Twisted Standard Model and its Krein structure -- in memoriam Manuele Filaci

TL;DR

This paper reviews Manuele Filaci's work on twisting the spectral triple of the Standard Model within noncommutative geometry, revealing a Krein space structure and connections to twistor symmetries.

Contribution

It systematically studies the inner product induced by various twists of the spectral triple, showing it forms a Krein space and relates to twistor symmetry groups.

Findings

The twisted spectral triple induces a Krein space structure.

The unitary group with respect to the twisted product includes twistor symmetries.

Multiple minimal twists of the Standard Model spectral triple exist.

Abstract

We review the contributions of Manuele Filaci - a PhD student from the university of Genova prematurely deceased a little more than a year ago - to the description of the Standard Model in noncommutative geometry. Building on Manuele's discovery that there exist various ways to minimally twist the spectral triple of the Standard Model, we study in a systematic way the inner product induced by the twist. Under loose assumptions, this product turns the Hilbert space of the spectral triple into a Krein space. For the Standard Model, the group of unitary with respect to the twisted product contains the symmetry group of twistors as a subgroup.