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In Freed and Neitzke (Surveys Diff Geom 26(1):51–155, 2021), D. Freed and A. Neitzke introduce an abstract framework for spectral networks on manifolds of dimension less than three and describe a stratified abelianization process for three-dimensional Chern–Simons invariants. Specifically, they demonstrate that the \( \operatorname {\mathrm {SL}}_2(\mathbb {C})\) -Chern–Simons invariant of a three-manifold M equipped with a principal \( \operatorname {\mathrm {SL}}_2(\mathbb {C})\) -bundle and a flat connection, can be computed in terms of the spin \(\mathbb {C}^*\) -Chern–Simons invariant of a ramified double cover of M. Moreover, the authors extend the construction of three-dimensional (3d) spectral networks to the groups \( \operatorname {\mathrm {GL}}_2(\mathbb {C})\) and \( \operatorname {\mathrm {PSL}}_2(\mathbb {C})\) .

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Three-Dimensional Networks

  • 章节

  • 出版模式：

    开放获取

  • 发表日期：

    2026年

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

摘要

In Freed and Neitzke (Surveys Diff Geom 26(1):51–155, 2021), D. Freed and A. Neitzke introduce an abstract framework for spectral networks on manifolds of dimension less than three and describe a stratified abelianization process for three-dimensional Chern–Simons invariants. Specifically, they demonstrate that the \( \operatorname {\mathrm {SL}}_2(\mathbb {C})\) -Chern–Simons invariant of a three-manifold M equipped with a principal \( \operatorname {\mathrm {SL}}_2(\mathbb {C})\) -bundle and a flat connection, can be computed in terms of the spin \(\mathbb {C}^*\) -Chern–Simons invariant of a ramified double cover of M. Moreover, the authors extend the construction of three-dimensional (3d) spectral networks to the groups \( \operatorname {\mathrm {GL}}_2(\mathbb {C})\) and \( \operatorname {\mathrm {PSL}}_2(\mathbb {C})\) .