Parallel Spinors and Holonomy
by
Ines Kath
Max-Planck-Institut für Mathematik

For a semi-Riemannian manifold of signature (p, q) the parallel transport along all loops at a fixed point generates the so-called holonomy group, which is a subgroup of O(p, q). The existence of parallel objects on the manifold implies algebraic conditions for this group and vice versa. We will study this relation for spinor fields. The result is well-known if the manifold is simply connected, irreducible and not locally symmetric. It is due to McK. Wang in the Riemannian case and was generalized by H. Baum and the author to the pseudo-Riemannian situation. Such a manifold admits parallel spinors if and only if the holonomy group is SU(r, s), Sp(r, s), G₂, G₂^*, G₂^C, Spin(7), Spin⁺(4, 3) or Spin(7)^C. In all these cases the number, the chirality and the causality of the parallel spinors are known. Furthermore, a Riemannian symmetric space cannot admit parallel spinors. Using de Rham's Theorem we obtain a description of Riemannian manifolds admitting parallel spinor fields in terms of the holonomy group. On the other hand the pseudo-Riemannian situation is much more complicated. Here de Rham's theorem does not reduce the study of manifolds to that of irreducibles ones. It is possible that the holonomy representation is reducible but does not decompose into irreducible holonomy representations of semi-Riemannian manifolds. Examples are manifolds with parallel orthogonal optical structures. We will explain the relation between these structures and the existence of parallel pure spinor fields. We will specialize this to signature (m, m) or (m, m+1) and will give a normal form for metrics with real parallel pure spinors in this case. Other results which will be presented concern a further pseudo-Riemannian phenomenon. Contrary to the Riemannian situation there are pseudo-Riemannian symmetric spaces which admit parallel spinors. We will study double extensions of Lie groups which yield a big class of examples for such symmetric spaces.

Date received: March 30, 2001