nLab
spectral action

Idea

The spectral action is a natural functional on the space of spectral triples.

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.

Applications

The spectral action has been proposed as an action functional for describing fundamental physics. See at higher category theory and physics the section The standard model and gravity.

References

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)

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)