Potts model: Duality, Uniformization and the Seiberg–Witten modulus

The introduction of a modulus z(K), analogous to u = 〈trφ2〉 in the N = 2 SUSY SU(2) gauge theory solved by Seiberg and Witten, and whose defining property is the invariance under the symmetry and duality transformations of the effective coupling K, reveals an intriguing correspondence between the D = 2 Ising and Potts models on the square lattice. The moduli spaces of both models, the spaces of inequivalent effective temperatures K, correspond to a three–punctured sphere M3 = P1(C)\{z = ±1,∞}. Furthermore, in both models, the locus of Fisher zeroes is given by the segment joining zc = −1 to zc = +1. This naturally leads to conjecture that the free energy of the Potts model may indeed be expressed in terms of hypergeometric functions and thus satisfy a Picard–Fuchs–like equation in the modulus z, as in the Ising case. The free energy F(z) is a polymorphic function on M3 and the analogue of the Kramers–Wannier self–duality relation is interpreted as a monodromy of the free energy around zc = 1.