Clifford Algebras and Their Representations

There are two approaches to spinor fields on a (pseudo-) Riemannian manifold $(M,g)$: the bundle of spinors is either defined as a bundle associated with the principal bundle of ‘spin frames’ or as a complex bundle $\Sigma \to M$ with a homomorphism $\tau :\mathcal{C}\ell \left(g\right)\to End\Sigma$ of bundles of algebras over $M$ such that, for every $x\in M$, the restriction of $\tau$ to the fiber over $x$ is equivalent to a spinor representation of a suitable Clifford algebra. By Hermitian and complex conjugation one obtains the homomorphisms ${\tau }^{†}:\mathcal{C}\ell \left(g\right)\to End{\bar{\Sigma }}^{\ast }$ and $\bar{\tau }:\mathcal{C}\ell \left(g\right)\to End\bar{\Sigma }$. These data define the bundles $\mathfrak{a}\left(\tau \right)$ and $\mathfrak{c}\left(\tau \right)$ of intertwiners of $\tau$ with ${\tau }^{†}$ and $\bar{\tau }$, respectively. It is shown that, given sections of $\mathfrak{a}\left(\tau \right)\to M$ and of $\mathfrak{c}\left(\tau \right)\to M$, any metric linear connection on $(M,g)$ defines a unique connection on the spinor bundle $\Sigma \to M$ relative to which these sections are covariantly constant. The connection defines a Dirac operator acting on sections of $\Sigma$. As an example, the trivial spinor bundle on hypersurfaces in ${\mathbb{R}}^m$ and the corresponding Dirac operator are described in detail.