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There are two special values of the coupling constant for which there
exist noncentral elements of SL(2,Z) that map N=4 Super Yang-Mills
theory with gauge group U(n) to itself. At these values, the field
theory can be compactified on a circle with duality-twisted boundary
conditions. The low-energy limit of this model directly probes the
S-duality operator. Augmented by an R-symmetry twist, and with
additional restrictions on the rank n, this low-energy limit appears to
be a nontrivial topological field theory. Upon further compactification
on a torus, the Hilbert space of the low-energy theory can be mapped,
using U-duality, to the finite dimensional space of minimal string
states on a three-dimensional manifold that is a torus fibre-bundle over
a circle. Using the string theory realization, I'll compare the
low-energy theory with Chern-Simons theory. Also, compactification on a
Riemann surface of higher genus suggests a relation between the
dimension of the Hilbert space of certain Chern-Simons theories on the
Riemann surface and the supertrace of the action induced by mirror
symmetry on the appropriate cohomology of the appropriate Hitchin space.
**

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