Geometric Langlands duality and the equations of Nahm and Bogomolny

This paper is intended as an introduction to the gauge theory approach [15] to the geometric Langlands correspondence. But, rather than a conventional overview, which I have attempted elsewhere [25, 26], the focus here is on understanding a very particular result, which I learned of from [13]. (Another standard reference on closely related matters is [17].) This introduction is devoted to describing the facts that we wish to explain. In § 2, gauge theory, in the form of new results about how duality acts on boundary conditions [11,12], will be brought to bear to explain them. Finally, some technical details are reserved for § 3. In § 3.3, we also briefly discuss the compactification of the relevant gauge theory to three dimensions, showing some novel features that appear to be relevant to recent work [6]. In § 3.5 we discuss the universal kernel of geometric Langlands from a gauge theory point of view.