Fermionic quantization and configuration spaces for the Skyrme and Faddeev-Hopf models

TL;DR

This paper computes the topology of configuration spaces in Skyrme and Faddeev-Hopf models, determining conditions for fermionic quantization of solitons based on the models' topological and algebraic properties.

Contribution

It provides the first detailed analysis of the fundamental group and rational cohomology of these configuration spaces, linking topology to quantization possibilities.

Findings

Fermionic quantization of Skyrmions requires target groups with symplectic or special unitary factors.

Fermionic quantization of Hopfions is always feasible.

Topological conditions for quantization are explicitly characterized.

Abstract

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is $S^2$. It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given.