Calabi-Yau Monopoles for the Stenzel Metric

TL;DR

This paper constructs the first nontrivial examples of Calabi-Yau monopoles on the Stenzel metric, explores their moduli space parametrized by mass, and analyzes their bubbling behavior in the large mass limit.

Contribution

It provides the first explicit construction and classification of symmetric Calabi-Yau monopoles on the Stenzel metric, including their moduli and asymptotic behavior.

Findings

Monopoles are uniquely determined by their mass parameter.

Explicit construction of an irreducible $SU(2)$ Hermitian-Yang-Mills connection.

Analysis of bubbling behavior in the large mass limit.

Abstract

We construct the first nontrivial examples of Calabi-Yau monopoles. Our main interest on these, comes from Donaldson and Segal's suggestion \cite{Donaldson2009} that it may be possible to define an invariant of certain noncompact Calabi-Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the $3$-sphere $T^* \mathbb{S}^3$ and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole which is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end an irreducible $SU(2)$ Hermitian-Yang-Mills connection on the Stenzel metric is constructed explicitly.