Arithmetic of Potts model hypersurfaces

TL;DR

This paper investigates the algebraic and arithmetic properties of Potts model hypersurfaces, comparing them with quantum field theory graph hypersurfaces, and explores their connections to periods and multiple zeta values.

Contribution

It introduces new insights into the Grothendieck class behavior and polynomial countability of Potts hypersurfaces, and links period computations to quantum field theory techniques.

Findings

Failure of the fibration condition in Potts hypersurfaces.

Period computations relate to multiple zeta values for certain models.

Potential non-mixed Tate periods in specific polymer chain configurations.

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.