### Abstract

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 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 nth 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).

Original language | English |
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Pages (from-to) | 577-606 |

Number of pages | 30 |

Journal | Nuclear Physics B |

Volume | 856 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Mar 11 |

### Keywords

- Gaseous BECs
- Homotopy theory
- Liquid crystal
- Superfluid helium 3
- Topological defect

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Nuclear Physics B*,

*856*(2), 577-606. https://doi.org/10.1016/j.nuclphysb.2011.11.003