FINITE-DIMENSIONAL APPROXIMATIONS OF SPECTRAL TRIPLES ON QUANTUM TORI

The asymptotic behavior of matrix models, as their dimension grows to infinity, is of common interest in mathematical physics. The formalization of the study of limits of finite-dimensional quantum spaces, endowed with some geometric structure, can be done within the larger framework of noncommutative metric geometry involving, in particular, various noncommutative analogues of the Gromov-Hausdorff distance. On the other hand, spectral triples have emerged as the preferred geometric structure to explore noncommutative analogues of Riemannian geometry. In this talk, we will present our distance function on the space of metric spectral triples, and see how it may be used to prove that spectral triples for fuzzy tori, a common family of examples of finite-dimensional quantum spaces, converge to some spectral triples on classical and even quantum tori.