Calabi–Yau Monopoles for the Stenzel Metric

Abstract

We construct the first nontrivial examples of Calabi–Yau monopoles. Our main interest in these comes from Donaldson and Segal’s suggestion (Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol 16, pp 1–41, 2011) 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 that 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.