TITLE: Examples of Clifford modules
QUESTION [17 upvotes]: For a Riemannian manifold $M$, a Clifford module is a bundle over $M$ that is, fiberwise, a representation of the Clifford algebra bundle $\mathrm{Cl}(TM)$ and has a connection compatible with this action.  The canonical examples of Clifford modules are

$\Lambda^* M$ (in two ways, with the grading formed by the degree of differential forms or with the hodge star).
The spinor bundle associated to a spin or spin$^c$ structure.
The Dolbeault complex $\Lambda^{0,*} TM$ for a complex manifold (which is really a special case of a spin$^c$ structure).

Are there other examples of Clifford modules that come up naturally?

REPLY [4 votes]: The following example is very specific (but generalizes to hypersurfaces in spin-manifolds). Consider an immersion $f\colon \Sigma\to\mathbb R^3,$ and the Riemannian metric induced by the immersion. Identify $\mathfrak{su}(2)=\mathbb R^3$. Thus, vectors in $\mathbb R^3$ act on $\mathbb C^2,$ which then is the spinor bundle for the euclidean 3-space. In this regard, tangent vectors to $\Sigma$ act on the trivial $\mathbb C^2$ bundle over $\Sigma.$ This bundle is equipped with the canonical trivial connection $d$ which is compatible with the Clifford action.
As a bundle with Clifford action, it can be identified with the spinor module,
but the connection $d$ is in general not the spinor connection of the induced Riemannian metric. In fact, the difference between these two connections is determined (and determines) the second fundamental form of the surface. A classical consequence of the above is the Weierstrass representation for minimal surfaces in euclidean 3-space.