The Critical Ising Model via Kac-Ward Matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^2g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer ${\overline{\partial}}$ -torsions of the Riemann surface in which G embeds.