Rational Maps, Monopoles and Skyrmions

TL;DR

This paper explores the connection between BPS monopoles and Skyrmions through rational maps, introducing a new ansatz for Skyrme fields to approximate known and new Skyrmions, and analyzing their vibrational modes.

Contribution

It introduces a novel rational map-based ansatz for Skyrme fields, enabling accurate approximations of Skyrmions up to baryon number nine and some new configurations, enhancing understanding of their vibrational modes.

Findings

Constructed approximations for all minimal energy Skyrmions up to baryon number nine.

Discovered a new baryon number seventeen Skyrme field with a buckyball structure.

Identified a Morse function on the space of rational maps relevant to monopole moduli spaces.

Abstract

We discuss the similarities between BPS monopoles and Skyrmions, and point to an underlying connection in terms of rational maps between Riemann spheres. This involves the introduction of a new ansatz for Skyrme fields. We use this to construct good approximations to several known Skyrmions, including all the minimal energy configurations up to baryon number nine, and some new solutions such as a baryon number seventeen Skyrme field with the truncated icosahedron structure of a buckyball. The new approach is also used to understand the low-lying vibrational modes of Skyrmions, which are required for quantization. Along the way we discover an interesting Morse function on the space of rational maps which may be of use in understanding the Sen forms on the monopole moduli spaces.