Topological charges and the genus of surfaces

TL;DR

This paper establishes a mathematical relationship between the topological charge of sine-Gordon solitons and the genus of associated surfaces, linking soliton configurations to surface topology.

Contribution

It introduces a novel relation n=2(g-1) connecting soliton charge to surface genus and explores the moduli space of these solutions.

Findings

Topological charge n relates to genus g via n=2(g-1).

Moduli space dimension B(g)=3(g-1) matches soliton configuration parameters.

Speculation on odd soliton states describing unoriented surfaces.

Abstract

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.