Global and local aspects of spectral actions

TL;DR

This paper explores the spectral action in noncommutative geometry, analyzing its expansions and convergence beyond the low-energy limit, with detailed calculations on the torus and gauge connections.

Contribution

It provides a detailed study of spectral action expansions, convergence criteria, and explicit heat kernel computations beyond the low-energy approximation.

Findings

Derived convergence criteria for spectral action expansions

Computed heat kernel on the torus up to second order in gauge connection

Analyzed limiting cases of spectral actions in commutative spectral triples

Abstract

The principal object in noncommutatve geometry is the spectral triple consisting of an algebra A, a Hilbert space H, and a Dirac operator D. Field theories are incorporated in this approach by the spectral action principle, that sets the field theory action to Tr f(D^2/\Lambda^2), where f is a real function such that the trace exists, and \Lambda is a cutoff scale. In the low-energy (weak-field) limit the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up the second order in gauge connection and consider limiting cases.