Gauge Theory and Skein Modules

TL;DR

This paper explores the structure of skein modules of 3-manifolds using 4d super-Yang-Mills theories, revealing new insights into their dimensions, dualities, and connections to geometric Langlands and Higgs bundle moduli spaces.

Contribution

It introduces an algorithm to determine skein module dimensions for manifolds with reduced holonomy and relates these to supersymmetric gauge theories and Langlands duality.

Findings

Dimensions often differ between Langlands-dual pairs beyond A-series.

The approach clarifies the relation to the geometric Langlands program.

Provides a physical interpretation of skein-valued curve counting.

Abstract

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.