Dirac operators on real spinor bundles of complex type

Let (M,g) be a pseudo-Riemannian manifold of signature (p,q). We compute the obstruction for a vector bundle S over (M,g) to admit a Dirac operator whose principal symbol induces on S the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible complex type. In order to do this, we use the theory of Lipschitz structures in signature p−q≡83,7 to reformulate the problem as the obstruction problem for (M,g) to admit a Spinαo structure with α=−1 if p−q≡83 or α=+1 if p−q≡87, where Spin+o(p,q)=Spin(p,q)⋅Pin2,0 and Spin−o(p,q)=Spin(p,q)⋅Pin0,2. This allows computing the obstruction in terms of the Karoubi Stiefel-Whitney classes of (M,g) and the existence of an auxiliary O(2) bundle with prescribed characteristic classes. Furthermore, we explicitly show how a Spinαo structure can be used to construct S and we give geometric characterizations (in terms of associated bundles) of the conditions under which the structure group of S reduces to certain natural subgroups of Spinαo. Finally, we prove that certain codimension two submanifolds of spin manifolds and certain products of tori with Grassmannians, which were not known to admit irreducible real spinor bundles, do admit Spinαo structures and therefore do admit real spinor bundles of irreducible complex type.