Type III sigma-spectral triples and quantum statistical mechanical systems

TL;DR

This paper explores the connection between spectral triples and quantum statistical mechanical systems in noncommutative geometry, introducing type III sigma-spectral triples as a bridge, with applications to number theory and geometry.

Contribution

It introduces and investigates type III sigma-spectral triples, linking spectral triples and quantum statistical systems in a novel way.

Findings

Type III sigma-spectral triples unify spectral and quantum statistical structures.

Explicit examples include Bost-Connes system and Riemann surfaces.

New insights into geometric reconstructions from noncommutative data.

Abstract

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.