On sections of complex line bundles over surfaces minimizing a Ginzburg-Landau energy

TL;DR

This paper extends Ginzburg-Landau vortex analysis from tangent vector fields on surfaces to sections of complex line bundles, revealing vortex locations for Q-tensors and higher-rank analogs on spheres.

Contribution

It generalizes existing results to complex hermitian line bundles and identifies vortex locations for advanced tensor fields on curved surfaces.

Findings

Vortex locations are characterized for Q-tensors on spheres.

Extension of vortex analysis to higher-rank tensor fields.

Results applicable to complex line bundle sections on Riemannian surfaces.

Abstract

In this work we extend some of the results of Ignat and Jerrard for Ginzburg-Landau vortices of tangent vector fields on two-dimensional Riemannian manifolds to the setting of complex hermitian line bundles. In particular, we elucidate the locations of vortices for the cases of Q-tensors and their higher-rank analogs on a sphere.