Clifford bundle

Let $V$ be a (real or complex) vector space equipped with a symmetric bilinear form $⟨ \cdot, \cdot ⟩$ . Recall that the Clifford algebra $C ℓ (V)$ is defined as a natural (unital associative) algebra that is generated by $V$ , adhering strictly to the relation

v^{2} = ⟨ v, v ⟩

for all $v$ in $V$ .
Similar to other tensor operations, this construction can be performed fiberwise on a smooth vector bundle. Let $E$ be a smooth vector bundle over a smooth manifold $M$ , and let $g$ be a smooth symmetric bilinear form on $E$ . The Clifford bundle of $E$ , denoted as $C ℓ (E)$ , is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of $E$ :

C ℓ (E) = \underset{x \in M}{∐} C ℓ (E_{x}, g_{x})

The topology of $C ℓ (E)$ is influenced by that of $E$ through an associated bundle construction.

A particularly interesting case is the tangent bundle $T M$ for a Lorentzian manifold:

C ℓ (M) = \underset{x \in M}{∐} C ℓ (T_{x} M, g_{x})