Existence of Hyperbolic Calorons

TL;DR

This paper proves the existence of hyperbolic calorons by constructing specific vortex solutions in an Abelian Higgs model, extending the understanding of self-dual Yang--Mills solutions on hyperbolic spaces.

Contribution

It establishes the existence of minimal action charge-$N$ calorons on hyperbolic space through explicit vortex solution construction, linking gauge theory and vortex equations.

Findings

Existence of hyperbolic calorons with arbitrary charge $N$

Construction of prescribed vortex solutions of Witten type equations

Extension of self-dual Yang--Mills solutions to hyperbolic geometry

Abstract

Recent work of Harland shows that the $SO(3)$-symmetric, dimensionally-reduced, charge-$N$ self-dual Yang--Mills calorons on the hyperbolic space $ H^3\times S^1$ may be obtained through constructing $N$-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper we establish the existence of such minimal action charge-$N$ calorons by constructing arbitrarily prescribed $N$-vortex solutions of the Witten type equations.