Continued fractions and fermionic representations for characters of M(p,p′) minimal models

We present fermionic sum representations of the characters χτ, s(p, p′) of the minimal M(p,p′) models for all relatively prime integers p′>p for some allowed values of r and s. Our starting point is biomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1/2 chain of anisotropy −Δ=−cos(π(p/p′)). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of {p/p′} (and p/p′), where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the Hermitian irreducible representations of SUq- (2) (and SUq+ (2)) with q−=exp(iπ{p/p}) (and q+=exp(iπ(p/p′))). We also establish the duality relation M(p,p′) ↔ M(p′−p,p′) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.