nLab spectral action

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 to a 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

• higher category theory and physics the section The standard model and gravity.

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.

Related concepts

• Connes-Lott-Chamseddine model

References

General On spectral triples

The notion of spectral triple and of spectral action was introduced in

• Alain Connes, Gravity coupled with matter and foundation of non-commutative geometry (arXiv:hep-th/9603053)

A discussion specifically of the spectral action is in

• Ali Chamseddine, Alain Connes, The Spectral Action Principle arXiv:hep-th/9606001,

Earlier articles on this include

• Ali Chamseddine, Remarks on the Spectral Action Principle (arxiv:hep-th/9806046)

Phenomenological models

In the article

• Alain Connes, Noncommutative Geometry and the standard model with neutrino mixing, JHEP0611:081,2006 (arXiv:hep-th/0608226)

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).

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

• Ali Chamseddine, An Effective Superstring Spectral Action (arXiv:hep-th/9705153)

A detailed derivation of how ordinary spectral triples arise as point particle limits of vertex operator algebras for 2d SCFTs is in

• Daniel Roggenkamp, Katrin Wendland, Limits and Degenerations of Unitary Conformal Field Theories (arXiv:hep-th/0308143)

A summary of this is in

• Daniel Roggenkamp, Katrin Wendland, Decoding the geometry of conformal field theories (arXiv:0803.0657)

Also

• Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo, Spectral Triples and the Super-Virasoro Algebra Communications in Mathematical Physics, Volume 295, Number 1 / April 2010 (Springer pdf)

A brief indication of some ideas of Yan Soibelman and Maxim Kontsevich on this matter is at

• Urs Schreiber, Spectral triples and graph field theory

Details are in

• Yan Soibelman, Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry , in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2001)