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  "body": "\n## Summary\n\nA Clifford module is a module over a [[Clifford algebra|A38B1QFK20]].\nIt is a vector space on which the vectors of a quadratic space act as linear maps.\nThe square of each such map is a scalar fixed by the quadratic form: minus its value, or its value, depending on the sign convention.\nSpinor spaces are the irreducible Clifford modules.\nBundles of Clifford modules over Riemannian manifolds carry Dirac operators, which makes Clifford modules the algebraic input of index theory.\nTheir classification shows a periodicity of period $8$ over the reals and $2$ over the complex numbers, closely related to Bott periodicity.\n\n## Explanations\n\n### L1 human\n\nThe complex numbers are the real plane with a rule saying that $i \\cdot i = -1$.\nA Clifford module generalises this: several directions $e_1, \\ldots, e_n$ act on a space, each one squaring to $-1$, and different ones anticommuting.\nFor one direction, the plane $\\C$ with $e_1$ acting as multiplication by $i$ is a Clifford module.\nFor two directions, the quaternions with $e_1$ and $e_2$ acting as left multiplication by $i$ and $j$ form one.\nThese modules are the spaces on which spinors live.\n",
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