A Kahler-Einstein inspired anzatz for Spin(7) holonomy metrics and its solution

TL;DR

This paper introduces a new ansatz for Spin(7) metrics inspired by Kahler-Einstein geometry, leading to explicit solutions that reveal connections to Calabi-Yau, hyperkahler, and tri-Sassakian structures.

Contribution

It proposes a novel ansatz for Spin(7) metrics based on G2 structures and derives explicit solutions linking to well-known geometric cones and bundles.

Findings

Derived a specific Spin(7) metric solution as a Calabi-Yau cone.

Identified the metric as a hyperkahler Swann bundle.

Calculated torsion classes for the G2 structure.

Abstract

We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor space of any compact quaternion Kahler space is Kahler-Einstein. The reduction of the holonomy to a subgroup of Spin(7) gives non linear system relating three unknown functions of one variable. We obtain a particular solution and we find that the resulting metric is a Calabi-Yau cone over an Einstein-Sassaki manifold which means that the holonomy is reduced to SU(4). Another coordinate change show us that our metrics are hyperkahler cones known as Swann bundles, thus the holonomy is reduced to Sp(2) and the cone is tri-Sassakian. We revert our argument and state that the Swann bundle define a closed G2 structure by reduction along an isometry. We calculate the torsion classes for such structure explicitly.