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Arithmetic of Potts Model Hypersurfaces

Marcolli, Matilde and Su, Jessica (2013) Arithmetic of Potts Model Hypersurfaces. International Journal of Geometric Methods in Modern Physics, 10 (4). Art. No. 1350005. ISSN 0219-8878. doi:10.1142/S0219887813500059. https://resolver.caltech.edu/CaltechAUTHORS:20130507-112308486

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Abstract

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the "fibration condition" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1142/S0219887813500059DOIArticle
http://www.worldscientific.com/doi/abs/10.1142/S0219887813500059PublisherArticle
http://arxiv.org/abs/1112.5667arXivDiscussion Paper
Additional Information:© 2013 World Scientific Publishing Company. Received 4 April 2012. Accepted 25 July 2012. Published 14 December 2012. This paper is based on the results of the second author’s summer research project, supported by the Summer Undergraduate Research Fellowship program at Caltech. The first author was partly supported by NSF Grants DMS-0901221, DMS-1007207, DMS-1201512, and PHY-1205440. The authors thank Paolo Aluffi and Bill Dubuque for useful conversations.
Funders:
Funding AgencyGrant Number
Caltech Summer Undergraduate Research Fellowship (SURF)UNSPECIFIED
NSFDMS-0901221
NSFDMS-1007207
NSFDMS-1201512
NSFPHY-1205440
Subject Keywords:Potts models; points over finite fields; zeta function; mixed Tate motives
Issue or Number:4
Classification Code:AMSC: 82B20, 11G25
DOI:10.1142/S0219887813500059
Record Number:CaltechAUTHORS:20130507-112308486
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130507-112308486
Official Citation:ARITHMETIC OF POTTS MODEL HYPERSURFACES MATILDE MARCOLLI, JESSICA SU DOI: 10.1142/S02198878135000591350005
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38324
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:07 May 2013 18:44
Last Modified:09 Nov 2021 23:36

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