Abe homotopy classification of topological excitations under the topological influence of vortices Original Research Article

Topological excitations are usually classified by the nth homotopy group image. However, for topological excitations that coexist with vortices, there are cases in which an element of image cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of image 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 image on image. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group image. The nth Abe homotopy group image is defined as a semi-direct product of image and image. In this framework, the action of image on image is understood as originating from noncommutativity between image and image. 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 image, where image is an n-dimensional sphere and K is a discrete subgroup of image. We show that the influence of vortices on a topological excitation exists only if n is even and K includes a nontrivial element of image.