Real Part of Twisted-by-Grading Spectral Triples

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.