 Mass-deformed spherical sigma model

A spherical sigma model in polar fields $\varphi ,\alpha$ has kinetic metric $r(d{\varphi }^2+{\sin }^2\varphi d{\alpha }^2)/2$ in the usual one-half kinetic convention. The displayed positive potential selects the two polar vacua. For $r>0$ it supports static and rotating charged kinks, and a topological theta term shifts their canonical internal charge without changing the bulk equations.