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

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.

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