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