Gauge transformations of spectral triples with twisted real structures

TL;DR

This paper explores how gauge transformations can be applied to spectral triples with twisted real structures, focusing on their coupling to gauge fields and implications for models beyond the Standard Model.

Contribution

It introduces a framework for gauge transformations in twisted spectral triples, analyzing the twisted first-order condition and its role in extending particle physics models.

Findings

Spectral triple based on left-right symmetric algebra reduces to Standard Model under certain conditions.

Twisted real structures can implement conformal transformations without twisting noncommutative 1-forms.

Modified inner fluctuations influence gauge coupling in twisted spectral triples.

Abstract

Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative 1-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyse the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyond the Standard Model, and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically-motivated assumptions the spectral triple based on the left-right symmetric algebra should reduce to that of the Standard Model of fundamental particles and interactions, as in the untwisted case.