On the existence of minima in the Skyrme model

Abstract

Well-separated Skyrme solitons of arbitrary degree attract after a suitable relative rotation in space and iso-space, provided the orders of the solitons' leading multipoles do not differ by more than two. I summarise the derivation of this result, obtained jointly with Manton and Singer, and discuss to what extent its combination with earlier results of Esteban allows one to deduce the existence of minima of the Skyrme energy functional.Talk given at the workshop on Integrable Theories, Solitons and Duality Sao Paulo, July 2002.

The Interaction Energy of Well-Separated Skyrme Solitons

“…However, we have not been able to develop this observation into a mathematical proof. Further thoughts and speculations in this direction can be found in [17].…”

The inevitability of sphalerons in field theory

“…It appears that in the basic Skyrme model, the global minimum of the energy is attained in each sector C B , but there is no proof of this. Calculations suggest that it is not favourable for a Skyrme field of baryon number B to break up into infinitely separated subclusters whose baryon numbers add up to B, but these calculations have loopholes [35,36].…”

“…It appears that in the basic Skyrme model, the global minimum of the energy is attained in each sector C B , but there is no proof of this. Calculations suggest that it is not favourable for a Skyrme field of baryon number B to break up into infinitely-separated subclusters whose baryon numbers add up to B, but these calculations have loopholes [31,32]. The topology of the space of maps U : S 3 → S 3 is quite rich, so within each sector C B there are non-contractible loops and spheres of various dimensions, and as a result, there are numerous unstable solutions of the field equations, i.e.…”