TITLE: How to characterize Dirac's gamma matrices in differential geometry?
QUESTION [6 upvotes]: I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 Lorentz (obeying the [1, -1, -1, -1] metric) index and 2 spinor indices (obeying the flat metric [1, 1, 1, 1]).
The problem is that such a tensor seems to be ill-defined in book definitions, I was unable to identify it as a tensor product of elements of the tangent space and cotangent space of some manifold.
Separating the Lorentz and spinor indices, it is clear that the Lorentz index stands for a 1-rank tensor (i.e. vector or one-form, depending on raise/lower index) of the spacetime manifold, while the spinor indices stand for a 2-rank tensor of a flat manifold (or a Clifford algebra?).
I also know that the Lorentz and spinor indices are two representations of the Lorentz group, but still, how to characterize that in terms of manifolds and differential geometry?

REPLY [2 votes]: When you have a vector space $V$ with a symmetric non-degenerate bi-linear form $g$ you can ask for the existence of an algebra generated by the elements of $V$ which obey the following relation. 
$$vu+uv=-2g(u,v)$$
Such a property for the algebra multiplication is needed in the famous paper of Dirac, you can find it here, in which he is trying to find a differential operator which is the square root of the Laplacian on the flat Lorentzian space. The relation is the result of a very easy manipulation of the symbol of the operator (which we know is of order zero) an the fact that it is the root of the Laplacian. 
what he founds, by manipulating the Pauli matrices, are matrices assigned only for the "orthonormal" basis $e_i$. Which is enough to create all algebra. because it shows that in 4 dimensional case this algebra can be realized as an sub-algebra of $M_4(C)$ (by dimension argument it actually is exactly $M_4(C)$.
Finding this matrices not only gives the algebra but also gives an irreducible representation of the Clifford algebra. The space of $C^4$ on which $M_4(C)$ acts is the space of spinors. (Note that he was able to find the root of the Laplacian as a differential operator by paying the price of moving into higher rank space).
So Dirac gamma matrices are representations for a basis of Lorentz vector space which satisfies the Clifford multiplication relation. Note that there are other basis for clifford algebra like Weyl and Majorana basis (see wikipeia page)
To generalize this on general manifold, one has to do this process on cotangent space of every single point of the manifold. This is doable for any manifold CL(T*M), However, it is not always possible to have a vector bundle (which is called spinor bundle if it exists) on which the Clifford algebra acts irreducibly. It turns out that the existence of the spinor bundle has a topological obstruction. The second Whitney class should vanish and in this case the manifold is called spin manifold. Then one can go ahead and define the Dirac operator.