Alexander Berkovich

Fermionic counting of RSOS states and Virasoro character formulas for the unitary minimal series M(v,v + 1): Exact results

Abstract The Hilbert space of an RSOS model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a0, a1,…, aL}, with aj-integers restricted by 1 ≤ aj ≤ v, | aj − aj+1 |=1, a0 ≡ s, aL ≡ r. In this paper we introduce different basis which, as shown here, has the same dimension as that of an RSOS model. This basis appears naturally in the Bethe ansatz calculations of the spin (v−1)/2 XXZ model. Following McCoy et al., we call this basis fermionic (FB). Our first theorem Dim(FB) = Dim(RSOS − basis) can be succinctly expressed in terms of some identities for binomial coefficients. Remarkably, these binomial identities can be q-deformed. Here, we give a simple proof of these q-binomial identities in the spirit of Schurs proof of the Rogers-Ramanujan identities. Notably, the proof involves only the elementary recurrences for the q-binomial coefficients and a few creative observations. Finally, taking the limit L → ∞ in these q-identities, we derive an expression for the character formulas of the unitary minimal series M(v,v + 1) “Bosonic Sum ≡ Fermionic Sum”. Here, Bosonic Sum denotes Rocha-Caridi representation (Xr,s=1v,v+1 (q)) and Fermionic Sum stands for the companion representation recently conjectured by the McCoy group.

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.

Rogers--Schur--Ramanujan Type Identities for the M(p,p') minimal models of Conformal Field Theory

Abstract:We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M(p,p′). The proof uses the continued fraction decomposition of p′/p introduced by Takahashi and Suzuki for the study of the Bethes Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. We use this method to get fermionic representations of the characters for many classes of r and s.

Polynomial identities, indices, and duality for theN=1 superconformal modelSM(2, 4v)

We prove polynomial identities for theN=1 superconformal modelSM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses theq-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations forq-series on the fermionic side. We use these polynomials to demonstrate a dual relation underq→q−1 betweenSM(2, 4v) andM(2v−1, 4v). We also introduce a genralization of the Witten index which is expressible in terms of the Rogers false theta functions.

N = 2 supersymmetry and Bailey pairs

We demonstrate that the Bailey pair formulation of Rogers—Ramanujan identities unifies the calculations of the characters of N = 1 and N = 2 supersymmetric conformal field theories with the counterpart theory with no supersymmetry. We illustrate this construction for the M(3,4) (Ising) model where the Bailey pairs have been given by Slater. We then present the general unitary case. We demonstrate that the model M(p,p + 1) is derived from M(p − 1, p) by a Bailey renormalization flow and conclude by obtaining the N = 1 model SM(p,p + 2) and the unitary N = 2 model with central charge c = 3(1 − 2/p).

Generalizations of the Andrews-Bressoud identities for the N = 1 superconformal model SM(2, 4ν)

We present generalized Rogers-Ramanujan identities which relate the Fermi and Bose forms of all the characters of the superconformal model SM(2, 4@n). In particular, we show that to each bosonic form of the character there is an infinite family of distinct fermionic q-series representations.