Holonomy and Skyrme's model

TL;DR

This paper explores the role of holonomy in generalized Skyrme models, analyzing configuration spaces of fields and flat connections, and establishing the existence of minimizers for various invariants.

Contribution

It introduces new invariants for fields with finite Skyrme energy, characterizes their configuration spaces, and emphasizes the significance of holonomy in these models.

Findings

Path components of configuration spaces are described.

Invariants are well-defined for non-smooth fields with finite energy.

Existence of minimizers for all invariant values is proven.

Abstract

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact three-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat $ L^2_{loc} $ connection; the local developing maps for such connections need not be continuous.