# Contents

## Idea

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).

## Application to particle phenomenology

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.

## References

### General On spectral triples

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

### Phenomenological models

In the article

previous models of the standard model of particle physics were 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).

### Lift to 2-spectral triples

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

Revised on March 19, 2014 04:45:47 by Urs Schreiber (89.204.130.240)