On Spin(7) holonomy metric based on SU(3)/U(1)

TL;DR

This paper constructs and analyzes new $Spin(7)$ holonomy metrics with cohomogeneity one, revealing regular solutions and their topological features, including asymptotic behaviors and harmonic forms, based on the choice of a $U(1)$ subgroup.

Contribution

It introduces a family of $Spin(7)$ holonomy metrics with a manifest symmetry and explores conditions for regularity and topology based on subgroup choices.

Findings

Discovery of a regular $Spin(7)$ metric with specific $U(1)$ subgroup

Identification of asymptotically locally conical (ALC) $Spin(7)$ metrics

Existence of an $L^2$-normalisable harmonic 4-form

Abstract

We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest $\Sigma_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space ${\bf CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric.