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

TL;DR

This work constructs and analyzes Spin$(7)$ holonomy metrics of cohomogeneity one with principal orbit $SU(3)/U(1)$ by exploiting a free $U(1)$ in the Cartan subalgebra to reveal the Weyl symmetry $ ext{Σ}_3$. Through octonionic self-duality of the spin connection, the authors derive first-order gradient-flow equations and identify a superpotential, enabling explicit ALC gravitational instantons. A subset of these metrics avoids orbifold singularities by a special $U(1)$ embedding, resulting in a new regular Spin$(7)$ metric with ${f CP}(2)$ as a singular orbit; perturbative and global analyses show a rich family of deformations and topology controlled by embedding data $(n_1,n_2,n_3)$. They also construct an $L^2$-normalisable harmonic 4-form on the ALC background, providing a tool for supersymmetric M2-brane physics on these backgrounds. Together, the results expand the landscape of explicit Spin$(7)$ manifolds and furnish new geometric ingredients for M-theory compactifications and brane constructions.

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 $Σ_3=W(SU(3))$ (= 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.