Methods of holonomy theory for Ricci-flat Riemannian manifolds

Abstract

Compact, Ricci-flat Riemannian manifolds often arise in physical applications, either as a technical device or as models of ‘‘internal’’ space. The idea of extending the holonomy group of such a manifold to a larger gauge group (‘‘embedding the connection in the gauge group’’) plays a fundamental role in the ‘‘manifold compactification’’ approach to superstring phenomenology, and the work of Gepner suggests that this idea may have equally fundamental analogs in other approaches. The holonomy theory of simply connected Ricci-flat manifolds has recently been the subject of much mathematical work, but physicists are mainly interested in the case of multiply connected manifolds. The purpose of this paper is to present some techniques for understanding the holonomy theory of compact, multiply connected Ricci-flat manifolds. These lead to a general classification theorem.