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

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 us to 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 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-normalizable harmonic 4-form in the background of the ALC metric.