1. Prologue 1.1. Introduction The rich mathematical structure of the two-dimensional conformal field theories of Belavin, Polyakov, and Zamolodchikov [9, 18] is afforded by the presence of infinite dimensional Lie algebras such as the Virasoro algebra in their symmetries. Indeed, the presence of these algebras ensures the solvability of the theories. In the conformal field theories known as the minimal models, which are denoted Mp'p where 1 p p' with p and p' coprime, the spectra are expressible in terms of the Virasoro characters xp]p where 1 r p and 1 s pf. The irreducible highest weight module corresponding to the character xP}P n a s central charge cp,p and conformal dimension AP,P given by: (1.1) ?* = 1 - 6{P '-,P)\ A? f - {P 'T - PS)2 - {P ' ~ P? ppf , /^ppf In [19, 36, 20], it was shown that xp,P = qAr's xp]p where the (normalised) character xp]p is given by: 1 oo Q_2) v p ' p / = V ^ / \2pp' + \(p'r-ps) _ (Ap+r)(Ap'+s)\ ^ ° ° A=-oo and as usual, (q)^ = UZii1 ~ ^ ) - N o t e t h a t Xp]p' = x£_r,p/_s- An expression such as (1.2) is known as bosonic because it arises naturally via the construction of a Fock space using bosonic generators. Submodules are factored out from the Fock space using an inclusion-exclusion procedure, thereby leading to an expression involving the difference between two constant-sign expressions. However, there exist other expressions for xp,p that provide an intrinsic phys- ical interpretation of the states of the module in terms of quasiparticles. These expressions are known as fermionic expressions because the quasiparticles therein are forbidden to occupy identical states. Two of the simplest such expressions arise when p' 5 and p 2\ 00 n n2 °° „n(n+l) /i o\ 2,5 V ^ Q 2,5 V ^ Q n=0 W n n=0 W n Here, (q)o = 1 and (q)n = nr=i(l #*) ^or n 0- Equating expressions of this fermionic type with the corresponding instances of (1.2) yields what are known as bosonic-fermionic g-series identities. In this paper, we give fermionic expressions for all Xr,p I n doing so, we obtain a bosonic-fermionic identity for each character A.r,s * 1
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