QKZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime

An integral solution to the quantum Knizhnik-Zamolodchikov ($q$KZ) equation with $|q|=1$ is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest neighbor correlator with an arbitrary coupling constant, and general correlators in the XXX and XY limits

Three-point function of semiclassical states at weak coupling

We give the derivation of the previously announced analytic expression for the correlation function of three heavy non-BPS operators in N=4 super-Yang-Mills theory at weak coupling. The three operators belong to three different su(2) sectors and are dual to three classical strings moving on the sphere. Our computation is based on the reformulation of the problem in terms of the Bethe Ansatz for periodic XXX spin-1/2 chains. In these terms the three operators are described by long-wave-length excitations over the ferromagnetic vacuum, for which the number of the overturned spins is a finite fraction of the length of the chain, and the classical limit is known as the Sutherland limit. Technically our main result is a factorized operator expression for the scalar product of two Bethe states. The derivation is based on a fermionic representation of Slavnov's determinant formula, and a subsequent bosonisation.Comment: 28 pages, 5 figures, cosmetic changes and more typos corrected in v

Fermionic solution of the Andrews-Baxter-Forrester model II: proof of Melzer's polynomial identities

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters $\chi_{b,a}^{(r-1,r)}(q)$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers--Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.Comment: 28 pages, Latex, 7 Postscript figure

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

(l,q)(l,q)-Deformed Grassmann Field and the Two-dimensional Ising Model

In this paper we construct the exact representation of the Ising partition function in the form of the $SL_q(2,R)$-invariant functional integral for the lattice free $(l,q)$-fermion field theory ($l=q=-1$). It is shown that the $(l,q)$-fermionization allows one to re-express the partition function of the eight-vertex model in external field through functional integral with four-fermion interaction. To construct these representations, we define a lattice $(l,q,s)$-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At $l=q=-1, s=1$ we obtain the lattice $(l,q)$-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over $(q,s)$-Grassmann variables is expressed through the $(q,s)$-deformed Pfaffian which is equal to square root of the determinant of some matrix at $q=\pm 1, s=\pm 1$.Comment: 24 pages, LaTeX; minor change

Off-Critical Logarithmic Minimal Models

We consider the integrable minimal models ${\cal M}(m,m';t)$, corresponding to the $\varphi_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m, m'\to\infty$, $m/m'\to p/p'$ where $p, p'$ are coprime and the limit is taken through coprime values of $m,m'$. We view these off-critical minimal models ${\cal M}(m,m';t)$ as the continuum scaling limit of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. Applying Corner Transfer Matrices to the Forrester-Baxter RSOS models in Regime III, we argue that taking first the thermodynamic limit and second the {\it logarithmic limit\,} yields off-critical logarithmic minimal models ${\cal LM}(p,p';t)$ corresponding to the $\varphi_{1,3}$ perturbation of the critical logarithmic minimal models ${\cal LM}(p,p')$. Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ${\cal LM}(p,p')$. We also calculate the logarithmic limit of certain off-critical observables ${\cal O}_{r,s}$ related to One Point Functions and show that the associated critical exponents $\beta_{r,s}=(2-\alpha)\,\Delta_{r,s}^{p,p'}$ produce all conformal dimensions $\Delta_{r,s}^{p,p'}<{(p'-p)(9p-p')\over 4pp'}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(p s-p' r)^2< 8p(p'-p)$. The exponent $2-\alpha ={p'\over 2(p'-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $\Delta_t={1-\alpha\over 2-\alpha}={2p-p'\over p'}=\Delta_{1,3}^{p,p'}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\cal O}_{r,s}$ diverge at criticality.Comment: 18 pages, 5 figures; version 3 contains amplifications and minor typographical correction

Mixing of Ground States in Vertex Models

We consider the analogue of the 6-vertex model constructed from alternating spin n/2 and spin m/2 lines, where $1\leq n<m$. We identify the transfer matrix and the space on which it acts in terms of the representation theory of $U_q(sl_2)$. We diagonalise the transfer matrix and compute the S-matrix. We give a trace formula for local correlation functions. When n=1, the 1-point function of a spin m/2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the 1-point functions of the pure spin m/2 model with different ground states. The mixing ratios are calculated exactly and are expressed in terms of irreducible characters of $U_q(sl_2)$ and the deformed Virasoro algebra.Comment: 12 pages, LaTeX, typos correcte

Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4\nu)

We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.Comment: 41 pages, harvmac, no figures; new identities, proofs and comments added; misprints eliminate

Bulk correlation functions in 2D quantum gravity

We compute bulk 3- and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c<1, following and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground ring generators deformed by Liouville and matter ``screening charges''. A by-product general formula for the matter 3-point OPE structure constants is derived. We also consider a ``diagonal'' CFT of 2D quantum gravity, in which the degenerate fields are restricted to the diagonal of the semi-infinite Kac table. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points.Comment: 14 pages, 2 figure