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.

associated to( en.wikipedia.org/wiki/Associated_bundle ) principal bundles (connections on which give gauge fields) over the manifold. See for example the chapters on fiber bundles and connections in Nakahara's book "Geometry, Topology and Physics", but it's explained in many other books too. $\endgroup$6more comments