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N=4 Super Yang Mills theory has a large set of interesting half BPS,
conformally invariant boundary conditions. S-duality acts in surprising
ways on that set, mapping even elementary boundary conditions, as
Neumann or Dirichlet, to rather intricate ones. We will give detailed
examples of S-dual pairs of boundary conditions for U(1), SU(2) gauge
groups, and introduce a general recipe for any gauge group. Besides the
physical interest, the subject has various mathematical applications,
including symplectic duality (aka 3d mirror symmetry) and geometric
Langlands, and can be formulated as an interesting mathematical problem
in its own sake.
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