发表机构
Uppsala University（乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过$SU(2)$ $\mathcal{N}=2^*$理论的$\mathfrak{q}$-非阿贝尔化，证明穿孔环面$GL_2$ skein代数同构于球形DAHA，并利用有框墙交叉统一了其多种表示。
AI 中文摘要
类$\mathcal{S}$的$A_{N-1}$型4维$\mathcal{N}=2$理论在半Omega背景中的线算子为$SL_N$ skein代数提供了物理模型。本文将这一对应关系推广到$GL_N$ skein代数，并证明了穿孔环面的$GL_2$ skein代数同构于$GL_2$球形双仿射Hecke代数${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$。该构造基于$SU(2)$ $\mathcal{N}=2^*$规范理论的$\mathfrak{q}$-非阿贝尔化，它将skein代数实现为与Seiberg-Witten曲线相关的量子环面代数。Coulomb支流的不同区域给出了${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$的不同表示，范围从Fenchel-Nielsen（弱耦合）图中的Macdonald $q$-差分模到Fock-Goncharov（强耦合）图中的簇型实现。这些描述之间的转换由$\mathcal{N}=2^*$理论的香草BPS谱通过有框墙交叉控制，这为球形DAHA的多种表示提供了统一的物理框架。
英文摘要
Line operators of 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $A_{N-1}$ in a half Omega background provide a physical model for $SL_N$ skein algebras. In this paper we extend this correspondence to $GL_N$ skein algebras and prove that the $GL_2$ skein algebra of the punctured torus is isomorphic to the $GL_2$ spherical double affine Hecke algebra ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. The construction is based on $\mathfrak{q}$-nonabelianization for the $SU(2)$ $\mathcal{N}=2^*$ gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. These range from the Macdonald $q$-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the $\mathcal{N}=2^*$ theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.
Comments50 pages
