  We show that the topological charge of the n-soliton solution of the
sine-Gordon equation n is related to the genus g > 1 of a constant negative
curvature compact surface described by this configuration. The relation is
n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1)
corresponds precisely to the freedom to choosing the configuration with n
solitons of arbitrary positions and velocities. We speculate also that the odd
soliton states will describe the unoriented surfaces.

High Energy Physics - Theory;Differential Geometry;Differential Geometry