Fermionic Character Sums and the Corner Transfer Matrix

We present a “natural finitization” of the fermionic q-series (certain generalizations of the Rogers-Ramanujan sums) which were recently conjectured to be equal to Virasoro characters of the unitary minimal conformal field theory (CFT) M(p, p+ 1). Within the quasi-particle interpretation of the fermionic q-series this finitization amounts to introducing an ultraviolet cutoff, which – contrary to a lattice spacing – does not modify the linear dispersion relation. The resulting polynomials are conjectured (proven, for p=3,4) to be equal to corner transfer matrix (CTM) sums which arise in the computation of order parameters in regime III of the r=p+1 RSOS model of Andrews, Baxter, and Forrester. Following Schur’s proof of the Rogers-Ramanujan identities, these authors have shown that the infinite-lattice limit of the CTM sums gives what later became known as the RochaCaridi formula for the Virasoro characters. Thus we provide a proof of the fermionic q-series representation for the Virasoro characters for p=4 (the case p=3 is “trivial”), in addition to extending the remarkable connection between CFT and off-critical RSOS models. We also discuss finitizations of the CFT modular-invariant partition functions. 1 Address after Sept. 1, 1993: School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel. 1. Prelude This paper is concerned with generalizations of the following identities: