Explicit ambient metrics and holonomy

TL;DR

This paper constructs explicit examples of conformal structures with special ambient metrics, revealing new cases with holonomy groups like G_2 and Spin(4,3), advancing understanding of ambient metric holonomy.

Contribution

It provides explicit solutions to Ricci-flat ambient metric equations for specific conformal structures, including pp-waves and distributions, linking them to exceptional holonomy groups.

Findings

Explicit Ricci-flat ambient metrics for conformal structures

Examples with holonomy G_2 and Spin(4,3)

Solutions include conformal pp-waves and distributions

Abstract

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\mathbf{G}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.