A lot of notions in differential geometry have direct meaning in Physics. For example:

• A Riemannian metric is a way to encode distances on a manifold and in Physics it is the gravitational field. The curvature of the Levi-Civita connection gives the strength of the gravitation in a certain sense,
• A principal $G$-connection is a object that allows us to do parallel transport conveniently with respect to an action of a certain Lie group $G$, and in Physics it is a gauge field, that is a field that is related to a fundamental interaction, for instance a principal $U(1)$-connection can be seen as the electromagnetic field. The curvature of the connection gives the field strength, in a way.

I would like to have an interpretation of what is a spinor field (when the manifold on which we are working admits a spin structure) in classical differential geometry, that is a section of the spinor bundle. By classical differential geometry I mean typical manifolds, not supermanifolds. This is because, for me, spinors in the theory of supermanifolds, play a different role, since in a way they are "odd spacetime coordinates". I am interested in the geometry of classical fields: a spinor field represents "matter" (fermions) whereas gauge fields (that is, principal connections) represent "forces" (bosons). But this is Physics. I am interested in a mathematical interpretation like:

• Riemannian metric = gravitational field = a way to measure distances,
• Principal connection = gauge field = a way to do parallel transport,
• Spinor field = matter field = what in Mathematics?

So my questions are:

In classical differential geometry (that is, ordinary manifolds), how can we interpret geometrically spinor fields? How can we interpret the spin connection and its curvature?

Thanks.

EDIT: In a comment below I was saying that spinor geometry is of fundamental importance to the Atiyah-Singer theorem. So perhaps this gives a lead to other people to help me with the interpretation of spinors in classical differential geometry.