The vortex: complex Hopf bundle and Morse theory

Abstract

The authors interpret the vortex solution of Nielsen-Olesen (1973) as a complex vector bundle associated to the second Hopf sphere bundle (analogously to consider the kink of the first Hopf bundle); the peculiarity of the soliton behaviour of the two-dimensional vortex stems from the non-trivial character of the fibration; the electromagnetic and scalar (Higgs) field are the connection and the section in this bundle respectively. Properties of this mathematical construction have their natural physical translation; for example the complex structure of the sphere S2 leads to a closed (Kahler) two-form, which has physical implications, and the fact that the vortex can be considered as the square root of the tangent bundle to the sphere implies a spinor nature for the vortex. A Morse theory of critical points suggests some Atiyah-Singer type of theorems, which have bearing on the stability of the multi-vortex solutions. They finish by a geometrical interpretation of the fractional charges found recently.