Fermionic solution of the Andrews-Baxter-Forrester model I: unification of TBA and CTM methods

The problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto the problem of evaluating the grand-canonical partition function of a gas of charged particles obeying certain fermionic exclusion rules. We thus obtain a new {\em fermionic} method to compute the local height probabilities of the model. Combined with the original {\em bosonic} approach of Andrews, Baxter and Forrester, we obtain a new proof of (some of) Melzer's polynomial identities. In the infinite limit these identities yield Rogers--Ramanujan type identities for the Virasoro characters $\chi_{1,1}^{(r-1,r)}(q)$ as conjectured by the Stony Brook group. As a result of our working the corner transfer matrix and thermodynamic Bethe Ansatz approaches to solvable lattice models are unified.