A spinor bundle is a specific type of vector bundle that arises in the context of differential geometry and the theory of spinors, particularly in relation to Riemannian and pseudo-Riemannian manifolds. Here’s a more in-depth explanation: ### Context In the study of geometrical structures on manifolds, one often encounters vector bundles, which are collections of vector spaces parameterized by the points of a manifold.
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A chosen spin structure on a spin manifold defines the complex spinor bundle associated to the spin representation. In dimension its vector bundle rank is . Its chirality matrix is in the negative Clifford multiplication convention. It satisfies and anticommutes with each , giving and an odd Dirac operator. In odd dimension there is no such canonical chiral splitting of the irreducible spinor bundle.