Krein Spectral Triples and the Fermionic Action

Article

Abstract

Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces also exhibit a Krein space structure.

Keywords

Lorentzian manifolds Noncommutative geometry Gauge theories 

Mathematics Subject Classification (2010)

53C50 58B34 70S15 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  3. 3.SISSA (Scuola Internazionale Superiore di Studi Avanzati)TriesteItaly

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