Gauge theory, ramification, and the geometric Langlands program

Abstract

In the gauge theory approach to the geometric Langlands program, ramification can be described in terms of "surface operators," which are supported on two-dimensional surfaces somewhat as Wilson or 't Hooft operators are supported on curves. We describe the relevant surface operators in N = 4 super Yang-Mills theory, and the parameters they depend on, and analyze how S-duality acts on these parameters. Then, after compactifying on a Riemann surface, we show that the hypothesis of S-duality for surface operators leads to a natural extension of the geometric Langlands program for the case of tame ramification. The construction involves an action of the affine Weyl group on the cohomology of the moduli space of Higgs bundles with ramification, and an action of the affine braid group on A-branes or B-branes on this space.

Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories

“…surface operators. Surface operators in N = 4 gauge theories were extensively studied in [15] (see [16,17] for some similar work in N = 2 theories). In the context of the AGT relation, surface operators have been studied in several papers [1,11,[18][19][20][21][22].…”

Mirror Symmetry, Hitchin's Equations, and Langlands Duality

“…However, quantum mechanically, there is one more natural variable η (analogous to the usual θ-angles of gauge theory), as described in section 2.3 of Gukov & Witten (2006). With the complete set of parameters (α, β, γ, η) at hand, it is possible to formulate a natural duality statement, according to which…”

“…, with a certain map between the parameters, described in section 2.4 of Gukov & Witten (2006). The main point of this map is that (α, η) = ( L η, − L α).…”

“…Moreover, there is a Hitchin fibration, and most of the usual statements about the unramified case -those that we have explained and those that we have omitted here -have close analogs. For a much more detailed explanation, and references to the original literature, see Gukov & Witten (2006).…”

“…(4.3) we seem to be in the opposite case of semi-simple residue. In Gukov & Witten (2006), it is explained that, with some care, mirror symmetry for M H (G, C ′ ; α, β, γ, η) is actually a sufficient framework to understand geometric Langlands for a connection with a simple pole of any residue. For example, the case of a nilpotent residue can be understood by setting L α = L γ = 0 (or γ = η = 0 in the dual description).…”