[Paper Review] Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Frédéric Latrémolière|arXiv (Cornell University)|Feb 7, 2021

Advanced Operator Algebra ResearchMathematics84 references6 citations

TL;DR

This paper constructs spectral triples on fuzzy tori—finite-dimensional C*-algebras arising from twisted group C*-algebras of finite cyclic groups—and proves their convergence to a canonical spectral triple on quantum tori in the sense of spectral propinquity. The convergence is established via modular tunnels and metrical correspondences, demonstrating that fuzzy tori approximate quantum tori not only as quantum metric spaces but also as noncommutative differentiable manifolds, including convergence of state spaces and dynamics generated by Dirac operators.

ABSTRACT

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

Motivation & Objective

• To formalize the heuristic idea that fuzzy tori approximate quantum tori as noncommutative Riemannian manifolds.
• To construct explicit spectral triples on fuzzy tori that mirror natural constructions on classical and quantum tori.
• To establish convergence of these spectral triples to a spectral triple on the quantum torus in the sense of the spectral propinquity.
• To show that the convergence includes not only the quantum metric structure but also the dynamics generated by the Dirac operators.
• To provide a rigorous framework for matrix models in quantum field theory and string theory as approximations of continuum noncommutative geometries.

Proposed method

• Constructs spectral triples on fuzzy tori using Dirac operators derived from the dual action of finite abelian groups on twisted group C*-algebras.
• Defines a dense *-subalgebra of the fuzzy torus algebra and ensures the Dirac operator commutes with the representation up to bounded operators.
• Introduces modular tunnels between the spectral triples on fuzzy tori and the limit spectral triple on the quantum torus using a joint Hilbert module structure over C⊕C.
• Uses a mean value theorem for spectral triples to control the difference in norms between approximating and limit Dirac operators.
• Employs a sequence of finite-rank projections V_f^n and a cut-off function f with rapidly decaying Fourier transform to control approximation error.
• Establishes convergence via the spectral propinquity by bounding the extent of the modular tunnel and metrical correspondence within arbitrary ε > 0.

Experimental results

Research questions

• RQ1 Can spectral triples on fuzzy tori be constructed in a way that mirrors the natural Dirac operators on classical and quantum tori?
• RQ2 Does the sequence of spectral triples on fuzzy tori converge to a spectral triple on the corresponding quantum torus in a meaningful geometric sense?
• RQ3 Is the convergence of spectral triples stronger than mere quantum metric convergence, including convergence of state spaces and dynamics?
• RQ4 Can the convergence be quantified using the spectral propinquity, a noncommutative analogue of Gromov-Hausdorff convergence for spectral triples?
• RQ5 Do the Dirac operators on fuzzy tori generate quantum dynamics that converge to those on the quantum torus?

Key findings

• The constructed spectral triples on fuzzy tori converge to a canonical spectral triple on the quantum torus in the sense of the spectral propinquity.
• The convergence is quantified: for any ε > 0, there exists N ∈ ℕ such that for all n ≥ N, the spectral propinquity between the fuzzy torus triple and the quantum torus triple is at most ε.
• The state spaces of the spectral triples, equipped with their quantum metrics, converge as metric spaces under the Gromov-Hausdorff distance.
• The dynamics generated by the Dirac operators on fuzzy tori converge to those on the quantum torus, as shown by uniform approximation of the one-parameter groups of unitaries.
• The modular tunnel construction ensures that the metrical correspondence between the spectral triples has extent at most ε, confirming convergence in the metrical sense.
• The proof relies on a finite-rank approximation of the Dirac operator using a cut-off function f with ℓ1 Fourier transform, ensuring control over the error in the norm and the operator norm difference.