Some Exact Solutions of the Semilocal Popov Equations with Many Flavors

TL;DR

This paper derives and solves exact solutions for semilocal Popov vortex equations with many flavors in 2+1D Chern-Simons theories on a sphere, revealing new solutions expressed via rational functions and linking to known lump configurations.

Contribution

It introduces new exact solutions to semilocal Popov equations with multiple flavors, transforming them into semilocal Liouville equations and expressing solutions through rational functions.

Findings

Found several families of exact solutions.

Solutions expressed in terms of rational functions.

Some solutions reduce to $CP^{M-1}$ lump configurations.

Abstract

In 2+1 dimensional nonrelativistic Chern-Simons gauge theories on $S^2$ which has a global $SU(M)$ symmetry, the semilocal Popov vortex equations are obtained as Bogomolny equations by minimizing the energy in the presence of a uniform external magnetic field. We study the equations with many flavors and find several families of exact solutions. The equations are transformed to the semilocal Liouville equations for which some exact solutions are known. In this paper, we find new exact solutions of the semilocal Liouville equations. Using these solutions, we construct solutions to the semilocal Popov equations. The solutions are expressed in terms of one or more arbitrary rational functions on $S^2$. Some simple solutions reduce to $CP^{M-1}$ lump configurations.