Spinor bundle

Remarks

The bundle $S$ has rank $2^{\floor{n/2}}$. Clifford 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 $X \cdot X \cdot \psi = -g(X, X) \psi$. The 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 $e_1, \dots, e_n$. If $n$ is even, $S = S^+ \oplus S^-$ splits into half-spinor bundles of rank $2^{n/2 - 1}$.

Example: 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 $\R^n \to \C^{2^{\floor{n/2}}}$.

Non-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. The element $-1 \in \operatorname{Spin}(n)$ acts as the identity in $\kappa'$ but as $-\id$ in the spin representation $\kappa$.