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Spectral triples and quantum statistical mechanical systems are two importantconstructions in noncommutative geometry. In particular, both lead tointeresting reconstruction theorems for a broad range of geometric objects,including number fields, spin manifolds, graphs. There are similarities betweenthe two structures, and we show that the notion of type III sigma-spectraltriple, introduced recently by Connes and Moscovici, provides a natural bridgebetween them. We investigate explicit examples, related to the Bost-Connesquantum statistical mechanical system and to Riemann surfaces and graphs.

Type III sigma-spectral triples and quantum statistical mechanical systems