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Signature change by a morphism of spectral triples

Gaston Nieuviarts

TL;DR

The work establishes a formal bridge between twisted spectral triples and pseudo-Riemannian (Krein) geometries via a $K$-morphism that maps a spectral triple to its dual. In even dimensions, a local signature change is realized through a parity-like operator induced by the twist, with the unitary $K$ driving the transformation $D^{K} = K D$ and the $K$-inner product $\langle\cdot,\cdot\rangle_{K} = \langle\cdot, K\cdot\rangle$ linking Krein and twisted inner products. The authors show that the four generalized triple types are interrelated by the duality, preserving fluctuations, fermionic actions, and spectral actions under duality; the $K$-morphism provides an involutive, structure-preserving map between the dual triples. They further specialize to even-dimensional manifolds to realize signature change via spacelike reflections and discuss the explicit 4D Lorentzian case, where a dual spectral triple yields a Lorentzian Dirac operator consistent with a causal structure. Overall, the paper proposes a coherent, algebraic mechanism for Lorentzian embedding in noncommutative geometry through signature-changing dualities, with potential implications for the noncommutative Standard Model and Lorentz-invariant fermionic actions.

Abstract

We present a connection between twisted spectral triples and pseudo-Riemannian spectral triples, rooted in the fundamental interplay between twists and Krein products. A concept of morphism of spectral triples is introduced, transforming one spectral triple into its dual. In the case of even-dimensional manifolds, we demonstrate how this construction implements a local signature change via the parity operator induced by the twist. Consequently, the signature change transformation is governed solely by the unitary operator that implements the twist. This unitary is a central element of our approach, as it is directly linked to the Krein product, the twist, and the parity operator that implements the signature change.

Signature change by a morphism of spectral triples

TL;DR

The work establishes a formal bridge between twisted spectral triples and pseudo-Riemannian (Krein) geometries via a -morphism that maps a spectral triple to its dual. In even dimensions, a local signature change is realized through a parity-like operator induced by the twist, with the unitary driving the transformation and the -inner product linking Krein and twisted inner products. The authors show that the four generalized triple types are interrelated by the duality, preserving fluctuations, fermionic actions, and spectral actions under duality; the -morphism provides an involutive, structure-preserving map between the dual triples. They further specialize to even-dimensional manifolds to realize signature change via spacelike reflections and discuss the explicit 4D Lorentzian case, where a dual spectral triple yields a Lorentzian Dirac operator consistent with a causal structure. Overall, the paper proposes a coherent, algebraic mechanism for Lorentzian embedding in noncommutative geometry through signature-changing dualities, with potential implications for the noncommutative Standard Model and Lorentz-invariant fermionic actions.

Abstract

We present a connection between twisted spectral triples and pseudo-Riemannian spectral triples, rooted in the fundamental interplay between twists and Krein products. A concept of morphism of spectral triples is introduced, transforming one spectral triple into its dual. In the case of even-dimensional manifolds, we demonstrate how this construction implements a local signature change via the parity operator induced by the twist. Consequently, the signature change transformation is governed solely by the unitary operator that implements the twist. This unitary is a central element of our approach, as it is directly linked to the Krein product, the twist, and the parity operator that implements the signature change.
Paper Structure (15 sections, 32 theorems, 91 equations, 1 table)

This paper contains 15 sections, 32 theorems, 91 equations, 1 table.

Table of Contents

  1. Introduction
  2. Spectral triples
  3. Twisted spectral triples
  4. Introduction to twisted spectral triples
  5. Twisted product when $\rho$ is an inner automorphism
  6. Pseudo-Riemannian spectral triples
  7. Connection for even real spectral triples
  8. Presentation of the connection
  9. $K$-morphism of generalized spectral triples
  10. Connection for even-dimensional manifolds
  11. Signature change through the parity operator
  12. The spectral triples of the connection
  13. Signature change by the $K$-morphism
  14. The 4-dimensional Lorentzian manifold
  15. Summary and outlook

Key Result

Proposition 3.3

If $\langle\, . \, , \, . \, \rangle_\rho$ is Hermitian, then $\rho$ is $\mathcal{B}(\mathcal{H})$-regular. Conversely, if $\rho$ is not $\mathcal{B}(\mathcal{H})$-regular, then $\langle\, . \, , \, . \, \rangle_\rho$ is not Hermitian.

Theorems & Definitions (91)

  • Definition 2.1: Spectral triple
  • Definition 3.1: Twisted spectral triple
  • Definition 3.2: $\mathcal{B}(\mathcal{H})$-regular automorphism
  • Proposition 3.3
  • Proof 1
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6: twist by grading
  • Proposition 3.7
  • Proof 2
  • ...and 81 more