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 | ||||||||||||
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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. | ||||||||||||
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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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