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

[2607.01681] How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale Skip to main content Search Submit Donate Log in Mathematics > Operator Algebras arXiv:2607.01681 (math) [Submitted on 2 Jul 2026 ( v1 ), last revised 5 Oct 2026 (this version, v3)] Title: How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale Authors: Frederic Latremoliere View PDF HTML (experimental) 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. 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, we instead preserve the $C^*$-algebraic category by generalizing the commutator formula. 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. We also require the L-seminorm to control both the twisted commutator and the displacement of the twist from the identity. 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. Comments: 46 pages. The definition of the twist is relaxed somewhat, without needing any change in the proofs, but to make it compatible with a subsequent paper on C*-crossed products which uses this technique. Previous version correction: Corrected the finite Dirac operators by symmetrizing them to ensure self-adjointness Subjects: Operator Algebras (math.OA) MSC classes: 46L89, 46L30, 58B34 Cite as: arXiv:2607.01681 [math.OA] (or arXiv:2607.01681v3 [math.OA] for this version) https://doi.org/10.48550/arXiv.2607.01681 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Frederic Latremoliere [ view email ] [v1] Thu, 2 Jul 2026 04:15:22 UTC (45 KB) [v2] Thu, 27 Aug 2026 03:36:23 UTC (47 KB) [v3] Mon, 5 Oct 2026 08:10:31 UTC (48 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OA < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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