2011/12/23 by Matilde Marcolli, Marcolli, Matilde, Jessica T. Su +2
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech) #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1112.5667
19 pages, latex, 2 pdf figures
arxiv created 2011/12/23 · openalex publication_date 2011/12/23 · arxiv updated 2011/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.