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

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