Geometric Langlands And The Equations Of Nahm And Bogomolny

TL;DR

The paper presents a gauge-theoretic framework for understanding aspects of geometric Langlands by leveraging twisted $\mathcal N=4$ super Yang–Mills, the Bogomolny and Nahm equations, and dual boundary conditions. It builds a concrete bridge between the cohomology of Hecke-modification moduli $\overline{\mathcal N}(\rho)$ and dual group representations $R^\vee$ through the $\widehat{\mathcal A}$- and $\widehat{\mathcal B}$-models, showing how a principal $SL_2$ action emerges from the grading of cohomology and how universal characteristic classes correspond to Lie-algebra elements under duality. The work details how Wilson and ‘t Hooft operators are related via OPEs, how fusion is compatible with dual actions, and how a three-dimensional theory with a universal kernel unifies boundary-condition dualities across the Langlands correspondence. A key insight is that Nahm’s equations and regular Nahm poles encode the dual boundary data, yielding a unique dual description and a natural map between cohomological and representation-theoretic structures. Overall, the article provides a rigorous gauge-theory route to geometric Langlands, clarifying the role of dualities, boundary conditions, and three-dimensional reductions while connecting universal kernels to boundary theories like $T(G)$.

Abstract

Geometric Langlands duality relates a representation of a simple Lie group $G^\vee$ to the cohomology of a certain moduli space associated with the dual group $G$. In this correspondence, a principal $SL_2$ subgroup of $G^\vee$ makes an unexpected appearance. Why this happens can be explained using gauge theory, as we will see in this article, with the help of the equations of Nahm and Bogomolny. (Based on a lecture at Geometry and Physics: Atiyah 80, Edinburgh, April 2009.)

Figures (6)

• Figure 1: Depicted here is $W=S^2\times I$ in $G$ gauge theory with a marked point $w$ at which an 't Hooft operator is inserted. Dirichlet boundary conditions are imposed at the right boundary and Neumann boundary conditions at the left.
• Figure 2: $W=S^2\times I$ in $G^\vee$ gauge theory with a marked point at which is included an external charge in the representation $R^\vee$. The boundary conditions are dual to those of fig. \ref{['octo']}. At the left are Dirichlet boundary conditions modified with a regular Nahm pole, while at the right are more complicated boundary conditions associated with the universal kernel of geometric Langlands.
• Figure 3: Drawn here is an 't Hooft or Wilson line operator that runs in the time direction (shown vertically) at a fixed position in $W$. A small two-surface $S$ (sketched here as a circle) is supported at a fixed time and is linked with $L$.
• Figure 4: $S^2\times I$ with $n$ marked points (only a few of which have been labeled) at which 't Hooft or Wilson operators have been inserted.
• Figure 5: A two-sphere $S$ (at fixed time) surrounding all of the marked points $w_\alpha\in S^2\times I$. (In this example, there are three marked points). $S$ is homologous to a sum of two-spheres $S_\alpha$, each of them linking just one of the $w_\alpha$.