Inner Fluctuations in Noncommutative Geometry without the first order condition

TL;DR

This paper extends the concept of inner fluctuations in noncommutative geometry to cases lacking the first-order condition, introducing a quadratic term and a semi-group structure, with applications to spectral models beyond the Standard Model.

Contribution

It generalizes inner fluctuations to noncommutative spectral triples without the first-order condition, revealing a semi-group structure dependent only on the algebra.

Findings

Introduces a quadratic term in inner fluctuations.

Establishes a semi-group of fluctuations extending the unitary group.

Applies to noncommutative spectral models beyond the Standard Model.

Abstract

We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model.