Hopf instantons and the Liouville equation in target space

Abstract

We generalise recent results on Hopf instantons in a Chern--Simons and Fermion theory in a fixed background magnetic field. We find that these instanton solutions have to obey the Liouville equation in target space. As a consequence, these solutions are given by a class of Hopf maps that consist of the composition of the standard Hopf map with an arbitrary rational map.Comment: Latex file, 11 pages, no figure

New topological structures of Skyrme theory: baryon number and monopole number

“…As we have noticed, the Skyrme theory has the vacuum (29) independent ofn, and thisn adds the knot topology π 3 (S 2 ) to the vacuum. Of course, in the Skyrme theory we do not need the vacuum potential (30) to describe the vacuum. We only needn which describes the knot topology.…”

Particle creation via relaxing hypermagnetic knots

“…This class of fields and their zero modes were discussed in [18], so we have to review some of these results. In [18] the concept of Hopf maps was used, so let us briefly explain it. Hopf maps are maps S 3 → S 2 .…”

“…This implies that half-integer m corresponding to double-valued, square-root type maps G : S 2 → S 2 have to be allowed in order to take into account the cases when the number of zero modes is even. Again, the zero modes for gauge fields of the type (22) have been constructed explicitly in [18], and it was proven in [20] that these are all zero modes that exist for the given gauge fields.…”

“…(we use the complex variable z as a stereographic coordinate on S 2 ) which is indeed a map S 2 → S 2 with winding number m provided that g(0) = 0 and g(∞) = ∞, see [18]. Concretely, we assume lim…”

“…For this purpose we shall make use of some further results of [18], so let us briefly review them. It was shown in [18] that the spinor…”