c = 1 − 6(n − 1)2/n, THEORIES COUPLED TO GRAVITY: A COMMENT ON THEIR POSSIBLE LATTICE MODELS REALIZATIONS

We discuss the recently proposed logarithmic violation of scaling for c = 1 − 6(n − 1)2/n theories in the light of lattice models. We study for this purpose the Q state Potts model in its antiferromagnetic regime eK − 1 = −Q1/2, coupled to gravity. Setting Q1/2 = 2 cos π/t, this model is known to have a generic central charge c = 1 − 6(t − 1)2/t. Summing over all possible planar graphs allows us to make connection with Kostov's solution of IRF models, and to calculate the genus zero properties along the critical line. Except for n = 1, 2 we do not get indications of logarithmic violations. The apparent regularity of the thermodynamic properties (γ str = −(n − 1) = integer ) is explained by a discontinuity of the free energy of the Potts model when Q crosses the Beraha numbers [Formula: see text], n ≥ 3 in the antiferromagnetic region. Such behavior was recently observed for some regular lattices. The logarithmic terms for n = 2, c = −2 appear simply because a derivative with respect to Q has to be taken to define a non-vanishing partition function. Only the n = 1, c = 1 logarithmic terms seem to have a non-trivial origin.