Recent mathematical developments in the Skyrme model

TL;DR

This paper reviews recent mathematical advances in the Skyrme model, covering geometrical aspects, analytical approximations, topology, dynamics, and semi-classical quantization, aiming to make these developments accessible to physicists.

Contribution

It provides a comprehensive overview of recent mathematical developments in the Skyrme model, including new analytical methods, topological insights, and dynamical studies.

Findings

Analytical approximation to minimum energy configurations for B=2

Insights into multibaryon minima using rational maps

Application of gradient flow method to Skyrmion dynamics

Abstract

In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the definition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy configuration in any sector of fixed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B=2. We then describe the work done on the multibaryon minima using rational maps, on the topology of the configuration space and the possible implications of Morse theory. Next we turn to recent work on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, specifically the gradient flow method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering.