)abelianization thus provides very strong motivation for using spectral networks in the study of higher-rank Teichmüller spaces. In particular, we show how one can use spectral networks and abelianization to induce Fock–Goncharov coordinates or Fenchel–Nielsen coordinates on spaces of flat connections. Suitable references demonstrating the link between spectral networks and higher-rank Teichmüller theory include (Gaiotto et al. Annales Henri Poincare 14:1643–1731;2013. https://doi.org/10.1007/s00023-013-0239-7 . arXiv: 1204.4824 [hep-th], Hollands and Kidwai, Adv Theor Math Phys 22(7):1713–1822;2019. ISSN: 1095-0761. https://doi.org/10.4310/ATMP.2018.v22.n7.a2 , Hollands and Neitzke, Lett Math Phys 106(6):811–877;2016. https://doi.org/10.1007/s11005-016-0842-x . arXiv: 1312.2979 [math.GT], Hollands and Neitzke, Commun Math Phys. 380(1):131–186;2020. https://doi.org/10.1007/s00220-020-03875-1 . arXiv: 1906.04271 [hep-th], Nho, Family Floer theory, non-abelianization, and Spectral Networks. Preprint. arXiv:2307.04213 [math.SG]; 2023, Nikolaev, Sel Math New Ser 27:35;2021. Id/No 78. ISSN: 1022–1824. https://doi.org/10.1007/s00029-021-00688-5 ).

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Non-abelianization and Abelianization

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  • 发表日期：

    2026年

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  • Clarence Kineider,
  • Georgios Kydonakis,
  • Eugen Rogozinnikov,
  • Valdo Tatitscheff,
  • Alexander Thomas

摘要

As an important application of the spectral networks we have so far introduced, we present in this chapter the construction of non-abelianization and abelianization maps. These maps describe the equivalence between flat connections on a rank n bundle over a Riemann surface S and flat connections on a line bundle over the spectral curve \(\Sigma \) , an n-sheeted ramified covering of S. The monodromies of the abelianized local system are cluster \(\mathcal {X}\) -coordinates. The process of (non-)abelianization thus provides very strong motivation for using spectral networks in the study of higher-rank Teichmüller spaces. In particular, we show how one can use spectral networks and abelianization to induce Fock–Goncharov coordinates or Fenchel–Nielsen coordinates on spaces of flat connections. Suitable references demonstrating the link between spectral networks and higher-rank Teichmüller theory include (Gaiotto et al. Annales Henri Poincare 14:1643–1731;2013. https://doi.org/10.1007/s00023-013-0239-7 . arXiv: 1204.4824 [hep-th], Hollands and Kidwai, Adv Theor Math Phys 22(7):1713–1822;2019. ISSN: 1095-0761. https://doi.org/10.4310/ATMP.2018.v22.n7.a2 , Hollands and Neitzke, Lett Math Phys 106(6):811–877;2016. https://doi.org/10.1007/s11005-016-0842-x . arXiv: 1312.2979 [math.GT], Hollands and Neitzke, Commun Math Phys. 380(1):131–186;2020. https://doi.org/10.1007/s00220-020-03875-1 . arXiv: 1906.04271 [hep-th], Nho, Family Floer theory, non-abelianization, and Spectral Networks. Preprint. arXiv:2307.04213 [math.SG]; 2023, Nikolaev, Sel Math New Ser 27:35;2021. Id/No 78. ISSN: 1022–1824. https://doi.org/10.1007/s00029-021-00688-5 ).