A Clifford module bundle is a mathematical construct that arises in the context of differential geometry and representation theory, particularly in relation to spin geometry and the manipulation of spinors. To understand what a Clifford module bundle is, let's break this down into a few components: 1. **Clifford Algebras:** A Clifford algebra is an algebra that is generated by a vector space equipped with a quadratic form.
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A Clifford module bundle is a vector bundle over a Riemannian manifold with a fibrewise Clifford multiplication. A compatible connection on a vector bundle satisfies for the Levi-Civita connection on covectors. On a graded Clifford module bundle, Clifford multiplication exchanges the two grading summands. The associated Dirac operator is . Its fibres are modules over the fibres of the Clifford algebra bundle.