The Topology of Nontopological Chern–Simons Vortices

The quantized magnetic flux Φ=−4π N(sign κ), N=0, ±1, ...of nontopological vortices in nonrelativistic Chern–Simons theory is related to the topological degree of the S2 → S2 mapping defined by lifting the problem to Riemann spheres. Regular solutions with finite degree only arise for rational functions whose topological degree, N, is the common number of zeros and poles on the Riemann sphere, also called their algebraic order.