Gauge Theory and Skein Modules
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910025977430016 |
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| author | Pei, Du |
| author_facet | Pei, Du |
| contents | We study skein modules of 3-manifolds by embedding them into the Hilbert spaces of 4d ${\cal N}=4$ super-Yang-Mills theories. When the 3-manifold has reduced holonomy, we present an algorithm to determine the dimension and the list of generators of the skein module with a general gauge group. The analysis uses a deformation preserving ${\cal N}=1$ supersymmetry to express the dimension as a sum over nilpotent orbits in its Lie algebra. We find that the dimensions often differ between Langlands-dual pairs beyond the A-series, for which we provide a physical explanation involving chiral symmetry breaking and 't Hooft operators. We also relate our results to the structure of $\mathbb{C}^*$-fixed loci in the moduli space of Higgs bundles. This approach helps to clarify the relation between the gauge-theoretic framework of Kapustin and Witten with other versions of the geometric Langlands program, explains why the dimensions of skein modules do not exhibit a TQFT-like behavior, and provides a physical interpretation of the skein-valued curve counting of Ekholm and Shende. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16213 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gauge Theory and Skein Modules Pei, Du High Energy Physics - Theory Algebraic Geometry Geometric Topology Quantum Algebra Representation Theory We study skein modules of 3-manifolds by embedding them into the Hilbert spaces of 4d ${\cal N}=4$ super-Yang-Mills theories. When the 3-manifold has reduced holonomy, we present an algorithm to determine the dimension and the list of generators of the skein module with a general gauge group. The analysis uses a deformation preserving ${\cal N}=1$ supersymmetry to express the dimension as a sum over nilpotent orbits in its Lie algebra. We find that the dimensions often differ between Langlands-dual pairs beyond the A-series, for which we provide a physical explanation involving chiral symmetry breaking and 't Hooft operators. We also relate our results to the structure of $\mathbb{C}^*$-fixed loci in the moduli space of Higgs bundles. This approach helps to clarify the relation between the gauge-theoretic framework of Kapustin and Witten with other versions of the geometric Langlands program, explains why the dimensions of skein modules do not exhibit a TQFT-like behavior, and provides a physical interpretation of the skein-valued curve counting of Ekholm and Shende. |
| title | Gauge Theory and Skein Modules |
| topic | High Energy Physics - Theory Algebraic Geometry Geometric Topology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2601.16213 |