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

TL;DR

This paper explores new Spin(7) holonomy metrics with cohomogeneity one, identifying exact solutions with SU(4) holonomy and analyzing asymptotic behaviors, including asymptotically conical and locally conical metrics.

Contribution

It introduces a two-parameter family of exact Spin(7) metrics with SU(4) holonomy and classifies asymptotic geometries, expanding the known landscape of special holonomy spaces.

Findings

Found exact SU(4) holonomy solutions with asymptotically conical geometry.

Identified two types of asymptotically locally conical metrics distinguished by circle stabilization.

Numerical evidence supports existence of two families of Spin(7) metrics, including deformations of Calabi hyperKahler and new line bundle metrics.

Abstract

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 twister 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 distingushed by the choice of $S^1$ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of 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 hyperKahler metric, the other is a new family of metrics on a line bundle over the twister space of CP(2).