Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

TL;DR

This paper demonstrates that spectral triples on fuzzy tori converge to spectral triples on quantum tori as the dimension increases, establishing fuzzy tori as finite-dimensional approximations of quantum tori in noncommutative geometry.

Contribution

It constructs spectral triples on fuzzy tori and proves their convergence to quantum tori spectral triples using spectral propinquity, linking finite approximations to infinite noncommutative manifolds.

Findings

Spectral triples on fuzzy tori converge to those on quantum tori.

Fuzzy tori approximate quantum tori as noncommutative differentiable manifolds.

Convergence includes state spaces and quantum dynamics.

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.