Let \(S\) be a real vector bundle over a connected pseudo-Riemannian manifold \((M,g)\) of signature \((p,q)\) and dimension \(d\). A Dirac operator \(D:\Gamma(S) \rightarrow \Gamma(S)\) is a first-order differential operator such that \(D^{2}\) has principal symbol: \[ \sigma(m,\xi)=g(\xi,\xi)\mathrm{Id}_{S} \ \ \ \ x\in M,\ \ \xi \in T^{*}M \ . \] The symbol of \(D\) induces a morphism of bundles \(\gamma:\mathrm{Cl}(M,g) \rightarrow \mathrm{End}(S)\) so that \((S, \gamma)\) becomes a bundle of irreducible real spinors.
Existence of irreducible real spinors on a bundle \(S\) is obstructed. The paper under review shows that, if \(p-q \equiv_{8} 3,7\), a bundle \(S\) has an irreducible Dirac operator if and only if \((M,g)\) admits an adapted \(\mathrm{Spin}^{0}\) structure. This is a principal \(\mathrm{Spin}^{0}_{p,q}\)-bundle \(Q\) over \(M\) endowed with a \(\tilde{\lambda}\)-equivariant map to the orthonormal coframe bundle of \((M,g)\), where \[ \mathrm{Spin}^{0}_{p,q}=\begin{cases} \mathrm{Spin}_{p,q}\mathrm{Pin}_{0,2} \ \ \ \ \ \ \text{if }p-q \equiv_{8} 3 \\ \mathrm{Spin}_{p,q}\mathrm{Pin}_{2,0} \ \ \ \ \ \ \text{if }p-q \equiv_{8} 7 \end{cases} \] and \(\tilde{\lambda}:\mathrm{Spin}^{0}_{p,q}\rightarrow \mathrm{0}(p,q)\) is a certain group homomorphism.

The authors characterize pseudo-Riemannian manifolds that admit an adapted \(\mathrm{Spin}^{0}_{p,q}\) structure in terms of the existence of principal \(\mathrm{O}(2)\)-bundles with certain Karoubi Stiefel-Whiteney classes.