  We study the current in a multi-channel quantum wire and the magnetization in
the multi-channel Kondo problem. We show that at zero temperature they can be
written simply in terms of contour integrals over a (two-dimensional)
hyperelliptic curve. This allows one to easily demonstrate the existence of
weak-coupling to strong-coupling dualities. In the Kondo problem, the curve is
the same for under- and over-screened cases; the only change is in the contour.
