Parallel surface defects, Hecke operators, and quantum Hitchin system

TL;DR

This work builds an N=2 gauge-theory framework to realize the geometric Langlands correspondence for Hitchin systems in the presence of ramification, via two half-BPS surface defects: a regular monodromy defect and a canonical defect. It shows that the regular defect yields twisted ${\mathcal D}$-modules on ${\rm Bun}_{G_{\bf C}}(\mathcal C;S)$ whose spectral data are governed by opers constructed from the canonical defect, thereby producing common eigenfunctions for quantum Hitchin Hamiltonians. Parallel defects furnish a Hecke operator whose action on twisted coinvariants reproduces the Hecke eigensheaf property with eigenvalues given by the oper local system; this ties the eigenfunctions to Gaudin algebra via universal opers. The analysis uses gauge origami, brane/boundary dualities, and the $\Omega$-background to connect 4d N=2 theories with 2d/4d coupled systems, and culminates in demonstrating that the regular monodromy defect selects a basis of twisted coinvariants while the canonical defect encodes oper data that diagonalizes the Gaudin/Hitchin commuting Hamiltonians. The results unify N=2 gauge theory quantization with geometric Langlands via a concrete oper/Eigenstructure, and anchor the duality with GL-twisted N=4 theory and brane pictures. These insights advance a gauge-theoretic path to both automorphic D-modules and quantum integrable systems, with potential extensions to analytic Langlands and broader defect networks.

Abstract

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.

Figures (3)

• Figure 1: IIA brane picture for $Q$-observable
• Figure 2: IIA brane picture for canonical surface defect with (a) positive real FI parameter (b) negative real FI parameter
• Figure 3: $\mathbb{P}^1$ with four marked points $S=\{0,\mathfrak{q},1\,\infty\}$. The double circles represent the maximal marked points at $0$ and $\infty$, while the dots denote the minimal marked points at $\mathfrak{q}$ and $1$. The $A$-cycle is drawn in blue and the $B$-cycle is drawn in red.