Spin$^o$ structures and semilinear (s)pinor bundles

C. I. Lazaroiu, C. S. Shahbazi

Published 2018-09-24Version 1

We introduce and explore a new type of extended spinorial structure, based on the groups $\mathrm{Spin}^o_+(V,h)=\mathrm{Spin}(V,h)\cdot\mathrm{Pin}_{2,0}$ and $\mathrm{Spin}^o_-(V,h)=\mathrm{Spin}(V,h)\cdot \mathrm{Pin}_{0,2}$, which can be defined on unorientable pseudo-Riemannian manifolds $(M,g)$ of any dimension $d$ and any signature $(p,q)$. In signature $p-q\equiv_8 3,7$, $(M,g)$ admits a bundle $S$ of irreducible real Clifford modules if and only if the orthonormal coframe bundle of $(M,g)$ admits a reduction to a $\mathrm{Spin}^{o}_{\alpha}$ structure with $\alpha = -1$ if $p-q \equiv_{8} 3$ or $\alpha = +1$ if $ p-q \equiv_{8} 7$. We compute the topological obstruction to existence of such structures, obtaining as a result the topological obstruction for $(M,g)$ to admit a bundle of irreducible real Clifford modules $S$ of complex type. We explicitly show how $\mathrm{Spin}^o_{\alpha}$ structures can be used to construct $S$ and give geometric characterizations (in terms of certain associated bundles) of the conditions under which the structure group of $S$ (viewed as a vector bundle) reduces to certain natural subgroups of $\mathrm{Spin}^o_{\alpha}$. Finally, we discuss some examples of manifolds admitting $\mathrm{Spin}^{o}_{\alpha}$ structures which are constructed using real Grassmannians and certain codimension two submanifolds of spin manifolds.