Spectral Action from Anomalies

TL;DR

This paper derives the spectral action from fermionic anomalies in a gauge and gravitational background, linking it to noncommutative geometry and phase transitions in the early universe.

Contribution

It demonstrates how the spectral action naturally arises from fermionic anomalies and connects gauge, Higgs, and gravitational fields within a unified framework.

Findings

Spectral action cancels fermionic anomalies.

Links between spectral action and phase transitions in early universe.

Proposes a dilaton realization involving fermion collective modes.

Abstract

Starting from a theory of fermions moving in a fixed gauge and gravitational background we implement the scale invariance of the theory. Upon quantization the theory is anomalous but the anomaly can be cancelled by the addition of another term to the action. This term comes out to be basically the Chamseddine Connes spectral action introduced in the context of noncommutative geometry. An alternative realization of the dilaton may involve a collective scalar mode of all fermions accumulated in a {scale-noninvariant} dilaton action. The entire spectral action describes gauge and Higgs fields coupled with gravity. Here this action is coupled with a dilaton and we discuss how it relates to the transition from the radiation to the electroweak broken phase via condensation of Higgs fields.