Surface operators in four-dimensional topological gauge theory and Langlands duality

TL;DR

The paper develops a 2-categorical framework for surface and line operators in the GL-twisted N=4 gauge theory, systematically analyzing the t-dependent theories (t=i, t=1, t=0) and their Montonen–Olive dualities. By reducing to 3d, it identifies familiar and novel 3d TFTs (Rozansky–Witten, gauged RW, B-type gauge theories) and computes the categories of boundary and bulk operators as module categories over monoidal defect categories, such as D^b(Coh(G_C)) and D^b(Coh(C^*)). It provides explicit descriptions of the 2-category of surface operators at t=i (abelian and nonabelian) and outlines the t=1 and t=0 pictures, including dualities, boundary conditions, and the corresponding brane and line-operator categories. These constructions illuminate a 2-categorical form of geometric Langlands duality and offer a path toward a quantum geometric Langlands framework via the rich interplay between 4d TFTs, their 3d reductions, and associated 2d brane categories.

Abstract

We study surface and line operators in the GL-twisted N=4 gauge theory in four dimensions. Their properties depend on the parameter t which determines the BRST operator of theory. For t=i we propose a complete description of the 2-category of surface operators in terms of module categories. We also determine the monoidal category of line operators which includes Wilson lines as special objects. For t=1 and t=0 we only discuss surface and line operators in the abelian case. Applications to the categorification of the local geometric Langlands duality and its quantum version are briefly described. In the appendices we discuss several 3d and 2d topological field theories with gauge fields. In particular, we explain a relationship between the category of branes in the gauged B-model and the equivariant derived category of coherent sheaves.

Figures (8)

• Figure 1: Morphisms in the category of boundary conditions correspond to local operators sitting at the junction of two segments of the boundary.
• Figure 2: Composition of morphisms is achieved by merging the insertion points of the local operators. We use $\cdot$ to denote this operation.
• Figure 3: A wall separating theories ${\mathbb X}$ and ${\mathbb Y}$ is equivalent to a boundary of theory $\bar{{\mathbb X}}\times{\mathbb Y}$.
• Figure 4: 1-morphisms of the 2-category of 2d TFTs correspond to walls, and composition of 1-morphisms corresponds to fusing walls. This operation is denoted $\otimes$.
• Figure 5: Composition of 2-morphisms of the 2-category of 2d TFTs is achieved by fusing the walls on which they are inserted. The corresponding operation is denoted $\otimes$.