Arithmetic of Potts model hypersurfaces

@article{Marcolli2011ArithmeticOP,
  title={Arithmetic of Potts model hypersurfaces},
  author={Matilde Marcolli and Jessica T. Su},
  journal={arXiv: Mathematical Physics},
  year={2011}
}
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… 

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References

SHOWING 1-10 OF 48 REFERENCES
Parametric Feynman integrals and determinant hypersurfaces
The purpose of this paper is to show that, under certain combinatorial conditions on the graph, parametric Feynman integrals can be realized as periods on the complement of the determinant
A motivic approach to phase transitions in Potts models
Feynman motives of banana graphs
We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as
Matroids motives, and a conjecture of Kontsevich
Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of
A K3 in $\phi^{4}$
Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field $\F_q$ is a (quasi-) polynomial
Cohomology of graph hypersurfaces associated to certain Feynman graphs
To any Feynman graph (with 2n edges) we can associate a hypersurface $X\subset\PP^{2n-1}$. We study the middle cohomology $H^{2n-2}(X)$ of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer
The Massless Higher-Loop Two-Point Function
We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to
On the periods of some Feynman integrals
We study the related questions: (i) when Feynman amplitudes in massless $\phi^4$ theory evaluate to multiple zeta values, and (ii) when their underlying motives are mixed Tate. More generally, by
Feynman motives and deletion-contraction relations
We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions
An introduction to motivic zeta functions of motives
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale,
...
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