How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale

Abstract

We prove that the classical and the quantum flat torus can be rigorously approximated at a differential level by finite-dimensional fuzzy tori within the framework of the spectral propinquity, a distance up to unitary equivalence which we extend in this paper to the space of twisted, metric spectral triples. Standard attempts to establish this convergence are traditionally obstructed by the intrinsic non-locality of discrete calculus and the subsequent failure of the Leibniz rule. While contemporary alternatives such as spectral truncations circumvent this issue by abandoning C^*-algebras in favor of operator systems, our approximations instead remain within the C^*-algebraic category, generalizing the commutator formula used to define the Lipschitz seminorm induced by spectral triples. To this end, we introduce a relaxed notion of a twisted spectral triple where the twist is a linear map acting as a discretized Riesz transform that encapsulates the non-locality of the discrete world. Self-adjointness is obtained by adjoining the backward difference to the forward difference and representing this doubled calculus with the creation and annihilation operators of the CAR algebra. The maximum of the norm of the twisted commutator and norm of the displacement of the twist from the identity becomes our Lipschitz seminorm, from which the quantum metric structure of quantum compact metric spaces (QCMS) on fuzzy tori is constructed. Our proposal to connect twisted spectral triples with QCMS recovers the usual connection between metric spectral triples and QCMS. By extending the spectral propinquity to this generalized setting of twisted spectral triples with possibly unbounded twists, we prove that fuzzy tori equipped with their natural two-sided discrete calculus converge to an amplification of the standard flat Dirac triple, while the underlying twists converge to the identity.

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BibTeX RIS

BibTeX

@misc{latremoliere2026approximate,
  title = {{How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale}},
  author = {Frederic Latremoliere},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.01681},
  eprint = {2607.01681},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.01681},
}

RIS

TY  - UNPB
TI  - How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale
AU  - Frederic Latremoliere
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2607.01681
UR  - https://arxiv.org/abs/2607.01681
AB  - We prove that the classical and the quantum flat torus can be rigorously approximated at a differential level by finite-dimensional fuzzy tori within the framework of the spectral propinquity, a distance up to unitary equivalence which we extend in this paper to the space of twisted, metric spectral triples. Standard attempts to establish this convergence are traditionally obstructed by the intrinsic non-locality of discrete calculus and the subsequent failure of the Leibniz rule. While contemporary alternatives such as spectral truncations circumvent this issue by abandoning C^*-algebras in favor of operator systems, our approximations instead remain within the C^*-algebraic category, generalizing the commutator formula used to define the Lipschitz seminorm induced by spectral triples. To this end, we introduce a relaxed notion of a twisted spectral triple where the twist is a linear map acting as a discretized Riesz transform that encapsulates the non-locality of the discrete world. Self-adjointness is obtained by adjoining the backward difference to the forward difference and representing this doubled calculus with the creation and annihilation operators of the CAR algebra. The maximum of the norm of the twisted commutator and norm of the displacement of the twist from the identity becomes our Lipschitz seminorm, from which the quantum metric structure of quantum compact metric spaces (QCMS) on fuzzy tori is constructed. Our proposal to connect twisted spectral triples with QCMS recovers the usual connection between metric spectral triples and QCMS. By extending the spectral propinquity to this generalized setting of twisted spectral triples with possibly unbounded twists, we prove that fuzzy tori equipped with their natural two-sided discrete calculus converge to an amplification of the standard flat Dirac triple, while the underlying twists converge to the identity.
ER  -

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