Special Kähler geometries of N=4 superYang-Mills

The low energy effective theory on the moduli space of vacua of 4d superYang-Mills (sYM) theory defines a special Kähler geometry. For simple sYM gauge algebras, g, we classify all compatible special Kähler structures by showing that they are in one-to-one correspondence with certain equivalence classes of integral symplectic representations of the Weyl group of g. We further demonstrate that, for principal Dirac pairing, these equivalence classes are in one-to-one correspondence with the S-duality orbits of the global structures of the corresponding g sYM gauge theory, after a mistake in the field theory literature is corrected. This provides a low-energy test of S-duality. We also discuss twisted product geometries made from factors with special Kähler structures with non-principal Dirac pairings.