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Finite spectral triples for the fuzzy torus

John W. Barrett, James Gaunt

Abstract

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

Finite spectral triples for the fuzzy torus

Abstract

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

Paper Structure

This paper contains 35 sections, 8 theorems, 195 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The torus
  3. Spectral triples
  4. The results
  5. Summary
  6. Acknowledgements
  7. The flat torus
  8. Poisson structure
  9. The non-commutative torus
  10. Normalised monomials
  11. Clock and shift algebra
  12. Transformations
  13. Translations
  14. Modular transformations
  15. Correspondence
  16. ...and 20 more sections

Key Result

Lemma 1

The flat tori $(T^2,\widetilde{g})$ and $(T^2,\widetilde{g}')$ are isometric iff there exists $B\in{\mathrm{GL}}(2,{\mathbb Z})$ such that $g'=B^\bullet g$.

Figures (6)

  • Figure 1: A 3D plot of the positive eigenvalues \ref{['eq:fuzzyevBB']} with $N=100$.
  • Figure 2: A side by side comparison of the positive eigenvalues \ref{['eq:fuzzyevBB']} and \ref{['eq:classevBB']}, with $N=100$ and $k,l>0$.
  • Figure 3: Contour plot for the positive eigenvalues \ref{['eq:fuzzyevBB']} with $N=100$.
  • Figure 4: Histogram of multiplicity against the eigenvalues \ref{['eq:fuzzyevBB']} and \ref{['eq:commtorusev']}, with $N=100$ and bin width $2$.
  • Figure 5: Contour plot for the spectrum \ref{['eq:fuzzyevsq']} with $N=100$.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Lemma 1
  • proof
  • Definition 1
  • Example 1
  • Lemma 2: Weyl
  • proof
  • Definition 2
  • Example 2
  • Definition 3
  • Lemma 3
  • ...and 33 more