However strange may seem, the fact that the Dirac operator is used in Physics makes all the definitions different and messy. I would like to ask about the definition of the Dirac operator in the setting of Pure Mathematics (Riemannian geometry):

• Which underlying structure (metrics, bundles...) must have a manifold so that the Dirac operator makes sense?

• As a function, which are the domain a range of the Dirac operator (perhaps the set of sections of a certain bundle?)

• Is there an intuitive explanation of what does the Dirac operator?

I will be grateful for any suggestion.

Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator in the setting of pure mathematics:

• Which underlying structure (metrics, bundles, etc.) a manifold have in order to be able to define a Dirac?

• What are the domain a range of the Dirac operator (perhaps the set of sections of a certain bundle?)

• Is there an intuitive explanation of what the Dirac operator does? (say, along the lines of "the Laplacian represents diffusion, as in the heat equation $\partial_tu=\Delta u$).

I will be grateful for any suggestions.