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The spectral action is a natural functional on the space of spectral triples. It is essentially a regularized heat kernel expansion of the Dirac operator-like operator in the spectral triple.
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).
The spectral action has been proposed as an action functional for describing fundamental physics (the standard model of particle physics). See at
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 claim that the spectral action for something like a 2-spectral triple does reproduce the effective background action of string theory is in
A detailed derivation of how ordinary spectral triples arise as point particle limits of vertex operator algebras for 2d SCFTs is in
A summary of this is in
Also
A brief indication of some ideas of Yan Soibelman and Maxim Kontsevich on this matter is at
Details are in