Abe homotopy classification of topological excitations under the topological influence of vortices

TL;DR

This paper introduces the Abe homotopy group to classify topological excitations in the presence of vortices, accounting for their influence and noncommutative interactions, extending traditional homotopy classification methods.

Contribution

The paper proposes the Abe homotopy group as a new classification tool for topological excitations affected by vortices, incorporating noncommutative effects and vortex creation-annihilation processes.

Findings

Abe homotopy group is a semi-direct product of c6_1 and c6_n.

Physical charges are described by conjugacy classes of c6_n.

Vortex influence on excitations occurs only for even n with specific subgroup conditions.

Abstract

Topological excitations are usually classified by the $n$th homotopy group $\pi_n$. However, for topological excitations that coexist with vortices, there are case in which an element of $\pi_n$ cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of $\pi_n$ corresponding to the charge of a topological excitation may change when the topological excitation circumnavigates a vortex. This phenomenon is referred to as the action of $\pi_1$ on $\pi_n$. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group $\kappa_n$. The $n$th Abe homotopy group $\kappa_n$ is defined as a semi-direct product of $\pi_1$ and $\pi_n$. In this framework, the action of $\pi_1$ on $\pi_n$ is understood as originating from noncommutativity between $\pi_1$ and $\pi_n$. We show that a physical charge of a topological excitation can be described in terms of the conjugacy class of the Abe homotopy group. Moreover, the Abe homotopy group naturally describes vortex-pair creation and annihilation processes, which also influence topological excitations. We calculate the influence of vortices on topological excitations for the case in which the order parameter manifold is $S^n/K$, where $S^n$ is an $n$-dimensional sphere and $K$ is a discrete subgroup of $SO(n+1)$. We show that the influence of vortices on a topological excitation exists only if $n$ is even and $K$ includes a nontrivial element of $O(n)/SO(n)$.