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

✍ Scribed by Hiroaki Kanno; Yukinori Yasui

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 150 KB
• Volume: 43
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(02)00018-9

✦ Synopsis

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( )) (=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.