The algebra of Wilson-'t Hooft operators

TL;DR

This work analyzes the algebra of Wilson–'t Hooft operators in a holomorphic–topological twist of N=4 SYM, leveraging Montonen–Olive duality to constrain their OPE and testing predictions for G=SU(2) via explicit weak-coupling computations. It develops a rich framework linking line operators to K-theory of DG-categories, Hecke transformations, and the Geometric Satake correspondence, with a concrete realization in terms of the Hitchin moduli space ${\mathcal M}_{Higgs}(G,C)$ and the affine Grassmannian. The paper performs detailed OPE computations at weak coupling for $G=PSU(2)$, identifying bulk and bubbled contributions from monopole bubbling, and verifies S-duality predictions by matching state spaces with $L^2$ Dolbeault cohomology on Schubert cells and their bundles. It further explores the structure of line operators as functors on branes, the $K^0$-theory framework, and the higher-categorical perspective on defects in TFTs, linking physical constructions to deep mathematical objects such as $D^b_{eq}(\\Lambda_G)$ and the geometric Satake equivalence. The results illuminate how dualities organize the spectrum and fusion of nonlocal operators and provide computational recipes for extending the OPE analysis to more general gauge groups.

Abstract

We study the Operator Product Expansion of Wilson-'t Hooft operators in a twisted N=4 super-Yang-Mills theory with gauge group G. The Montonen-Olive duality puts strong constraints on the OPE and in the case G=SU(2) completely determines it. From the mathematical point of view, the Montonen-Olive duality predicts the L^2 Dolbeault cohomology of certain equivariant vector bundles on Schubert cells in the affine Grassmannian. We verify some of these predictions. We also make some general observations about higher categories and defects in Topological Field Theories.