Irreducible Ginzburg–Landau Fields in Dimension 2

Article

Abstract

Ginzburg–Landau fields are the solutions of the Ginzburg–Landau equations which depend on two positive parameters, \(\alpha \) and \(\beta \). We give conditions on \(\alpha \) and \(\beta \) for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in \(\mathbb {R}^2\), spheres, tori, etc.) with de Gennes–Neumann boundary conditions. We also prove that, for each such manifold and all positive \(\alpha \) and \(\beta \), the Ginzburg–Landau free energy is a Palais–Smale function on the space of gauge equivalence classes; Ginzburg–Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg–Landau fields is compact.

Keywords

Ginzburg–Landau equations Gauge theory Moduli spaces 

Mathematics Subject Classification

70S15 35Q56 58J32 

Notes

Acknowledgements

I wish to thank my advisor, Tom Parker, for his advice during the preparation of this paper. I greatly benefited from the discussions with Paul Feehan and Manos Maridakis about the Łojasiewicz–Simon inequality. I am also grateful for the help of Benoit Charbonneau.

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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