Real Part of Twisted-by-Grading Spectral Triples

TL;DR

This paper explores the real part of twisted spectral triples, especially when twisted by grading, and illustrates the findings with the standard model's spectral triple.

Contribution

It adapts the notion of the real part of spectral triples to the twisted case, revealing its behavior depending on KO dimension and applying it to the standard model.

Findings

Real part is twisted or intersected with opposite algebra depending on KO dimension.

Application to the spectral triple of the standard model.

Provides a framework for understanding twisted spectral triples in physics.

Abstract

After a brief review on the applications of twisted spectral triples to physics, we adapt to the twisted case the notion of real part of a spectral triple. In particular, when one twists a usual spectral triple by its grading, we show that - depending on the $KO$ dimension - the real part is either twisted as well, or is the intersection of the initial algebra with its opposite. We illustrate this result with the spectral triple of the standard model.