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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]], and let $\\mathrm{Cl}(M)$ be its Clifford bundle, whose fibre at $x$ is the [[Clifford algebra|A38B1QFK20]] of $(T_x M, g_x)$ with $v \\cdot v = -g(v, v)$.\nThe [[Levi-Civita connection|CAZYGE1T4E]] induces a connection on $\\mathrm{Cl}(M)$.\nA *Dirac bundle* over $M$ is a real or complex vector bundle $S \\to M$ with these data and properties.\n\n1. $S$ is a bundle of left [[Clifford modules|31Q0RDPZ00]] over $\\mathrm{Cl}(M)$.\nThat is, a smooth bundle map $\\mathrm{Cl}(M) \\otimes S \\to S$, $\\varphi \\otimes \\psi \\mapsto \\varphi \\cdot \\psi$, makes each fibre $S_x$ a module over $\\mathrm{Cl}(T_x M)$.\n2. $S$ has a Riemannian or Hermitian metric $\\ip{\\cdot}{\\cdot}$ such that $\\ip{e \\cdot \\psi_1}{e \\cdot \\psi_2} = \\ip{\\psi_1}{\\psi_2}$ for every unit vector $e \\in T_x M$ and all $\\psi_1, \\psi_2 \\in S_x$.\n3. $S$ has a [[connection|5ZYRRVE8BW]] $\\nabla$ that preserves this metric.\n4. $\\nabla_X(\\varphi \\cdot \\psi) = (\\nabla_X \\varphi) \\cdot \\psi + \\varphi \\cdot \\nabla_X \\psi$ for every vector field $X$, every section $\\varphi$ of $\\mathrm{Cl}(M)$ and every section $\\psi$ of $S$.\n\nThe *Dirac operator* of $S$ is $D \\psi = \\sum_{i=1}^n e_i \\cdot \\nabla_{e_i} \\psi$ for a local orthonormal frame $e_1, \\ldots, e_n$.\n\n## Remarks\n\nClause 2 is equivalent to the skew-adjointness of Clifford multiplication by each tangent vector.\nClauses 2 to 4 make $D$ formally self-adjoint, and its square satisfies $D^2 = \\nabla^* \\nabla + \\mathfrak{R}$ for an explicit curvature endomorphism $\\mathfrak{R}$.\nThe [[spinor bundle|PYYQQKAZ0C]] of a spin manifold, with the spin connection, is a Dirac bundle, and its Dirac operator is the Atiyah–Singer operator.\n\nExample: let $S = \\Lambda^* T^*M$ with the metric and connection induced by $g$, and let $v \\cdot \\omega = v^\\flat \\wedge \\omega - \\iota_v \\omega$.\nThen $v \\cdot v \\cdot \\omega = -(v^\\flat \\wedge \\iota_v \\omega + \\iota_v(v^\\flat \\wedge \\omega)) = -g(v, v) \\, \\omega$, so clause 1 holds, and clauses 2 to 4 hold because the Levi-Civita connection is metric and commutes with wedge and contraction.\nIts Dirac operator is $\\sum_i (e_i^\\flat \\wedge \\nabla_{e_i} - \\iota_{e_i} \\nabla_{e_i}) = d + \\delta$, whose square is the [[Hodge Laplacian|PWG3GKZG80]].\n\nNon-example: on $M = \\R^n$ with $n \\ge 2$, take the same bundle and Clifford multiplication, but the connection $\\nabla'_X = \\nabla_X + dx_1(X) \\, A$, where $A$ is Clifford multiplication by $e_1 = \\partial_1$.\nSince $A$ is skew-adjoint, $\\nabla'$ still preserves the metric, so clause 3 holds.\nClause 4 fails: for $X = e_1$ and $\\varphi = e_2$, it would require $A(e_2 \\cdot \\psi) = e_2 \\cdot A\\psi$, but $e_1 e_2 = -e_2 e_1 \\ne e_2 e_1$.\n\n## Explanations\n\n### L1 human\n\nA Dirac bundle is a bundle of Clifford modules with a metric and a connection that are compatible with Clifford multiplication, so that it carries a well-behaved Dirac operator.\n",
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