Instantons on sine-cones over Sasakian manifolds

TL;DR

This paper explores the geometry of sine-cones over Sasakian manifolds, showing they admit special torsion structures and constructing instanton solutions for Yang-Mills fields in these complex, higher-dimensional spaces.

Contribution

It demonstrates that sine-cones over Sasaki-Einstein and 3-Sasakian manifolds are KT and HKT manifolds, and constructs explicit instanton solutions on these spaces.

Findings

Sine-cones over Sasaki-Einstein manifolds are KT manifolds.

Sine-cones over 3-Sasakian manifolds can be deformed to HKT structures.

Explicit instanton solutions are constructed on these conical manifolds.

Abstract

We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are K"ahler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-K"ahler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-K"ahler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.