Folding in the Skyrme model

There are only three stable singularities of a differentiable map between three-dimensional manifolds, namely folds, cusps and swallowtails. A Skyrme configuration is a map from space to $SU2,$ and its singularities correspond to the points where the baryon density vanishes. In this article we consider the singularity structure of Skyrme configurations. The Skyrme model can only be solved numerically. However, there are good analytic ansätze. The simplest of these, the rational map ansatz, has a nongeneric singularity structure. This leads us to introduce a nonholomorphic ansatz as a generalization. For baryon numbers 2, 3, and 4, the approximate solutions derived from this ansatz are closer in energy to the true solutions than any other ansatz solution. We find that there is a tiny amount of negative baryon density for baryon number 3, but none for 2 or 4. We comment briefly on the relationship to Bogomolny–Prasad–Sommerfield monopoles.