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Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers

Marcolli, Matilde and Tabuada, Gonçalo (2017) Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers. Journal of Noncommutative Geometry, 11 (1). pp. 1-49. ISSN 1661-6952. doi:10.4171/JNCG/11-1-1.

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In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

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Additional Information:© 2017 EMS Publishing House. Received 20 March, 2015. M. Marcolli was partially supported by the NSF grants DMS-1007207, DMS-1201512, and PHY-1205440. G. Tabuada was partially supported by a NSF CAREER Award.
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Subject Keywords:Quantum statistical mechanical systems, Gibbs states, zeta function, polylogarithms, Tannakian categories, Weil numbers, motives, Weil restriction
Issue or Number:1
Classification Code:Mathematics Subject Classification (2010). 11M06, 14C15, 82B10.
Record Number:CaltechAUTHORS:20170330-165516108
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Official Citation:Marcolli Matilde, Tabuada Gonçalo: Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers. J. Noncommut. Geom. 11 (2017), 1-49. doi: 10.4171/JNCG/11-1-1
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:75562
Deposited By: George Porter
Deposited On:31 Mar 2017 15:05
Last Modified:15 Nov 2021 16:34

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