Content-Type: multipart/mixed; boundary="-------------0311131245216" This is a multi-part message in MIME format. ---------------0311131245216 Content-Type: text/plain; name="03-499.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-499.keywords" Conformal Field Theory, Lattice models, relativistic field theory, solvable models, quantum Knizhnik-Zamolodchikov equation, spin chains, form factors ---------------0311131245216 Content-Type: application/x-tex; name="faroproc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="faroproc.tex" \documentclass[12pt]{article} %\voffset -.50in %\hoffset -.19in %\oddsidemargin 0in \evensidemargin 0in %\marginparwidth .75in \marginparsep 7pt \topmargin 0in %\headheight 12pt \headsep .25in %\footheight 18pt \footskip .35in %\textheight 9.5in \textwidth 6.5in \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{epsfig} \usepackage{times} \usepackage{epsf} %\input amstex %\documentstyle{amsppt} %\NoRunningHeads %\TagsAsMath %\TagsOnRight %\NoBlackBoxes \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\non{ \nonumber } \def\b{\beta} \def\th{\theta} \def\t{\text{t}} \def\s{\text{s}} \def\a{\alpha} \def\si{\sigma} \def\la{\lambda} \def\e{\epsilon} \def\half{\textstyle{\frac 1 2}} \begin{document} \begin{center} {\bf Connecting lattice and relativistic models via conformal field theory.} \end{center} \phantom{a} \vspace{1.5cm} \centerline{H. E. Boos \footnote{on leave of absence from the Institute for High Energy Physics, Protvino, 142284, Russia}} \centerline{\it Max-Planck Institut f{\"u}r Mathematik} \centerline{\it Vivatsgasse 7, 53111 Bonn, Germany} \phantom{a} \vspace{0.5cm} \phantom{a} \centerline{V. E. Korepin } \centerline{\it C.N.~Yang Institute for Theoretical Physics} \centerline{\it State University of New York at Stony Brook} \centerline{\it Stony Brook, NY 11794--3840, USA} \phantom{a} \vspace{0.5cm} \phantom{a} \centerline{F.A. Smirnov \footnote{Membre du CNRS} } \centerline{\it LPTHE, Tour 16, 1-er {\'e}tage, 4, pl. Jussieu} \centerline{\it 75252, Paris Cedex 05, France} \vspace{1cm} \vskip2em \begin{abstract} \noindent We consider the quantum group invariant XXZ-model. In infrared limit it describes Conformal Field Theory (CFT) with modified energy-momentum tensor. The correlation functions are related to solutions of level -4 of qKZ equations. We describe these solutions relating them to level 0 solutions. We further consider general matrix elements (form factors) containing local operators and asymptotic states. We explain that the formulae for solutions of qKZ equations suggest a decomposition of these matrix elements with respect to states of corresponding CFT. \end{abstract} \newpage \section {Quantum group invariant XXZ-model.} Let us recall some well known facts concerning XXZ-model and its continuous limit. Usually XXZ-model is considered as thermodynamic limit of finite spin chain. Consider the space $\(\mathbb{C}^2\)^{\otimes N}$. The finite spin chain in question is described by the Hamiltonian: \begin{align} H_{XXZ}=\sum\limits _{k=1}^N(\si ^1_k\si _{k+1}^1+\si ^2_k\si _{k+1}^2+ \Delta\si ^3_k\si _{k+1}^3)\label{hxxz} \end{align} where the periodic boundary conditions are implied: $\si _{N+1}=\si _1$. We consider the critical case $|\Delta |<1$ and parametrize it as follows: $$\Delta =\cos \pi \nu$$ It is well-known that in the infrared limit the model describes Conformal Field Theory (CFT) with $c=1$ and coupling constant equal to $\nu$. The correlation functions in the thermodynamic limit were found by Jimbo and Miwa \cite{jm}. It is equally matter of common knowledge that the model is closely related to the R-matrix: \begin{align} R(\b, \nu)= %\frac { R_0(\b)} {\b+\pi i} \left( \begin{array}{cccc} a(\b)&0&0&0\\ 0&b(\b)&c(\b)&0\\ 0&c(\b)&b(\b)&0\\ 0&0&0&a(\b) \end{array} \right) \label{R-m} \end{align} where \begin{align} &a(\b)= R_0(\b),\quad b(\b)= -R_0(\b)\frac {\sinh \nu \b} {\sinh \nu (\pi i-\b)}\non \\ &c(\b)=R_0(\b)\frac {\sinh \nu\pi i}{\sinh \nu(\pi i -\b)} \non \\ & R_0(\b)=\exp \left\{ i\int\limits _0^{\infty} \frac { \sin (\b k)\sinh\frac {\pi k(\nu -1)}{2\nu} } { k\sinh\frac {\pi k}{2\nu}\cosh \frac{\pi k}{2 } } \right\} \non \end{align} The coupling constant $\nu $ will be often omitted from $R(\b ,\nu)$. The relation between R-matrix and XXZ-model is explained later. From the point of view of mathematics the R-matrix (\ref{R-m}) is the R-matrix for two-dimensional evaluation representations of the quantum affine algebra $U_q(\widehat{sl}_2)$. The latter algebra contains two sub-algebras $U_q(sl_2)$. Let us perform a gauge transformation with the R-matrix in order to make the invariance with respect to one of them transparent: \begin{align} &\mathcal{R}(\b_1,\b_2,\nu)= e^{\frac {\nu} 2\b _1\si ^3}\otimes e^{\frac {\nu} 2\b _2\si ^3}\ R(\b_1-\b _2,\nu) \ e^{-\frac {\nu} 2\b _1\si ^3}\otimes e^{-\frac {\nu} 2\b _2\si ^3}= \non\\&= \frac {R_0(\b_1-\b_2)}{2\sinh \nu (\pi i-\b_1+\b_2)} \(e^{\nu (\b_1-\b _2)} R_{21}^{-1}(q)-e^{\nu (\b_2-\b _1)}R_{12}(q)\) \end{align} where $$q=e^{2i\pi(\nu+1)}$$ Adding $1$ to $\nu$ is important since we will use fractional powers of $q$. Here $R(q)$ is usual R-matrix for $U_q(sl_2)$: $$ R_{12}(q)= \left( \begin{array}{cccc} q^{\frac1 2}&0&0&0\\ 0&1&q^{\frac1 2}-q^{-\frac1 2}&0\\ 0&0&1&0\\ 0&0&0&q^{\frac1 2} \end{array} \right) $$ We want to use this quantum group symmetry. Unfortunately, the Hamiltonian (\ref{hxxz}) is not invariant with respect to the action of the quantum group which is represented in the space $\(\mathbb{C}^2\)^{\otimes N}$ by \begin{align} &S^3=\sum\limits _{k=1}^N \si ^3_k\non\\ &S^{\pm}=\sum\limits _{k=1}^N q^{-\frac{\si _1^{3}}4}\cdots q^{-\frac{\si_{k-1} ^{3}}4} \si ^{\pm}_kq^{\frac{\si_{k+1} ^{3}}4}\cdots q^{\frac{\si_{N} ^{3}}4}\non \end{align} A solution of this problem of quantum group invariance was found by Pasquier and Saleur \cite{ps}. They proposed to consider another integrable model on the finite lattice with Hamiltonian corresponding to open boundary conditions: \begin{align} H_{RXXZ}=\sum\limits _{k=1}^{N-1}(\si ^1_k\si _{k+1}^1+\si ^2_k\si _{k+1}^2+ \Delta\si ^3_k\si _{k+1}^3) +i\sqrt{1-\Delta}\ (\si _1^3-\si _N^3) \label{hrxxz} \end{align} This Hamiltonian is manifestly invariant under the action of quantum group on the finite lattice. After the thermodynamic limit one obtains a model with the same spectrum as original XXZ, but different scattering (this point will be described later). The infrared limit corresponds to CFT with modified energy-momentum tensor of central charge $$c=1-\frac {6\nu ^2}{1-\nu}$$ especially interesting when $\nu$ is rational and additional restriction takes place. In the present paper we shall consider RXXZ-model. We shall propose formulae for correlators for this model showing their similarity with correlators for XXX-model. The latter can be expressed in terms of values of Riemann zeta-function at odd natural arguments. We shall obtain an analogue of this statement for RXXZ-model. Let us say few words about hypothetic relation of XXZ and RXXZ models in thermodynamic limit. The argument that this limit should not depend on the boundary conditions must be dismissed in our situation since we consider a critical model with long-range correlations. Still we would expect that the following relation between two models in infinite volume exists. The quantum group $U_q(sl_2)$ acts on infinite XXZ-model and commute with the Hamiltonian. Consider a projector $\mathcal{P}$ on the invariant subspace. We had XXZ-vacuum $|\text{vac}\rangle _{XXZ}$. We suppose that the RXXZ-model is obtained by projection, in particular: $$|\text{vac}\rangle _{RXXZ}=\mathcal{P}|\text{vac}\rangle _{XXZ}$$ The correlators in RXXZ-model are $$ {\ }_{RXXZ}\langle \text{vac}|\mathcal{O}|\text{vac}\rangle _{RXXZ}= {\ }_{XXZ}\langle \text{vac}|\mathcal{P}\mathcal{O}\mathcal{P}|\text{vac}\rangle _{XXZ} $$ which can be interpreted in two ways: either as correlator in RXXZ-model or as correlator of $U_q(sl_2)$-invariant operator $\mathcal{P}\mathcal{O}\mathcal{P}$ in XXZ-model. This assumption explains the notation RXXZ standing for Restricted XXZ-model. So, we assume that in the lattice case a phenomenon close to the one taking place in massive models occurs \cite{rs}. Let us explain in some more details the set of operators in XXZ model for which we are able to calculate the correlators in simple form provided the above reasoning holds. Under $\mathcal{O}$ we understand some local operator of XXZ-chain, i.e. a product of several local spins $\si ^a_k$, $a=1,2,3$. Under the above action of quantum group these spins transform with respect to 3-dimensional adjoint representation. The projection $\mathcal{P}\mathcal{O}\mathcal{P}$ extracts all the invariant operators, i.e. projects over the subspace of singlets in the tensor product of 3-dimensional representations. Let us explain more explicitly the relation between the the R-matrix and XXZ, RXXZ Hamiltonians. Both of them can be constructed form the transfer-matrix with different boundary conditions constructed via the monodromy matrix: $$ R_{01}(\la)R_{02}(\la )\cdots R_{0,N-1}(\la )R_{0,N}(\la )$$ In some cases it is very convenient to consider inhomogeneous model for which the monodromy matrix contains a fragment: $$R_{0k}(\la -\la _k)\cdots R_{0,k+n}(\la -\la _{k+n})$$ As we shall see many formulae become far more transparent for inhomogeneous case. \section{QKZ on level -4 and correlators.} The main result of Kyoto group \cite{jmmn,jm} is that the correlators in XXZ-model are related to solutions of QKZ-equations \cite{SKyo,FR} on level -4. We formulate the equations first and then explain the relation. The equations for the function $g(\b _1,\cdots,\b _{2n})\in \mathbb{C}^{\otimes 2n}$ are \begin{align} &R(\b_j -\b _{j+1})g(\b _1,\cdots ,\b _{j+1},\b _j,\cdots,\b _{2n}) =&\non\label{symm}\\ &= \ \ g(\b _1,\cdots ,\b _{j},\b _{j+1},\cdots,\b _{2n}) %\nonumber\\&\ \non\\ \end{align} \begin{align} &g(\b _1,\cdots ,\b _{2n-1},\b _{2n}+2\pi i) =%\non\\&= g(\b _{2n},\b _1,\cdots ,\b _{2n-1}) %\non \label{Rie} \end{align} For application to correlators a particular solution is needed which satisfies additional requirement: \begin{align} &g(\b _1,\cdots ,\b _{j},\b _{j+1},\cdots,\b _{2n})|_{\b _{j+1}=\b _j-\pi i}= s_{j,j+1}\otimes g(\b _1,\cdots ,\b _{j-1},\b _{j+2},\cdots,\b _{2n})\label{norm} \end{align} where $s_{j,j+1}$ is the vector $(\uparrow\downarrow)+(\downarrow\uparrow)$ in the tensor product of $j$-th and $(j+1)$-th spaces. The relation of these equations to correlators is conjectured by Jimbo and Miwa \cite{jm}. It cannot be proved for critical model under consideration as it was done for the XXZ-model with $|q|<1$ in \cite{jmmn}. However, later arguments based on Bethe Anzatz technique were proposed by Maillet and collaborators \cite{Maillet1,Maillet2} which can be considered as a proof of Jimbo and Miwa conjecture. Jimbo and Miwa find the solution needed \cite{jm} in the form: \begin{align} &g(\b _1,\cdots, \b _{2n})=\frac 1 {\sum e ^{\b_j}} \prod\limits _{i