Gauge Theory and Skein Modules

TL;DR

Gauge Theory and Skein Modules develops a gauge-theoretic framework for 3-manifold skein modules sk$(M_3;G)$ by embedding skein data into the Hilbert space of a topologically twisted 4d ${\cal N}=4$ theory and then applying a controlled ${\cal N}=1$ deformation. The deformation partitions vacua by nilpotent orbits, enabling a computable dimension formula that aggregates contributions from IR phases, electric/magnetic flux sectors, and cosmic-string worldsheet dynamics. The authors illustrate the method across families of groups (A–D–exceptional), provide detailed $T^3$ and $\Sigma\times S^1$ examples, and discuss Langlands duality and fixed-point structures in Hitchin moduli spaces as cross-checks. They also highlight non-TQFT-like features in skein dimensions and propose enriched skein theories ${\widetilde{\mathcal S}}$ that include ’t Hooft and dyonic lines to restore dualities. The work suggests deep links between skein theory, geometric Langlands, and Higgs-bundle geometry, with several directions for future refinement and generalization, including higher-genus manifolds and a broader category-theoretic perspective.

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. 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.

Figures (8)

• Figure 1: A boundary condition of the topologically twisted 4d theory on $M^3\times \mathbb{R}^+$ with insertions of Wilson lines along $L\subset M_3$ is illustrated on the upper left. Upon compactification on $M_3$, this gives a boundary condition $B_L$ for the quantum mechanics on $\mathbb{R}^+$ (lower part of the figure), which gives a state ${\left< {L} \right|}$ in the dual of its Hilbert space ${\mathcal{H}}(M_3)^\vee$ (upper right).
• Figure 2: The nine generators of sk$(T^3)$. The configuration on the left represent $2^3=8$ different configurations with possible insertions of Wilson loops along the three directions, ranging from the empty configuration to the one with all three insertions. They can be associated with $\mathbb{Z}_2$ phase. The need for the additional configuration on the right can be explain by the existence of the confinement phase.
• Figure 3: On the left, we illustrate the three regions with qualitatively different dynamics in the presence of a cosmic string. After compactifying on the meridian circle, the system at low energy can be viewed as a bulk Chern--Simons theory coupling to massless chiral fermions on the boundary.
• Figure 4: Two actions of operators on the Hilbert space of the cosmic string (grey) on $S^1$. In (a), a Wilson/Verlinde line $W_\mu$ (blue) is inserted along an $S^1$. Its action on any state in the sector ${\mathcal{H}}_\nu$ is given by the scalar multiplication with $\frac{S_{\mu\nu}}{S_{0\nu}}$. This can be understood from the bulk perspective illustrated in (b), where, in the radial quantization, a state in ${\mathcal{H}}_\nu$ is created by a point operator attached to the bulk Wilson line $W_\nu$. The action of $W_\mu$ is then given by its braiding with $W_\nu$. The sub-figure (c) depicts the action of a boundary monopole operator (red dot), coming from the lasso action of an 't Hooft operator labeled by a co-character $m$ of the gauge group $K$. This operator is on the end point of an "'t Hooft vortex line" ${\mathcal{V}}_m$ (red) carrying a magnetic flux, mapping the initial state to a twisted sector. From the bulk perspective illustrated in (d), the action can be understood as the fusion of $W_\mu$ with the vortex line ${\mathcal{V}}_m$.
• Figure 5: For ${\mathfrak{sl}}(2)$, the dominant weights at level $k$ are labeled by non-negative integers $\{0,1,\ldots , k\}$. The two colors for the weights in the figure reflect whether the central character is trivial (blue) or non-trivial (orange). The $\mathbb{Z}_2$ center acts by a reflection, along axes depicted by dashed lines for various $k$ in the figure. The action for the $g=2$ (hence $k=4$) case on weights is explicitly illustrated. To obtain $\Gamma^0$ (or $\Gamma^1$) for a given $k$, one just counts blue (or orange) dots from zero until the reflection axis.