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

Topological excitations are usually classified by the nth homotopy group π_n. However, for topological excitations that coexist with vortices, there are cases 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 π₁ on π_n. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group κ_n. The nth Abe homotopy group κ_n is defined as a semi-direct product of π₁ and π_n. In this framework, the action of π₁ on π_n is understood as originating from noncommutativity between π₁ 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 Sⁿ/K, where Sⁿ 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).