Noncommutative geometry, topology and the standard model vacuum

TL;DR

This paper reinterprets the Standard Model's finite spectral action using noncommutative geometry, employing algebraic K-theory and Morse theory to analyze vacuum solutions and their physical implications.

Contribution

It introduces a novel analysis of the Standard Model's finite spectral triple by shifting from spectral triples to Fredholm modules and applying noncommutative Morse theory.

Findings

First two algebra summands identified up to Morita equivalence.

New vacuum solution compatible with physical mass matrix.

Enhanced understanding of noncommutative geometric structure of the Standard Model.

Abstract

As a ramification of a motivational discussion for previous joint work, in which equations of motion for the finite spectral action of the Standard Model were derived, we provide a new analysis of the results of the calculations herein, switching from the perspective of Spectral triple to that of Fredholm module and thus from the analogy with Riemannian geometry to the pre-metrical structure of the Noncommutative geometry. Using a suggested Noncommutative version of Morse theory together with algebraic $K$-theory to analyse the vacuum solutions, the first two summands of the algebra for the finite triple of the Standard Model arise up to Morita equivalence. We also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix.