We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin structures based on certain groups which we call real Lipschitz groups. In the fiberwise-irreducible case, we classify the latter in all dimensions and signatures. As a simple application, we show that the supersymmetry generator of eleven-dimensional supergravity in "mostly plus" signature can be interpreted as a global section of a bundle of irreducible Clifford modules if and $\textit{only if}$ the underlying eleven-manifold is orientable and spin.

中文翻译：

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实旋量丛和实Lipschitz结构

我们获得了在任意维度和签名的伪黎曼流形 $(M,g)$ 上存在不可约实 Clifford 模块的拓扑障碍，并证明了 Clifford 模块束与所谓的实 Lipschitz 结构相关联。后者给出了基于某些群的自旋结构的概括，我们称之为真正的 Lipschitz 群。在纤维不可约的情况下，我们在所有维度和特征中对后者进行分类。作为一个简单的应用，我们展示了“主要加”签名中的 11 维超引力的超对称生成器可以解释为一组不可约的 Clifford 模块的全局部分 if 和 $\textit{only if}$流形是可定向和自旋的。