In Chern-Weil Theorym, characteristic classes appear as certain closed differential forms associated to vector bundles with connections (constructed via the curvature form of the connection).

When you deal with Clifford bundles, e.g. spinors (which are an interesting thing to study from the physics perspective, I hear), there is Getzler's symbol map that sends Clifford elements to differential forms; in this sense, the Clifford Algebra appears as a "quantized version" of the Grassmann Algebra.

When you follow the Heat Kernel Proof of the Atiyah-Singer Index Theorem, differential forms jump out of the semiclassical asymptotics of the heat kernel using Getzler's symbol mapping, and by some miracle, these are exactly the $\hat{A}$ genus and the Chern character, defined by Chern-Weil theory.

In Chern-Weil Theory, characteristic classes appear as certain closed differential forms associated to vector bundles with connections (constructed via the curvature form of the connection).

When you deal with Clifford bundles, e.g. spinors (which are an interesting thing to study from the physics perspective, I hear), there is Getzler's symbol map that sends Clifford elements to differential forms; in this sense, the Clifford Algebra appears as a "quantized version" of the Grassmann Algebra.

When you follow the Heat Kernel Proof of the Atiyah-Singer Index Theorem, differential forms jump out of the semiclassical asymptotics of the heat kernel using Getzler's symbol mapping, and by some miracle, these are exactly the $\hat{A}$ genus and the Chern character, defined by Chern-Weil theory.