Abstract
Topological excitations are usually classified by the nth homotopy group Ï n . However, for topological excitations that coexist with vortices, there are cases in which an element of Ï n cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of Ï 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 Ï 1 on Ï n . In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group Îș n . The nth Abe homotopy group Îș n is defined as a semi-direct product of Ï 1 and Ï n . In this framework, the action of Ï 1 on Ï n is understood as originating from noncommutativity between Ï 1 and Ï 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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