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  "body": "\n## Summary\n\nThe spinor bundle of a spin manifold is the complex [[vector bundle|P4DZM975BV]] associated with its [[spin structure|RSPZ9TCFQ8]] and the spin representation of the [[spin group|JJSXXJQGEH]].\nIts sections are the spinor fields.\nIt carries a Hermitian metric, a connection induced by the Levi-Civita connection and a Clifford multiplication by tangent vectors, and these define the Dirac operator.\n\n## Explanations\n\n### L1 human\n\nOn a spin manifold, each point has a small complex vector space of spinors, and these spaces fit together into a bundle.\nA tangent vector acts on spinors, and acting twice by a unit vector gives minus the identity.\nOn flat space $\\R^4$ the spinor bundle is the trivial bundle with fibre $\\C^4$, matching the four complex components of a spinor in Dirac's equation for the electron.\n",
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