The Andrews-Gordon identities and qq-multinomial coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the identities involves $q$-deformations of the coefficients of $x^a$ in the expansion of $(1+x+\cdots+ x^k)^L$. A combinatorial interpretation for these $q$-multinomial coefficients is given using Durfee dissection partitions. The fermionic side of the polynomial identities arises as the partition function of a one-dimensional lattice-gas of fermionic particles. In the limit $L\to\infty$, our identities reproduce the analytic form of Gordon's generalization of the Rogers--Ramanujan identities, as found by Andrews. Using the $q \to 1/q$ duality, identities are obtained for branching functions corresponding to cosets of type $({\rm A}^{(1)}_1)_k \times ({\rm A}^{(1)}_1)_{\ell} / ({\rm A}^{(1)}_1)_{k+\ell}$ of fractional level $\ell$.Comment: 31 pages, Latex, 9 Postscript figure

A-D-E Polynomial and Rogers--Ramanujan Identities

We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal G}$=A$_{n-1}$ \mbox{$(\ell\geq 2)$}, D$_{n-1}$ $(\ell\geq 2)$, E$_{6,7,8}$ $(\ell=2)$. In support of our conjectures we establish the correct behaviour under level-rank duality for $\cal G$=A$_{n-1}$ and show that the A-D-E Rogers--Ramanujan identities have the expected $q\to 1^{-}$ asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary cosets are also discussed briefly.Comment: 19 pages, Latex, 1 Postscript figur

Magnetic Correlation Length and Universal Amplitude of the Lattice E_8 Ising Model

The perturbation approach is used to derive the exact correlation length $\xi$ of the dilute A_L lattice models in regimes 1 and 2 for L odd. In regime 2 the A_3 model is the E_8 lattice realisation of the two-dimensional Ising model in a magnetic field h at T=T_c. When combined with the singular part f_s of the free energy the result for the A_3 model gives the universal amplitude $f_s \xi^2 = 0.061~728...$ as $h\to 0$ in precise agreement with the result obtained by Delfino and Mussardo via the form-factor bootstrap approach.Comment: 7 pages, Late

Lattice Ising model in a field: E8_8 scattering theory

Zamolodchikov found an integrable field theory related to the Lie algebra E$_8$, which describes the scaling limit of the Ising model in a magnetic field. He conjectured that there also exist solvable lattice models based on E$_8$ in the universality class of the Ising model in a field. The dilute A$_3$ model is a solvable lattice model with a critical point in the Ising universality class. The parameter by which the model can be taken away from the critical point acts like a magnetic field by breaking the \Integer_2 symmetry between the states. The expected direct relation of the model with E$_8$ has not been found hitherto. In this letter we study the thermodynamics of the dilute A$_3$ model and show that in the scaling limit it exhibits an appropriate E$_8$ structure, which naturally leads to the E$_8$ scattering theory for massive excitations over the ground state.Comment: 11 pages, LaTe

Characters of graded parafermion conformal field theory

The graded parafermion conformal field theory at level k is a close cousin of the much-studied Z_k parafermion model. Three character formulas for the graded parafermion theory are presented, one bosonic, one fermionic (both previously known) and one of spinon type (which is new). The main result of this paper is a proof of the equivalence of these three forms using q-series methods combined with the combinatorics of lattice paths. The pivotal step in our approach is the observation that the graded parafermion theory -- which is equivalent to the coset osp(1,2)_k/ u(1) -- can be factored as (osp(1,2)_k/ su(2)_k) x (su(2)_k/ u(1)), with the two cosets on the right equivalent to the minimal model M(k+2,2k+3) and the Z_k parafermion model, respectively. This factorisation allows for a new combinatorial description of the graded parafermion characters in terms of the one-dimensional configuration sums of the (k+1)-state Andrews--Baxter--Forrester model.Comment: 36 page

Lattice realizations of unitary minimal modular invariant partition functions

The conformal spectra of the critical dilute A-D-E lattice models are studied numerically. The results strongly indicate that, in branches 1 and 2, these models provide realizations of the complete A-D-E classification of unitary minimal modular invariant partition functions given by Cappelli, Itzykson and Zuber. In branches 3 and 4 the results indicate that the modular invariant partition functions factorize. Similar factorization results are also obtained for two-colour lattice models.Comment: 18 pages, Latex, with minor corrections and clarification

Bailey flows and Bose-Fermi identities for the conformal coset models (A1(1))N×(A1(1))N′/(A1(1))N+N′(A^{(1)}_1)_N\times (A^{(1)}_1)_{N'}/(A^{(1)}_1)_{N+N'}

We use the recently established higher-level Bailey lemma and Bose-Fermi polynomial identities for the minimal models $M(p,p')$ to demonstrate the existence of a Bailey flow from $M(p,p')$ to the coset models $(A^{(1)}_1)_N\times (A^{(1)}_1)_{N'}/(A^{(1)}_1)_{N+N'}$ where $N$ is a positive integer and $N'$ is fractional, and to obtain Bose-Fermi identities for these models. The fermionic side of these identities is expressed in terms of the fractional-level Cartan matrix introduced in the study of $M(p,p')$. Relations between Bailey and renormalization group flow are discussed.Comment: 28 pages, AMS-Latex, two references adde

Exceptional structure of the dilute A3_3 model: E8_8 and E7_7 Rogers--Ramanujan identities

The dilute A$_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing the 1-dimensional configuration sums in terms of fermionic sums which explicitly involve the E$_8$ root system. In the thermodynamic limit, these polynomial identities yield a proof of the E$_8$ Rogers--Ramanujan identity recently conjectured by Kedem {\em et al}. The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by $q\to 1/q$. In this case we find an A_1\times\mbox{E}_7 structure and prove a Rogers--Ramanujan identity of A_1\times\mbox{E}_7 type. Finally, in the critical $q\to 1$ limit, we give some intriguing expressions for the number of $L$-step paths on the A$_3$ Dynkin diagram with tadpoles in terms of the E$_8$ Cartan matrix. All our findings confirm the E$_8$ and E$_7$ structure of the dilute A$_3$ model found recently by means of the thermodynamic Bethe Ansatz.Comment: 9 pages, 1 postscript figur