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  "body": "\n## Definition\n\nLet $(M, g)$ be an oriented Riemannian manifold of dimension $n \\ge 2$ with a [[spin structure|RSPZ9TCFQ8]] $(P, \\xi)$, where $P$ is a principal $\\operatorname{Spin}(n)$-bundle.\nLet $\\Sigma_n$ be a complex vector space of dimension $2^{\\floor{n/2}}$ that is an irreducible module over the complexified [[Clifford algebra|A38B1QFK20]] $\\mathbb{C}\\ell_n$.\nLet $\\kappa \\colon \\operatorname{Spin}(n) \\to GL(\\Sigma_n)$ be the restriction of this module structure to $\\operatorname{Spin}(n) \\subset \\mathbb{C}\\ell_n$.\nIt is the complex spin representation of the spin group.\nThe *spinor bundle* of $(M, g, P, \\xi)$ is the associated complex vector bundle\n$$S = P \\times_\\kappa \\Sigma_n = (P \\times \\Sigma_n) / \\sim, \\qquad (p \\cdot a, v) \\sim (p, \\kappa(a) v) \\text{ for } a \\in \\operatorname{Spin}(n).$$\n\n## Remarks\n\nThe bundle $S$ has rank $2^{\\floor{n/2}}$.\nClifford multiplication of $\\mathbb{C}\\ell_n$ on $\\Sigma_n$ gives a bundle map $TM \\otimes S \\to S$, $X \\otimes \\psi \\mapsto X \\cdot \\psi$, with $X \\cdot X \\cdot \\psi = -g(X, X) \\psi$.\nThe Levi-Civita connection lifts to a connection on $S$, and the Dirac operator is $D\\psi = \\sum_i e_i \\cdot \\nabla_{e_i} \\psi$ for a local orthonormal frame $e_1, \\dots, e_n$.\nIf $n$ is even, $S = S^+ \\oplus S^-$ splits into half-spinor bundles of rank $2^{n/2 - 1}$.\n\nExample: on $\\R^n$ with its unique spin structure $P = \\R^n \\times \\operatorname{Spin}(n)$, the spinor bundle is the trivial bundle $\\R^n \\times \\Sigma_n$, and spinor fields are maps $\\R^n \\to \\C^{2^{\\floor{n/2}}}$.\n\nNon-example: the bundle $P \\times_{\\kappa'} \\C^n$ for the representation $\\kappa'$ of $\\operatorname{Spin}(n)$ through $\\rho \\colon \\operatorname{Spin}(n) \\to SO(n) \\subset GL(\\C^n)$ is the complexified tangent bundle $TM \\otimes \\C$, not the spinor bundle.\nThe element $-1 \\in \\operatorname{Spin}(n)$ acts as the identity in $\\kappa'$ but as $-\\id$ in the spin representation $\\kappa$.\n",
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