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  "body": "\n## Summary\n\nA Dirac bundle over a Riemannian manifold is a bundle of [[Clifford modules|31Q0RDPZ00]] with a metric and a connection that are compatible with Clifford multiplication.\nThese data are exactly what is needed to define a [[Dirac operator|3DXGJTEQZH]] that is formally self-adjoint and satisfies a general Bochner–Weitzenböck formula.\nSpinor bundles, the bundle of differential forms and the Clifford bundle itself are Dirac bundles.\nThe notion gives one framework for the de Rham, signature, Dolbeault and spin Dirac operators used in index theory.\n\n## Explanations\n\n### L1 human\n\nA Dirac operator needs three ingredients.\nThey are a way to multiply sections by tangent vectors with the Clifford rule, a way to measure lengths of sections, and a way to differentiate them.\nA Dirac bundle is a vector bundle carrying all three, fitted together so that the multiplication preserves lengths and the differentiation respects both.\nFor example, differential forms with multiplication $v \\cdot \\omega = v^\\flat \\wedge \\omega - \\iota_v \\omega$ form a Dirac bundle.\nIts Dirac operator is $d + d^*$, the square root of the Hodge Laplacian.\n",
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