Topological charges and the genus of surfaces

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.