平行表面缺陷、海克尔算子与量子希奇金系统
Parallel surface defects, Hecke operators, and quantum Hitchin system
AI总结：
研究四维超对称 gauge 理论中两种半 BPS 表面缺陷的数学结构，揭示其与海克尔算子及量子希奇金系统的联系。
摘要：
我们研究了四维具有 N=2 超对称性和 Ω_{ε1,ε2} 背景的 gauge 理论中两种半 BPS 表面缺陷——正则单值表面缺陷和规范表面缺陷。数学上，我们研究了 parabolic framed sheaves 在 P^2 上的模空间积分。利用 N=2 理论的分析方法，我们证明前者给出 twisted D-模，后者作为海克尔算子。在 ε2→0 极限下，簇分解表明正则单值表面缺陷具有 Hecke 特征 sheaf 性质。特征值由规范表面缺陷关联的 opers 给出。我们推导出在 N=2 gauge 理论框架中，分配给 opers 的 twisted D-模代表几何 Langlands 对应的谱方程。讨论了与拓扑扭曲的四维 N=4 理论的对偶性，其中两种表面缺陷映射为 Dirichlet 边界和 't Hooft 线缺陷。这与 N=4 理论对几何 Langlands 对应的早期工作一致。
英文摘要：
We examine two types of half-BPS surface defects $-$ regular monodromy surface defect and canonical surface defect $-$ in four-dimensional gauge theory with $\mathcal{N}=2$ supersymmetry and $Ω_{\varepsilon_1,\varepsilon_2}$-background. Mathematically, we investigate integrals over the moduli spaces of parabolic framed sheaves over $\mathbb{P}^2$. Using analytic methods of $\mathcal{N}=2$ theories, we demonstrate that the former gives a twisted $\mathcal{D}$-module on $\text{Bun}_{G_{\mathbb{C}}}$ while the latter acts as a Hecke operator. In the limit $\varepsilon_2 \to 0$, the cluster decomposition implies the Hecke eigensheaf property for the regular monodromy surface defect. The eigenvalues are given by the opers associated to the canonical surface defect. We derive, in our $\mathcal{N}=2$ gauge theoretical framework, that the twisted $\mathcal{D}$-modules assigned to the opers in the geometric Langlands correspondence represent the spectral equations for quantum Hitchin integrable system. A duality to topologically twisted four-dimensional $\mathcal{N}=4$ theory is discussed, in which the two surface defects are mapped to Dirichlet boundary and 't Hooft line defect. This is consistent with earlier works on the $\mathcal{N}=4$ theory approach to the geometric Langlands correspondence.
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