# Gauge transformations for twisted spectral triples

## TL;DR

This paper extends the concept of metric fluctuations and gauge transformations to twisted spectral triples, revealing their relation to Morita equivalence and addressing self-adjointness issues in the twisted setting.

## Contribution

It introduces a framework for gauge transformations in twisted spectral triples and links them to Morita equivalence, including conditions for preserving self-adjointness.

## Key findings

- Twisted gauge potentials follow from Morita equivalence.
- The gauge transformation law is naturally twisted.
- Conditions for self-adjointness are explicitly solved for minimal twists.

## Abstract

We extend to twisted spectral triples the fluctuations of the metric, as well as their gauge transformations. The former are bounded perturbations of the Dirac operator that arise when a spectral triple is exported between Morita equivalent algebras; the later are obtained by the action of the unitary endomorphisms of the module implementing the Morita equivalence. It is shown that the twisted gauged Dirac operators, previously introduced to generate an extra scalar field in the spectral description of the standard model of elementary particles, in fact follow from Morita equivalence between twisted spectral triples. The law of transformation of the gauge potentials turns out to be twisted in a natural way. In contrast with the non-twisted case, twisted fluctuations do not necessarily preserve the self-adjointness of the Dirac operator. For a self-Morita equivalence, some conditions are obtained in order to maintain self-adjointness, that are solved explicitly for the minimal twist of a Riemannian manifold.