Gauge transformations for twisted spectral triples

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.

References in corpus (5)

• Resilience of the Spectral Standard Model
• Type III and spectral triples
• Grand Unification in the Spectral Pati-Salam Model
• On twisting real spectral triples by algebra automorphisms
• Twisted reality condition for Dirac operators

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• Twisted Spectral Triples without the First-Order Condition
• Modular spectral triples and deformed Fredholm modules
• Quantum causality in $\kappa$-Minkowski and related constraints
• Twisting Noncommutative Geometries with Applications to High Energy Physics
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects
• Gauge transformations of spectral triples with twisted real structures
• Real Part of Twisted-by-Grading Spectral Triples
• A critical survey of twisted spectral triples beyond the Standard Model