### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics B*,

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

**Abe homotopy classification of topological excitations under the topological influence of vortices.** / Kobayashi, Shingo; Kobayashi, Michikazu; Kawaguchi, Yuki; Nitta, Muneto; Ueda, Masahito.

Research output: Contribution to journal › Article

*Nuclear Physics B*, vol. 856, no. 2, pp. 577-606. https://doi.org/10.1016/j.nuclphysb.2011.11.003

}

TY - JOUR

T1 - Abe homotopy classification of topological excitations under the topological influence of vortices

AU - Kobayashi, Shingo

AU - Kobayashi, Michikazu

AU - Kawaguchi, Yuki

AU - Nitta, Muneto

AU - Ueda, Masahito

PY - 2012/3/11

Y1 - 2012/3/11

N2 - Topological excitations are usually classified by the nth homotopy group πn. However, for topological excitations that coexist with vortices, there are cases in which an element of πn cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of πn 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 πn. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group κn. The nth Abe homotopy group κn is defined as a semi-direct product of π1 and πn. In this framework, the action of π1 on πn is understood as originating from noncommutativity between π1 and πn. 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).

AB - Topological excitations are usually classified by the nth homotopy group πn. However, for topological excitations that coexist with vortices, there are cases in which an element of πn cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of πn 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 πn. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group κn. The nth Abe homotopy group κn is defined as a semi-direct product of π1 and πn. In this framework, the action of π1 on πn is understood as originating from noncommutativity between π1 and πn. 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).

KW - Gaseous BECs

KW - Homotopy theory

KW - Liquid crystal

KW - Superfluid helium 3

KW - Topological defect

UR - http://www.scopus.com/inward/record.url?scp=83555172341&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=83555172341&partnerID=8YFLogxK

U2 - 10.1016/j.nuclphysb.2011.11.003

DO - 10.1016/j.nuclphysb.2011.11.003

M3 - Article

AN - SCOPUS:83555172341

VL - 856

SP - 577

EP - 606

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

IS - 2

ER -