TITLE: How and why did mathematicians develop spin-manifolds in differential geometry?
QUESTION [27 upvotes]: First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that this question is not appropriate for math.SE.
As far as I have searched in mathematical physics literature, historically, physicists such as Pauli and Dirac pioneered the concept of spin for a particle. Dirac developed his theory for spin-1/2 electrons by factorizing the Klein-Gordon equation to find a linear relativistic equation that is compatible with Schrodinger's wave equation but doesn't give negative probability density. He factorized the Klein-Gordon equation and found algebraic constraints that gave the Clifford algebra $Cl(1,3)$.
Now that's the algebraic part of the story. What I don't understand is how the concepts in Spin manifolds and Spin geometry were developed from the point of view of differential geometry and topology. 
Apparently, Elie Cartan was one of the pioneers and he has written a book  about it (I have read its first few chapters). But his language is very different from the language of differential geometry that we use today. He doesn't talk about covering spaces, vector bundles or connections the way that has become common in today's math literature. So, I find it very difficult to trace the chain of thoughts that has led physicists and mathematicians to develop spin geometry in its current language.
I'd like to know how our today's mathematical physics literature has been developed historically and what is the aim of using and further developing this sophisticated language.

REPLY [5 votes]: The concept of spinor is a fascinating one, not only from the mathematical point of view but also from the physical point of view: I find extremely remarkable that some of the fundamental particles of nature are actually described in terms of spinors, a fact that has been extensively verified experimentally. Therefore, spin geometry is not just an abstract mathematical theory, but the language in which describe at least some aspects of the universe.
Spinors were invented, or rather discovered, by E. Cartan 1913 in his work on the classification of complex irreducible representations of simple Lie groups, where he discovered the "spinorial" representations of orthogonal groups. Fifteen years later, P. Dirac independently invented spinors in his attempt to find a "relativistic wave equation". 
E. Cartan made the connection between both constructions much later, in 1937. However, he was dissatisfied with the status of the theory of spinors, as he was fearing that it would not be possible to define spinors globally on a manifold. Then, spin structures on manifolds came along (I am not sure exactly when and how) and it was clear that one can define spinors on a manifold through the associated bundle construction of a spin structure.
What I want to stress in this answer is the following: it has been widely accepted that the right mathematical set up to deal with spinors is that of spin structures on manifolds. However this is some times not the most convenient formalism: in particular it is too restrictive for many physical applications, especially in the context of String Theory (I can elaborate on this if needed). In order to obtain a general theory of spinors on a manifold one has to get rid of the concept of spin structure. Instead, one defines a spinor bundle $S$ as a bundle of (say irreducible) Clifford modules over the bundle of Clifford algebras $Cl(M,g)$ of your pseudo-Riemannian manifold $(M,g)$. This approach was already advocated by E. Schrodinger (in the language of that time), but it was soon forgotten when the description in terms of spin groups gained favor. Note that $Cl(M,g)$ is not obstructed, but, depending on the representation used, $S$ may be. Clearly, if $(M,g)$ admits a spin structure $P$, it admits a bundle of irreducible Clifford modules over $Cl(M,g)$ which can be constructed as an associated vector bundle to $P$. However, what about the other direction? The manifold $(M,g)$ needs to be spin in order to admit a bundle of irreducible Clifford modules over $(M,g)$? The answer to the latter question is big no. This can already be seen from the theory of spin$^c$ structures. There are manifolds that admit spin$^c$ structures that do not admit any spin structure, and such spin$^c$ structure can be used to construct bundles of irreducible complex Clifford modules over $Cl(M,g)$. Obtaining the general topological obstruction to the existence of a bundle of irreducible/faithful Clifford modules over $(M,g)$ is a subtle mathematical problem addressed only relatively recently in the literature in the work of T. Friedrich, A. Trautman, C. Lazaroiu and C. S. Shahbazi.