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

✍ Scribed by Hiroaki Kanno; Yukinori Yasui

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 148 KB
• Volume: 43
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(02)00021-9

✦ Synopsis

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U (1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first-order system of Spin(7) instanton equation. One is the flag manifold SU(3)/T 2 also known as the twistor space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distinguished by the choice of S 1 circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of the possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyper-Kähler metric, the other is a new family of metrics on a line bundle over the twistor space of CP(2).