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

Topological excitations are usually classified by the $n$th homotopy group $π_n$. However, for topological excitations that coexist with vortices, there are case in which an element of $π_n$ cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of $π_n$ 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 $π_1$ on $π_n$. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group $κ_n$. The $n$th Abe homotopy group $κ_n$ is defined as a semi-direct product of $π_1$ and $π_n$. In this framework, the action of $π_1$ on $π_n$ is understood as originating from noncommutativity between $π_1$ and $π_n$. 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 $O(n)/SO(n)$.

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