Exotic Holonomies E

It is proved that the Lie groups E (5) 7 and E (7) 7 represented in R 56 and the Lie group E C 7 represented in R 112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connnections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension. §1 Introduction The notion of the Holonomy of an affine connection was introduced originally byÉlie Cartan in the 1920s who used it as an important tool in his attempt to classify all locally symmetric manifolds. Over time, the holonomy group proved to be one of the most informative and useful characteristics of an affine connection and found many applications in both mathematics and physics. By definition, the holonomy of an affine connection on a connected manifold M is the subgroup of all linear automorphisms of T p M which are induced by parallel translation along p-based loops. In 1955, Berger [4] showed that the list of irreducibly acting matrix Lie groups which can, in principle, occur as the holonomy of a torsion-free affine connection is very restricted. Berger presented his classification of all possible candidates for irreducible holonomies in two parts. The first part contains all possible groups preserving a non-degenerate symmetric bilinear form, the second part consists of those groups which do not preserve such a form; the latter part was stated to be complete up to a finite number of missing terms and was given without a proof. Bryant [5] was the first to discover the incompleteness of the second part of Berger's list, and referred to the missing entries as exotic holonomies. Since then, several other families of exotic holonomies have been found [6, 7, 8]. In this paper we present one more family of exotic holonomies associated with various real forms of the complex 56-dimensional representation of E C 7 .