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Algebraic Versus Spectral Torsion

Ludwik Dąbrowski, Yang Liu, Sugato Mukhopadhyay

TL;DR

This work bridges spectral and algebraic notions of torsion in noncommutative geometry by examining the almost-commutative space $M\times\mathcal{Z}_2$. It shows that for the two-point space $\mathcal{Z}_2$ there is a unique connection whose algebraic torsion matches the spectral torsion functional, and extends to the full product via Mesland-Rennie calculus with a non-product perturbation that exactly reproduces the spectral torsion. The analysis reveals that junk forms in Connes' calculus can obscure torsion contributions, while the MR framework provides a pathway to full agreement with the spectral data, even under generalized internal maps $\phi$. Overall, the paper establishes a concrete correspondence between spectral and algebraic torsion in a fundamental NC geometry and highlights how different calculi influence this relationship.

Abstract

We relate the recently defined spectral torsion with the algebraic torsion of noncommutative differential calculi on the example of the almost-commutative geometry of the product of a closed oriented Riemannian spin manifold $M$ with the two-point space $\mathcal Z_2$.

Algebraic Versus Spectral Torsion

TL;DR

This work bridges spectral and algebraic notions of torsion in noncommutative geometry by examining the almost-commutative space . It shows that for the two-point space there is a unique connection whose algebraic torsion matches the spectral torsion functional, and extends to the full product via Mesland-Rennie calculus with a non-product perturbation that exactly reproduces the spectral torsion. The analysis reveals that junk forms in Connes' calculus can obscure torsion contributions, while the MR framework provides a pathway to full agreement with the spectral data, even under generalized internal maps . Overall, the paper establishes a concrete correspondence between spectral and algebraic torsion in a fundamental NC geometry and highlights how different calculi influence this relationship.

Abstract

We relate the recently defined spectral torsion with the algebraic torsion of noncommutative differential calculi on the example of the almost-commutative geometry of the product of a closed oriented Riemannian spin manifold with the two-point space .
Paper Structure (19 sections, 19 theorems, 148 equations)

This paper contains 19 sections, 19 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Algebraic and Analytic Preliminaries
  3. Spectral Triples
  4. Differential Calculi
  5. Connes' Construction
  6. Mesland-Rennie construction.
  7. Algebraic Torsion of Connections on $\Omega^1_D ( \mathcal{A})$
  8. The Almost Commutative Geometry of $M \times \mathcal{Z}_2$
  9. The two-point space $\mathcal{Z}_2$
  10. Tensor Product Construction
  11. The almost commutative $M \times \mathcal{Z}_2$
  12. Spectral Torsion Functional
  13. The Projection $\Psi$
  14. Junk Tensors $JT^2_D (\mathcal{A})$
  15. Main Results
  16. ...and 4 more sections

Key Result

Proposition 2.10

Keep the notations as above, and set $S_\sigma := \sigma_2 \circ m \circ S: \Omega^1_D ( \mathcal{A}) \to \Lambda^2_D (\mathcal{A})$ and $S_\Psi := (1- \Psi) \circ S : \Omega^1_D ( \mathcal{A}) \to T^2_D ( \mathcal{A})$, then

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2: DSZ24
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 37 more