  Dark soliton is one of most interesting nonlinear excitations in physical
systems, manifesting a spatially localized density "dip" on a uniform
background accompanied with a phase jump across the dip. However, the
topological properties of the dark solitons are far from fully understood. Our
investigation for the first time uncover a vector potential underlying the
nonlinear excitation whose line integral gives the striking phase jump. More
importantly, we find that the vector potential field has a topological
configuration in analogous to the Wess-Zumino term in a Lagrangian
representation. It can induce some point-like magnetic fields scattered
periodically on a complex plane, each of them has a quantized magnetic flux of
elementary $\pi$. We then calculate the Euler characteristic of the topological
manifold of the vector potential field and classify all known dark solitions
according to the index.
