A second interesting thing one can do is to construct a spin structure. Perhaps the most remarkable thing about this is that it is a very recognizable generalization to a formula_23-dimensional pseudo-Riemannian manifold of the conventional physics concept of spinors living on a (1,3)-dimensional Minkowski spacetime. The construction passes through a complexified Clifford algebra to build a Clifford bundle and a spin manifold. At the end of this construction, one obtains a system that is remarkably familiar, if one is already acquainted with Dirac spinors and the Dirac equation. Several analogies pass through to this general case. First, the spinors are the Weyl spinors, and they come in complex-conjugate pairs. They are naturally anti-commuting (this follows from the Clifford algebra), which is exactly what one wants to make contact with the Pauli exclusion principle. Another is the existence of a chiral element, analogous to the gamma matrix formula_24 which sorts these spinors into left and right-handed subspaces. The complexification is a key ingredient, and it provides "electromagnetism" in this generalized setting. The spinor bundle doesn't "just" transform under formula_25, the generalization of the Lorentz group formula_26, but under a bigger group, the complexified spin group formula_27 It is bigger in that it has a double covering by formula_28