Calabi-Yau Monopoles for the Stenzel Metric

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 $T^{\ast }{\mathbb{S}}^3$ 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.

References in corpus (2)

• Asymptotically conical Calabi-Yau manifolds, I
• Monopoles on the Bryant-Salamon $G_2$ Manifolds

Cited by in corpus (4)

• $SU(2{)}^2$-invariant $G_2$-instantons
• $SU(2{)}^2$-invariant gauge theory on asymptotically conical Calabi-Yau 3-folds
• Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds
• The DT-instanton equation on almost Hermitian 6-manifolds