Fermionic Character Sums and the Corner Transfer Matrix

TL;DR

This paper introduces a finitization of fermionic q-series related to Virasoro characters, proving their equivalence to corner transfer matrix sums in RSOS models, thus linking conformal field theory with statistical mechanics.

Contribution

It provides a proof for the fermionic q-series representation of Virasoro characters for p=4 and extends the connection between CFT and off-critical RSOS models.

Findings

Fermionic q-series finitization matches CTM sums in RSOS models.

Proof of Virasoro character formulas for p=4 case.

Extension of CFT and off-critical RSOS model connection.

Abstract

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 Rocha-Caridi 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.