Remarks on the Spectral Action Principle

TL;DR

This paper discusses the spectral action principle in noncommutative geometry, linking chiral fermions to anomaly cancellation, and proposes supersymmetrisation of the fermionic spectral action.

Contribution

It introduces consistency conditions for the spectral action related to anomaly cancellation and suggests a supersymmetric approach for the fermionic part.

Findings

Chiral fermions impose anomaly-related conditions on the spectral action.

Supersymmetrisation offers a potential form for the fermionic spectral action.

Conditions ensure gauge and gravitational anomaly absence.

Abstract

The presence of chiral fermions in the physical Hilbert space implies consistency conditions on the spectral action. These conditions are equivalent to the absence of gauge and gravitational anomalies. Suggestions for the fermionic part of the spectral action are made based on the supersymmetrisation of the bosonic part.