Derivation of the Kač formula in the Coulomb gas picture

Using the equivalence between a special class of Restricted Solid On Solid models (the so-called Am models) and the six-vertex model a relation between the partition functions of these models on a L×M torus is obtained. For the six-vertex model this partition function is known if one uses the Kosterlitz–Thouless renormalization connection with the Gaussian model. The critical dimensions of the operators of the theory that, according to the results of conformal symmetry, can be read off this partition function correspond to those known from the Coulomb Gas picture but contain the renormalized Gaussian coupling as a parameter that remains to be specified. We show that the expression of the Am partition function in terms of six-vertex partition functions is so restrictive that it enforces a specific value (depending on m) of the renormalized Gaussian coupling. Using this value it turns out that the critical dimension of the operators of the Am model are given by the Kač formula