Coset Construction of $Spin(7), G_2$ Gravitational Instantons

TL;DR

This paper constructs explicit Ricci-flat metrics with $Spin(7)$ and $G_2$ holonomy on certain non-compact manifolds using coset space methods and vector fields, expanding the known examples of special holonomy metrics.

Contribution

It introduces a coset construction approach to explicitly derive $Spin(7)$ and $G_2$ holonomy metrics on specific non-compact manifolds, providing new explicit solutions.

Findings

Explicit volume-preserving vector fields on $Sp(2)/Sp(1)$ and $SU(3)/U(1)$

Differential equations characterizing $Spin(7)$ and $G_2$ holonomy metrics

Development of a dual formulation to self-duality conditions for spin connections.

Abstract

We study Ricci-flat metrics on non-compact manifolds with the exceptional holonomy $Spin(7), G_2$. We concentrate on the metrics which are defined on ${\bf R} \times G/H$. If the homogeneous coset spaces $G/H$ have weak $G_2$, SU(3) holonomy, the manifold ${\bf R} \times G/H$ may have $Spin(7), G_2$ holonomy metrics. Using the formulation with vector fields, we investigate the metrics with $Spin(7)$ holonomy on ${\bf R}\times Sp(2)/Sp(1), {\bf R}\times SU(3)/U(1)$. We have found the explicit volume-preserving vector fields on these manifold using the elementary coordinate parameterization. This construction is essentially dual to solving the generalized self-duality condition for spin connections. We present most general differential equations for each coset. Then, we develop the similar formulation in order to calculate metrics with $G_2$ holonomy