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Abe homotopy classification of topological excitations under the topological influence of vortices. (English) Zbl 1246.82031

Summary: Topological excitations are usually classified by the \(n\)th homotopy group \(\pi_n\). However, for topological excitations that coexist with vortices, there are cases 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 \(^nS\) 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 \(O(n)/SO(n)\).

MSC:

82B26 Phase transitions (general) in equilibrium statistical mechanics
82D05 Statistical mechanics of gases
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82D50 Statistical mechanics of superfluids
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
55P15 Classification of homotopy type
55Q52 Homotopy groups of special spaces
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