Vortices and the Abel-Jacobi map

JOURNAL OF GEOMETRY AND PHYSICS

卷:76

Article

Rink, Norman A.

关键词: Gauge theory;    Solitons;    Moduli spaces;    Riemann surfaces;    Line bundles;   

DOI  :  10.1016/j.geomphys.2013.10.017

来源: Elsevier

【 摘 要 】

The abelian Higgs model on a compact Riemann surface Sigma supports vortex solutions for any positive vortex number d is an element of Z. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of Sigma, in symbols, Sym(d)(Sigma). This moduli space is Kahler with respect to the physically motivated metric whose geodesics describe slow vortex motion. In this paper we appeal to classical properties of Sym(d)(Sigma) to obtain new results for the moduli space metric. Our main tool is the Abel-Jacobi map, which maps Sym(d)(Sigma) into the Jacobian of Sigma. Fibres of the Abel-Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini-Study metric. Additional significance is given to the fibres of the Abel-Jacobi map by our second result: we show that if Sigma is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel-Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free