# Examples of Clifford modules

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?

• Have you tried looking in Lawson and Michelsohn "Spin Geometry"? Dec 3, 2012 at 16:22
• @Robert Bryant: I did but didn't find any examples beyond the ones I gave. Dec 3, 2012 at 23:23
• Those three examples, together with their twisted versions, really seem to be the main examples--poking about Berline, Getzler, and Vergne's "Heat kernals and Dirac operators," the other main reference, brought up nothing more. For what it's worth, perhaps the most general way Clifford modules arise in the wild, at least when $M$ is even-dimensional, comes from seeking "square roots" to generalised Laplacians: $E \to M$ admits a Clifford action and compatible connection iff if it admits a Dirac-type operator, i.e., a first order differential operator $D$ such that $D^2$ is Laplace-type. Dec 4, 2012 at 4:08
• In mathematical physics, particularly supergravity, the Rarita-Schwinger spinors give you another example. The physicists call these 'spin 3/2 fields', and they aren't a submodule of any of the examples you have listed. These fields have also been used to formulate a deformation theory of (spinnable) Einstein manifolds. Dec 4, 2012 at 11:04

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.