### Abstract

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L_{loc}
^{2} connection; the local developing maps for such connections need not be continuous.

Original language | English (US) |
---|---|

Pages (from-to) | 97-122 |

Number of pages | 26 |

Journal | Communications in Mathematical Physics |

Volume | 240 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 2003 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*240*(1-2), 97-122. https://doi.org/10.1007/s00220-003-0901-x

**Holonomy and Skyrme's Model.** / Auckly, Dave; Kapitanski, Lev.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 240, no. 1-2, pp. 97-122. https://doi.org/10.1007/s00220-003-0901-x

}

TY - JOUR

T1 - Holonomy and Skyrme's Model

AU - Auckly, Dave

AU - Kapitanski, Lev

PY - 2003/9

Y1 - 2003/9

N2 - In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat Lloc 2 connection; the local developing maps for such connections need not be continuous.

AB - In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat Lloc 2 connection; the local developing maps for such connections need not be continuous.

UR - http://www.scopus.com/inward/record.url?scp=0242473178&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0242473178&partnerID=8YFLogxK

U2 - 10.1007/s00220-003-0901-x

DO - 10.1007/s00220-003-0901-x

M3 - Article

AN - SCOPUS:0242473178

VL - 240

SP - 97

EP - 122

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1-2

ER -