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.Comment: 160 p
