Arithmetic of Potts Model Hypersurfaces
- Creators
- Marcolli, Matilde
- Su, Jessica
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.
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.Attached Files
Submitted - 1112.5667v1.pdf
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Additional details
- Eprint ID
- 38324
- DOI
- 10.1142/S0219887813500059
- Resolver ID
- CaltechAUTHORS:20130507-112308486
- Caltech Summer Undergraduate Research Fellowship (SURF)
- DMS-0901221
- NSF
- DMS-1007207
- NSF
- DMS-1201512
- NSF
- PHY-1205440
- NSF
- Created
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2013-05-07Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field