The conformal analog of Calabi–Yau manifolds

Abstract

This survey intends to introduce the reader to holonomy theory of Cartan connections. Special attention is given to the normal conformal Cartan connection, uniquely defined for a class of conformally equivalent metrics, and to its holonomy group – the ‘conformal holonomy group’. We explain the relation between conformal holonomy group and existence of Einstein metrics in the conformal class as well as the relation between conformal holonomy group and existence of conformal Killing spinors. In particular, we describe Lorentzian manifolds with conformal holonomy group in SU(1, m), which can be viewed as conformal analog of Calabi–Yau manifolds. Such Lorentzian metrics, known as Fefferman metrics, appear on S1-bundles over strictly pseudoconvex CR spin manifolds.