derived smooth geometry
Since a spectral triple encodes Riemannian geometry in a generalized context of noncommutative geometry, a functional on a space of spectral triples is comparable to the Einstein-Hilbert action functional on the space of ordinary Riemennian manifolds. And indeed, on spectral triples corresponding to ordinary Riemannian geometry the spectral action reduces to the Einstein-Hilbert action plus a series of integral over higher curvature invariants.
The notion of spectral triple and of spectral action was introduced in
A discussion specifically of the spectral action is in
Earlier articles on this include