derived smooth geometry
Types of quantum field thories
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 integrals over higher curvature invariants. Moreover, by an incarnation of the Kaluza-Klein mechanism, on spectral triples corresponding toa Riemannian manifold times a non-classical space of classical dimension 0, the spectral action to first order is an Einstein-Yang-Mills-Dirac theory and in fact Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory, hence also includes gauge theory (Yang-Mills theory).
Notice that a spectral triple which describes the worldline quantum mechanics of a point particle may be regarded as the point-particle degeneration limit of a 2d SCFT (a “2-spectral triple”, Roggenkamp-Wendland 03, Soibelman 11) which describes the worldsheet quantum field theory of a superstring. It has been argued (Chamseddine 97) that the spectral action of the spectral triple correspondingly reproduces that of the effective field theories of these string theories.
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
In the article
a previous version of Connes-Lott-Chamseddine model was refined in some technical fine print, and it was found that the KO-dimension of the classically 0-dimensional compactification spaces has to be 6 (just as in superstring theory). Also a concise bimodule-characterization of the particle content of the standard model was claimed (prop. 2.2).
A summary of this is in
Details are in