\documentstyle[twocolumn,eqsecnum,aps]{revtex}
\def\btt#1{{\tt$\backslash$#1}}
\def\BibTeX{\rm B{\sc ib}\TeX}

\newcommand{\nc}{\newcommand}
\nc{\rnc}{\renewcommand}
\nc{\bc}{\begin{center}}
\nc{\ec}{\end{center}}
\nc{\e}[1]{{\em#1\/}}
\rnc{\b}[1]{{\bf#1}}
\nc{\comment}[1]{}
\nc{\spos}[2]{\makebox(0,0)[#1]{${\scriptstyle#2}$}}
\nc{\be}{\begin{equation}}
\nc{\ee}{\end{equation}}
\nc{\bea}{\begin{eqnarray}}
\nc{\eea}{\end{eqnarray}}
\nc{\disp}{\displaystyle}
\nc{\ade}{\mbox{$A$-$D$-$E$}}
\nc{\calN}{{\cal N}}
\nc{\calC}{{\cal C}}
\nc{\calM}{{\cal M}}
\nc{\calS}{{\cal S}}
\nc{\phit}{\hat{\varphi}}
\nc{\chit}{\hat{\chi}}
\nc{\hcalN}{\hat{\calN}}
\nc{\hcalS}{\hat{\calS}}
\nc{\hS}{\tilde{S}}             %{\hat{S}}
\nc{\sigmad}{\sigma^\dagger}
\nc{\psid}{\psi^\dagger}
\def\smat#1{\mbox{\scriptsize{\mbox{$\pmatrix{#1}$}}}}
\def\hV{\hat{V}}
\def\hN{\hat{N}}
\def\hcalV{\hat{{\cal V}}}

\font\tenmsb=msbm10
\font\sevenmsb=msbm7
\font\fivemsb=msbm5
\newfam\msbfam
\textfont\msbfam=\tenmsb
\scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb
\def\Bbb#1{{\fam\msbfam\relax#1}}


\begin{document}
\draft
\preprint{HEP/123-qed}
\title{Integrable Boundaries, Conformal Boundary Conditions\\
and $A$-$D$-$E$ Fusion Rules
}
\author{Roger E. Behrend and Paul A. Pearce}
\address{
Department of Mathematics and Statistics\\
University of Melbourne, Parkville, Victoria 3052, Australia
}
\author{Jean-Bernard Zuber}
\address{
Service de Physique Th\'eorique\\
CEA-Saclay, 91191 Gif-sur-Yvette,
Cedex, France
}
\date{\today}
\maketitle
\widetext
\begin{abstract}
The $sl(2)$ minimal theories are
labelled by a Lie algebra pair $(A,G)$ where $G$ is of
$A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete
set of
conformal boundary conditions labelled by the nodes of the tensor product
graph $A\otimes G$.
The cylinder partition functions are given by fusion rules arising from the
graph fusion algebra of $A\otimes G$. We further conjecture that, for
each conformal boundary condition, an integrable boundary condition exists
as a solution of the boundary
Yang-Baxter equation for the associated lattice model. The theory is
illustrated using the $(A_4,D_4)$
or 3-state Potts model. % Some comments are made concerning the
%generalization to higher rank.
\end{abstract}
%\pacs{Valid PACS appear here.
%{\tt$\backslash$\string pacs\{\}} should always be input,
%even if empty.}
\pacs{PACS numbers:\ 11.25.Hf,\ 05.50.+q,\ 75.10.Hk}


\narrowtext

\section{INTRODUCTION}
\label{sec:level1}

The study of conformal boundary conditions~\cite{Cardy89} continues to be
an active area of research
with applications in statistical mechanics and string theory. The problem of
a general classification of conformal boundary conditions has seen a
revival of interest  recently. For theories %that are diagonal in the bulk
with a diagonal torus partition function
it is known that there is a conformal boundary condition associated to each
operator appearing in the
theory. Moreover, the fusion rules of these boundary operators
%the operator algebra
are just given by the bulk fusion algebra and thus %are related to
%the modular transformation by
by the Verlinde formula~\cite{Verlinde88}. In contrast, for non-diagonal
theories, the fusion rules are not
known in general and  it is not even known %in general
what constitutes a complete set of conformal boundary
conditions. %for non-diagonal theories.
Indeed, these questions have only been
resolved~\cite{AfflOS98,FuchsS98} very recently for the simplest
non-diagonal theory, namely, the
critical 3-state Potts model. In this letter we conjecture a complete set
of conformal boundary
conditions, fusion rules and cylinder partition functions for the $sl(2)$
minimal models.

The $sl(2)$ minimal models in the bulk are
classified~\cite{CapItzZub87} by a pair of simply laced Dynkin diagrams
$(A,G)$ of  type
\be
(A,G)=\cases{
(A_{h-1},A_{g-1})&\cr
(A_{h-1},D_{(g+2)/2}),\quad g\ \mbox{even}&\cr
(A_{h-1},E_6), \quad g=12 &\cr
(A_{h-1},E_7), \quad g=18&\cr
(A_{h-1},E_8), \quad g=30.}
\ee
Here $h$ and $g$ are the coprime Coxeter numbers of
$A$ and $G$ and the central charges are
\be
c=1-\disp{6(h-g)^2\over hg}.
\ee
\goodbreak

$$\mbox{}$$
\vspace{1.0in}
\mbox{}


We conjecture that for these theories a complete set of conformal
boundary conditions $i$ and the corresponding boundary operators $\phit_i$ are
labelled by $i\in(A,G)$
\be
\phit_i:\quad i=(r,a)\in (A,G)%=A\otimes G
\label{boundops}
\ee
where $r$, $a$ are nodes on the Dynkin diagram of $A$ and $G$ respectively. We
will use $G$ to denote the Dynkin diagram and the adjacency
matrix of this graph. We use $r, r_1, r_2$ to denote nodes of $A_{h-1}$;
$s, s_1, s_2$ for the nodes of
$A_{g-1}$; $a, a_1, a_2, b$ for the nodes of $G$ and $i,j$
to label nodes in the pair $(A,G)$.

We now introduce fused adjacency matrices (intertwiners) and graph
fusion matrices.
The fused adjacency matrices
%$G^s$
$V_s$ with $s=1,\ldots,g-1$ are defined recursively by the
$sl(2)$ fusion algebra
\be
%G^s=G^{s-1}G^2-G^{s-2}
V_s=V_2 V_{s-1} -V_{s-2}
\ee
subject to the initial conditions $V_1=I$ and $V_2=G$. %$G^1=I$ and $G^2=G$.
%Here the superscripts denote recursion level and not powers.
The matrices $V_s$ %$G^s$
are symmetric and mutually commuting with entries given by a
Verlinde-type formula
\be
%G^s_{a,b}
V_{sa}{}^b=(V_s)_a{}^b
=\sum_{m\in \mbox{\scriptsize Exp}(G)}{\hS_{sm}\over
\hS_{1m}}\;\Psi_{am}\Psi_{bm}^*.
\ee
where the columns of the unitary matrices $\hS$ and $\Psi$ are
the eigenvectors of
the adjacency matrices $A_{g-1}$ and $G$ respectively and the sum is over
the Coxeter exponents of $G$
with multiplicities.
%
We assume the graph $G$ has a distinguished endpoint node labelled $a=1$
such that
$\Psi_{1m}>0$ for all $m$.
This is at least the case for \ade\ graphs.
In this notation we define the fundamental intertwiner as
 $\hV_s{}^a=V_{s1}{}^a$.
%\be
%\hV_s{}^a=V_{s1}{}^a %G^s_{1a}
%=\sum_{m\in \mbox{\scriptsize Exp}(G)}
%{\hS_{sm}\over \hS_{1m}}\;\Psi_{1m}\Psi_{am}^*.\label{fundintw}
%\ee


The graph fusion matrices $\hat{N}_a$ with $a\in G$
 were introduced by Pasquier~\cite{Pas}.
These  are defined
by the  Verlinde-type formula~\cite{DFZPZ}
\be
\hN_{ab}{}^c=(\hat{N}_a)_b{}^c= \sum_{m\in \mbox{\scriptsize Exp}(G)}
{\Psi_{am} \Psi_{bm} \Psi^*_{cm}\over \Psi_{1m}},\qquad a,b,c\in G.
\ee
These matrices satisfy the matrix recursion relation
\be
G\hN_a=\sum_{b\in G}G_a{}^b\hN_b
\ee
and initial conditions $\hN_1=I$ and $\hN_2=G$ where 2 denotes the unique
node adjacent to 1.  The numbers $\hN_{ab}{}^c$ are
the structure constants %of the fusion algebra
of the graph fusion algebra
\be
\hN_a\hN_b=\sum_{c\in G}\hN_{ab}{}^c \hN_c.
\ee
All the entries of the fused
adjacency matrices $V_s$ %and graph fusion matrices $\hN_a$
are nonnegative integers.
For a proper choice of the eigenvectors  and of the
node $1$, the entries of the graph fusion matrices $\hN_a$
are also integers,
and with the exception of %the graph fusion matrices of
$D_{2n+1}$
%$D_{\mbox{\scriptsize odd}}$
and $E_7$, they are nonnegative.
%
A key identity relating the fused adjacency matrices and graph fusion
matrices is
\be
V_s \hN_a =\sum_{b\in G} V_{sa}{}^b \hN_b\ .
%G^s_{ab}=\sum_{c\in G} G^s_{1c}\hN_{ca}{}^b=\sum_{c\in G}\hN_{a^*b}{}^c
%G^s_{1c}
\label{keyid}
\ee



\section{Fusion Rules}

Let $i_1%=(r_1,a_1)
$, $i_2 %=(r_2,a_2)
$ and $i_3%=(r_3,a_3)
\in (A,G)$ and
consider the tensor product graph
$A\otimes G$ with
distinguished node $i=1$ given by $i=(r,a)=(1,1)$. Then we conjecture
that the fusion rules for the boundary operators (\ref{boundops}) are
\be
\phit_{i_1}\;\times\;\phit_{i_2}=\sum_{i_3\in
(A,G)}\calN_{i_1 i_2}{}^{i_3}\,\phit_{i_3}
\ee
where $\calN_{i_1}$ are just the graph fusion matrices associated with the
tensor product graph $A\otimes G$
\be
\calN_{i_1 i_2}{}^{i_3}=
\calN_{(r_1,a_1)(r_2,a_2)}{}^{(r_3,a_3)}=N_{r_1 r_2}{}^{r_3}
\hN_{a_1 a_2}{}^{a_3}
\ee
where $N_{r_1}$ are the graph fusion matrices for $A_{h-1}$.
Let $\varphi_{r,s}$ %k$ where $k=(r,s)$
be the primary chiral fields
with respect to the Virasoro algebra. Then the operators
$\phit_i=\phit_{r,a}$ are related to
$\varphi_{r,s}$ by the intertwining relation
\be
\sum_{b\in G} \phit_{r,b}\,(\hV^T \hV)_b{}^a
= \sum_{s\in A_{g-1}}\varphi_{r,s} \hV_s{}^a \label{opintertw}
\ee
where $\hV$ is the fundamental adjacency matrix intertwiner
%given in (\ref{fundintw})
defined in sec. I. By equality in (\ref{opintertw})  we mean that
the operators on either side satisfy the same algebra
under fusion.



We define a conjugation operator $C(a)=a^*$ to be the identity
except for $D_{4n}$ graphs where the eigenvectors
 $\Psi_{am}$ are complex and conjugation
corresponds to the
${\Bbb Z}_2$ Dynkin diagram automorphism. It then follows that
 $\hN_{a^*b}{}^c=\hN_{ca}{}^b$.
We conjecture that the coefficients of
the cylinder partition functions $Z_{i_1|i_2}$ of the $sl(2)$ minimal
theories are given by the
fusion product $\phit_{i_1}^\dagger\times\phit_{i_2}$, that is
\begin{mathletters}\label{CPFs}
\be
Z_{i_1|i_2}(q)=\sum_{i_3\in (A,G)} \calN_{i_1^* i_2}{}^{i_3}\;\chit_{i_3}(q).
\ee
More explicitly,
\bea
&&Z_{(r_1,a_1)|(r_2,a_2)}(q)\nonumber\\
&=&\sum_{(r_3,a_3)\in(A_{h-1},G)}
\calN_{(r_1,a_1^*) (r_2,a_2)}{}^{(r_3,a_3)}\; \chit_{r_3,a_3}(q)\label{cpf1}\\
&=&
\sum_{(r,s)\in (A_{h-1},A_{g-1})}\chi_{r,s}(q)  N_{r r_1}{}^{r_2}
%G^s_{a_1 a_2}\,
V_{sa_1}{}^{a_2}
\label{cpf2}
\eea
\end{mathletters}
where, in terms of Virasoro characters,
\be
\chit_{r,a}(q)
=\sum_{s\in A_{g-1}}\chi_{r,s}(q) \hV_s{}^a\ .
\ee
%Notice that the sums on $(r,s)$ run over the complete Kac table.
The equivalence of the two forms (\ref{cpf1}) and (\ref{cpf2}) of the
cylinder partition functions
follows from the identity (\ref{keyid})
with $a=1$. The result (\ref{CPFs}) is not
entirely new but generalizes
and encompasses several previous results~\cite{SaleurB89,Cardy89,PasqS90}.
Note that the matrices $N_r\otimes V_s$ form a representation of the
fusion algebra of the minimal model.

\section{Critical 3-state Potts}

As an example we consider the $\calM(A_4,D_4)$ or critical 3-state Potts
model. To avoid redundancy, we consider the folded
%This model is the orbifold dual of the
$(T_2,D_4)$ model as shown graphically in Figure~1.

The complete list~\cite{AfflOS98,FuchsS98} of conformal boundary
conditions, conjugate fields $\phit$ and
associated characters $\chit$ is
\bea
\begin{array}{rclrclcl}
A&=&(1,1)=(4,1)&\phit_{1,1}&=&I&\ &\chi_0+\chi_3\\
B&=&(1,3)=(4,3)&\phit_{1,3}&=&\psi&&\chi_{2/3}\\
C&=&(1,4)=(4,4)&\phit_{1,4}&=&\psi^\dagger&&\chi_{2/3}\\
BC&=&(2,1)=(3,1)&\phit_{2,1}&=&\epsilon&&\chi_{2/5}+\chi_{7/5}\\
AC&=&(2,3)=(3,3)&\phit_{2,3}&=&\sigma&&\chi_{1/15}\\
AB&=&(2,4)=(3,4)&\phit_{2,4}&=&\sigma^\dagger&&\chi_{1/15}\\
F&=&(1,2)=(4,2)\quad&\phit_{1,2}&=&\eta&&\chi_{1/8}+\chi_{13/8}\\
N&=&(2,2)=(3,2)&\phit_{2,2}&=&\xi&&\chi_{1/40}\!+\!\chi_{21/40}\nonumber
\end{array}
\eea

The fused adjacency matrices of $G=D_4$ are
\bea
V_1=V_5&=&\smat{1&0&0&0\cr 0&1&0&0\cr 0&0&1&0\cr 0&0&0&1},\quad
V_2=V_4=\smat{0&1&0&0\cr 1&0&1&1\cr 0&1&0&0\cr 0&1&0&0},\nonumber\\
V_3&=&\smat{0&0&1&1\cr 0&2&0&0\cr 1&0&0&1\cr 1&0&1&0}.
\eea

\begin{figure}[htb]
\nc{\sm}[1]{{\scriptstyle #1}}
\nc{\pos}[2]{\makebox(0,0)[#1]{$#2$}}
%\nc{\spos}[2]{\makebox(0,0)[#1]{$\sm{#2}$}}
\setlength{\unitlength}{8mm}
\begin{center}
\begin{picture}(9.5,11)
\put(2,1.5){\begin{picture}(4,3)
\multiput(0,0)(1,0){5}{\line(0,1){3}}\multiput(0,0)(0,1){4}{\line(1,0){4}}
\put(0.5,0.5){\spos{}{0+3}}\put(2.5,0.5){\spos{}{\frac{2}{5}+\frac{7}{5}}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(1.5,1.5){\spos{}{\frac{1}{40}+\frac{21}{40}}}\put(3.5,1.5){\spos{}{\frac{1}
{8}+\frac{13}{8}}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(0.5,2.5){\spos{}{\frac{2}{3}\:,\:\frac{2}{3}}}\put(2.5,2.5){\spos{}{\frac{1
}{15}\:,\:\frac{1}{15}}}
\end{picture}}
\put(7.5,1.5){\begin{picture}(2,3)
\multiput(0,0)(1,0){3}{\line(0,1){3}}\multiput(0,0)(0,1){4}{\line(1,0){2}}
\put(0.5,0.5){\spos{}{0+3}}\put(1.5,0.5){\spos{}{\frac{2}{5}+\frac{7}{5}}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(0.5,1.5){\spos{}{\frac{1}{8}+\frac{13}{8}}}\put(1.5,1.5){\spos{}{\frac{1}{4
0}+\frac{21}{40}}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(0.5,2.5){\spos{}{\frac{2}{3}\:,\:\frac{2}{3}}}\put(1.5,2.5){\spos{}{\frac{1
}{15}\:,\:\frac{1}{15}}}
\end{picture}}
\put(2,6){\begin{picture}(4,5)
\multiput(0,0)(1,0){5}{\line(0,1){5}}\multiput(0,0)(0,1){6}{\line(1,0){4}}
\put(0.5,0.5){\spos{}{0}}\put(2.5,0.5){\spos{}{\frac{7}{5}}}
\put(1.5,1.5){\spos{}{\frac{1}{40}}}\put(3.5,1.5){\spos{}{\frac{13}{8}}}
\put(0.5,2.5){\spos{}{\frac{2}{3}}}\put(2.5,2.5){\spos{}{\frac{1}{15}}}
\put(1.5,3.5){\spos{}{\frac{21}{40}}}\put(3.5,3.5){\spos{}{\frac{1}{8}}}
\put(0.5,4.5){\spos{}{3}}\put(2.5,4.5){\spos{}{\frac{2}{5}}}
\end{picture}}
\put(7.5,6){\begin{picture}(2,5)
\multiput(0,0)(1,0){3}{\line(0,1){5}}\multiput(0,0)(0,1){6}{\line(1,0){2}}
\put(0.5,0.5){\spos{}{0}}\put(1.5,0.5){\spos{}{\frac{7}{5}}}
\put(0.5,1.5){\spos{}{\frac{13}{8}}}\put(1.5,1.5){\spos{}{\frac{1}{40}}}
\put(0.5,2.5){\spos{}{\frac{2}{3}}}\put(1.5,2.5){\spos{}{\frac{1}{15}}}
\put(0.5,3.5){\spos{}{\frac{1}{8}}}\put(1.5,3.5){\spos{}{\frac{21}{40}}}
\put(0.5,4.5){\spos{}{3}}\put(1.5,4.5){\spos{}{\frac{2}{5}}}
\end{picture}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(2.5,0.9){\begin{picture}(3,0)\put(0,0){\line(1,0){3}}\multiput(0,0)(1,0){4}
{\spos{}{\bullet}}\end{picture}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(8,0.9){\begin{picture}(1.4,0)\put(0,0){\line(1,0){1}}\multiput(0,0)(1,0){2}
{\spos{}{\bullet}}\put(1.2,0){\circle{0.4}}\end{picture}}
\put(0.9,2){\begin{picture}(0,2)\put(0,0){\line(0,1){1}}\multiput(0,0)(0,1){2}
{\spos
{}{\bullet}}
\put(0,1){\line(1,2){0.5}}\put(0,1){\line(-1,2){0.5}}\multiput(-0.5,2)(1,0){2}
{\spos{}{\bullet}}\end{picture}}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\put(1.3,6.5){\begin{picture}(0,4)\put(0,0){\line(0,1){4}}\multiput(0,0)(0,1){5}
{\spos{}{\bullet}}\end{picture}}
\put(4,0.3){\pos{}{A_4}}\put(8.5,0.3){\pos{}{T_2}}
\put(0.3,3){\pos{}{D_4}}\put(0.8,8.5){\pos{}{A_5}}
\put(6.75,3){\pos{}{=}}\put(6.75,8.5){\pos{}{=}}
\put(4,5.25){\pos{}{\updownarrow}}\put(8.5,5.25){\pos{}{\updownarrow}}
\end{picture}
\end{center}
\caption{Folding and orbifold duality relating the tensor product graph
$T_2\otimes D_4$ to
$A_4\otimes D_4$ and $A_4\otimes A_5$. The conformal weights of the 8
conformal boundary conditions
of the
3-state Potts model appear in the boxes of the $T_2\otimes D_4$ theory.}
\end{figure}


\noindent
The unitary matrix which diagonalizes $D_4$ is
\be
\Psi ={1\over\sqrt{3}}
\mbox{\smat{
{1\over\sqrt{2}}&{1\over\sqrt{2}}&1&1\cr
\sqrt{3\over 2}&-\sqrt{3\over 2}&0&0\cr
{1\over\sqrt{2}}&{1\over\sqrt{2}}&\omega&\omega^2\cr
{1\over\sqrt{2}}&{1\over\sqrt{2}}&\omega^2&\omega
}}
\ee
where $\omega=\exp(2\pi i/3)$ is a primitive cube root of unity.
The graph fusion matrices of $D_4$ are
\bea
&&\hat{N}_1=\smat{1&0&0&0\cr 0&1&0&0\cr 0&0&1&0\cr 0&0&0&1},\quad
\hat{N}_2=\smat{0&1&0&0\cr 1&0&1&1\cr 0&1&0&0\cr 0&1&0&0},\nonumber\\
&&\hat{N}_3=\smat{0&0&1&0\cr 0&1&0&0\cr 0&0&0&1\cr 1&0&0&0},\quad
\hat{N}_4=\smat{0&0&0&1\cr 0&1&0&0\cr 1&0&0&0\cr 0&0&1&0} \ .
\eea
The graph fusion matrices of $T_2$ are
\be
N_1=N^1=\mbox{\smat{1&0\cr 0&1}},\quad N_2=N^2=\mbox{\smat{0&1\cr 1&1}}.
\ee
The intertwiner $\hV$ and  conjugation $C$ are
\be
\hV=\mbox{\smat{1&0&0&0\cr 0&1&0&0\cr 0&0&1&1\cr 0&1&0&0\cr 1&0&0&0}},\qquad
C=\mbox{\smat{1&0&0&0\cr 0&1&0&0\cr 0&0&0&1\cr 0&0&1&0}}.
\ee
The  conjugation operator $C$ acts on the right to raise and lower
indices in the fusion
matrices $\hat{N}^a=\hat{N}_a C$.

The complete fusion rules of boundary fields are given as follows:
\bea
&&\smat{
I&\epsilon&\eta&\xi&\psi&\sigma&\psid&\sigmad\cr
\epsilon&\epsilon^2&\epsilon\eta&\epsilon\xi&\epsilon\psi&\epsilon\sigma&
\epsilon\psid&\epsilon\sigmad\cr
\eta&\eta\epsilon&\eta^2&\eta\xi&\eta\psi&\eta\sigma&\eta\psid&\eta\sigmad\cr
\xi&\xi\epsilon&\xi\eta&\xi^2&\xi\psi&\xi\sigma&\xi\psid&\xi\sigmad\cr
\psi&\psi\epsilon&\psi\eta&\psi\xi&\psi^2&\psi\sigma&\psi\psid&\psi\sigmad\cr
\sigma&\sigma\epsilon&\sigma\eta&\sigma\xi&\sigma\psi&\sigma^2&\sigma\psid&
\sigma\sigmad\cr
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\psid&\psid\epsilon&\psid\eta&\psid\xi&\psid\psi&\psid\sigma&{\psid}^2&\psid\sigmad\cr
\sigmad&\sigmad\epsilon&\sigmad\eta&
\sigmad\xi&\sigmad\psi&\sigmad\sigma&\sigmad
\psid&{\sigmad}^2\cr }\nonumber\\
&=&\sum_{r=1}^2 \sum_{a=1}^4 N^r\otimes \hat{N}^a\; \phit_{r,a}\nonumber\\
&=&\!\!\smat{1&0&0&0&0&0&0&0\cr
						   0&1&0&0&0&0&0&0\cr
         0&0&1&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&0&0&0&0&1&0\cr
         0&0&0&0&0&0&0&1\cr
         0&0&0&0&1&0&0&0\cr
         0&0&0&0&0&1&0&0}I\!+\!\!
   \smat{0&1&0&0&0&0&0&0\cr
						   1&1&0&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&1&0&0&0&0\cr
         0&0&0&0&0&0&0&1\cr
         0&0&0&0&0&0&1&1\cr
         0&0&0&0&0&1&0&0\cr
         0&0&0&0&1&1&0&0}\epsilon\nonumber\\
&+&\!\!\smat{0&0&1&0&0&0&0&0\cr
						   0&0&0&1&0&0&0&0\cr
         1&0&0&0&1&0&1&0\cr
         0&1&0&0&0&1&0&1\cr
         0&0&1&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0}\eta\!+\!\!
   \smat{0&0&0&1&0&0&0&0\cr
						   0&0&1&1&0&0&0&0\cr
         0&1&0&0&0&1&0&1\cr
         1&1&0&0&1&1&1&1\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&1&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&1&0&0&0&0}\xi\nonumber\\
&+&\!\!\smat{0&0&0&0&1&0&0&0\cr
						   0&0&0&0&0&1&0&0\cr
         0&0&1&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         1&0&0&0&0&0&0&0\cr
         0&1&0&0&0&0&0&0\cr
         0&0&0&0&0&0&1&0\cr
         0&0&0&0&0&0&0&1}\psi\!+\!\!
   \smat{0&0&0&0&0&1&0&0\cr
						   0&0&0&0&1&1&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&1&0&0&0&0\cr
         0&1&0&0&0&0&0&0\cr
         1&1&0&0&0&0&0&0\cr
         0&0&0&0&0&0&0&1\cr
         0&0&0&0&0&0&1&1}\sigma\nonumber\\
&+&\!\!\smat{0&0&0&0&0&0&1&0\cr
						   0&0&0&0&0&0&0&1\cr
         0&0&1&0&0&0&0&0\cr
         0&0&0&1&0&0&0&0\cr
         0&0&0&0&1&0&0&0\cr
         0&0&0&0&0&1&0&0\cr
         1&0&0&0&0&0&0&0\cr
         0&1&0&0&0&0&0&0}\psid\!+\!\!
   \smat{0&0&0&0&0&0&0&1\cr
						   0&0&0&0&0&0&1&1\cr
         0&0&0&1&0&0&0&0\cr
         0&0&1&1&0&0&0&0\cr
         0&0&0&0&0&1&0&0\cr
         0&0&0&0&1&1&0&0\cr
         0&1&0&0&0&0&0&0\cr
         1&1&0&0&0&0&0&0}\sigmad\nonumber
\eea

In total, we find twelve distinct cylinder partition
functions~\cite{Cardy89,AfflOS98}
\begin{eqnarray}
Z_{A|A}(q)&=&\chit_{1,1}(q)=\chi_{1,1}(q)+\chi_{1,5}(q)\nonumber\\
Z_{A|B}(q)&=&\chit_{1,4}(q)=\chi_{1,3}(q)\nonumber\\
Z_{A|AB}(q)&=&\chit_{2,4}(q)=\chi_{3,3}(q)\nonumber\\
Z_{A|BC}(q)&=&\chit_{2,1}(q)=\chi_{3,5}(q)+\chi_{3,1}(q)\nonumber\\
Z_{A|F}(q)&=&\chit_{1,2}(q)=\chi_{4,2}(q)+\chi_{4,4}(q)\nonumber\\
Z_{A|N}(q)&=&\chit_{2,2}(q)=\chi_{2,2}(q)+\chi_{2,4}(q)\nonumber\\
&=&Z_{AB|F}(q)\nonumber\\
Z_{AB|AB}(q)&=&\chit_{1,1}(q)+\chit_{2,1}(q)\nonumber\\
&=&\chi_{1,1}(q)+\chi_{3,5}(q)+\chi_{3,1}(q)+\chi_{1,5}(q)\nonumber\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
Z_{AB|AC}(q)&=&\chit_{1,4}(q)+\chit_{2,3}(q)=\chi_{3,3}(q)+\chi_{1,3}(q)\nonumber\\
Z_{AB|N}(q)&=&\chit_{2,2}(q)+\chit_{1,2}(q)\nonumber\\
&=&\chi_{2,2}(q)+\chi_{2,4}(q)+\chi_{4,2}(q)+\chi_{4,4}(q)\nonumber\\
Z_{F|F}(q)&=&\chit_{1,1}(q)+\chit_{1,3}(q)+\chit_{1,4}(q)\nonumber\\
&=&\chi_{1,1}(q)+\chi_{1,5}(q)+2\chi_{1,3}(q)\nonumber\\
Z_{F|N}(q)&=&\chit_{2,1}(q)+\chit_{2,3}(q)+\chit_{2,4}(q)\nonumber\\
&=&\chi_{3,5}(q)+\chi_{3,1}(q)+2\chi_{3,3}(q)\nonumber\\
Z_{N|N}(q)&=&\chit_{1,1}(q)+\chit_{2,1}(q)+\chit_{1,3}(q)\nonumber\\
&&\quad\mbox{}+\chit_{1,4}(q)+\chit_{2,3}(q)+\chit_{2,4}(q)\nonumber\\
&=&\chi_{1,1}(q)+\chi_{3,5}(q)+\chi_{3,1}(q)+\chi_{1,5}(q)\nonumber\\
&&\quad\mbox{}+2\chi_{3,3}(q)+2\chi_{1,3}(q)\nonumber
\end{eqnarray}
\goodbreak
Here we restrict to Virasoro characters with $r+s$ even.
The symmetry
\be
Z_{(r_1,a_1)|(r_2,a_2)}(q)=Z_{(r_2,a_2)|(r_1,a_1)}(q)
\ee
follows because the characters do not distinguish between a field $\phit$
and its conjugate
$\phit^\dagger$.

\section{Integrable Boundary Weights}

\nc{\ru}[1]{\rule[-#1ex]{0ex}{#1ex}}
\nc{\B}[5]{\setlength{\unitlength}{4mm}
\begin{array}{@{}c@{}}
\begin{picture}(2.2,2.2)(-0.1,0.16)
\multiput(1.6,0.1)(0,0.3){7}{\line(0,1){0.2}}
\put(0.6,1.1){\line(1,-1){1}}\put(0.6,1.1){\line(1,1){1}}
\put(1.8,2.2){\spos{tl}{#5}}\put(0.45,1.1){\spos{r}{#4}}
\put(1.8,0){\spos{bl}{#3}}\end{picture}\end{array}}

We conjecture that conformal boundary conditions for $sl(2)$ models can be
realized as integrable boundary conditions for the associated lattice
models~\cite{ABF84}. For the
$(A_{g-1},A_g)$ theories the integrable boundary weights have been
obtained~\cite{BehrP96}, as solutions to the boundary Yang-Baxter equation,
by a fusion construction.
This method generalizes~\cite{BehrP96} to the \ade\ models using the
appropriate fusion
process~\cite{ZhouP94}.
The solutions to the boundary Yang-Baxter
equation are naturally labelled by a pair $(r,a)$
and are constructed
by starting at $a$ and fusing $r-1$ times.
For $(A_4,D_4)$% or critical 3-state Potts
, the non-zero boundary weights
are given
explicitly by
\begin{eqnarray*}
\ru{3.5}A,B,C&\!=\!&(1,a):\B{1}{1}{a}{2}{a}=1,\quad a=1,3,4\\
\ru{3.5}F&\!=\!&(1,2):\B{1}{2}{2}{1}{2}=\!\B{1}{2}{2}{3}{2}=
\B{1}{2}{2}{4}{2}=1\\
\ru{3.5}BC&\!=\!&(2,1):\B{2}{1}{2}{3}{2}=\!\B{2}{1}{2}{4}{2}=\rho_1(u)\;,\;
\B{2}{1}{2}{1}{2}=\rho_1(-u)\\
\ru{3.5}AC&\!=\!&(2,3):\B{2}{3}{2}{1}{2}=\!\B{2}{3}{2}{4}{2}=\rho_1(u)\;,\;
\B{2}{3}{2}{3}{2}=\rho_1(-u)\\
\ru{4}AB&\!=\!&(2,4):\B{2}{4}{2}{1}{2}=\!\B{2}{4}{2}{3}{2}=\rho_1(u)\;,\;
\B{2}{4}{2}{4}{2}=\rho_1(-u)\\
N&\!=\!&(2,2):\left\{\begin{array}{@{\:}l@{}}
\ru{3.5}
\B{2}{2}{a}{2}{b}
=\rho_2(u),\quad a\ne b,\quad a,b=1,3,4\\
\B{2}{2}{a}{2}{a}=
\rho_3(u),\quad a=1,3,4
\end{array}
\right.
\end{eqnarray*}
with $u$ the spectral parameter, $\lambda=\pi/6$, $\xi$ arbitrary and
$$
\begin{array}{@{}l@{}}
\ru{3}\displaystyle\rho_1(u)\!=\!
\frac{\sin(u\!-\!\lambda\!-\!\xi)\:
\sin(u\!-\!\lambda\!+\!\xi)}{\sin^2\!\lambda},\qquad
\rho_2(u)\!=\!
\frac{\sin\!2u}{\sin\!2\lambda}\\
\ru{3}\displaystyle\rho_3(u)\!=\!
\frac{2\sin(u\!-\!\xi)\sin(u\!+\!\xi)\!+\!
\sin(u\!-\!2\lambda\!-\!\xi)
\sin(u\!-\!2\lambda\!+\!\xi)}{\sin^2\!2\lambda}.
\end{array}
$$
The new boundary condition~\cite{AfflOS98} $N$ is found to be
antiferromagnetic in nature. The value of $u$ should be set to its
isotropic value $u=\lambda/2$ and $\xi$ chosen appropriately
to obtain the conformal boundary conditions.


\section{Conclusion}


In conclusion
we have proposed a set of conjectures that extend
the theory of conformal boundaries in a consistent way.
The structure of the partition functions is dictated
by a new fusion algebra.
We comment that the conjecture  (\ref{CPFs}c)
is independent of the choice of endpoint node and
eigenvectors and is
meaningful for $D_{2n+1}$  and $E_7$, even though
a proper understanding of the  fusion matrices in (2.4b)
%with negative signs
is missing.  We expect the
 extension to higher rank~\cite{DFZLZZ} to be straightforward.
A much more comprehensive
version of this work will be published elsewhere.

\begin{references}
\bibitem{Cardy89}J. L. Cardy, Nucl.\  Phys.\ {\bf B324}, 581 (1989).
\bibitem{Verlinde88}E. Verlinde, Nucl.\  Phys.\ {\bf B300}, 360 (1988).
\bibitem{AfflOS98}I. Affleck, M. Oshikawa and H. Saleur, Boundary Critical
Phenomena in the Three-State Potts Model, cond-mat/9804117.
\bibitem{FuchsS98}J. Fuchs and C. Schweigert, Completeness of Boundary
Conditions for the Critical Three-State Potts Model,
hep-th/9806121.
\bibitem{CapItzZub87}A. Cappelli, C. Itzykson and J.-B. Zuber,
%Nucl.\ Phys.\ {\bf B280}, 445 (1987).
Comm.\ Math.\ Phys.\ {\bf 113}, 1 (1987).
\bibitem{Pas}V. Pasquier, {Mod\`eles Exacts Invariants Conformes},
Th\`ese d'Etat, Orsay, 1988.
\bibitem{DFZPZ}P. Di Francesco, and J.-B. Zuber, in Recent Developments in
Conformal Field Theories, eds.
S. Randjbar-Daemi, E. Sezgin and J.-B. Zuber, World Scientific (1990); P.
Di Francesco, Int.\ J. Mod.\
Phys.\ {\bf A7}, 407 (1992).
\bibitem{SaleurB89}H. Saleur and M. Bauer, Nucl.\ Phys.\ {\bf B320}, 591
(1989).
\bibitem{PasqS90}V. Pasquier and H. Saleur, Nucl.\ Phys.\ {\bf B330}, 523
(1990).
\bibitem{ABF84}G. E. Andrews, R. J. Baxter and P. J. Forrester, J. Stat.\
Phys.\ {\bf 35}, 193 (1984);
P. J. Forrester and R. J. Baxter, J. Stat.\ Phys.\ {\bf 38}, 435 (1985);
V. Pasquier, Nucl.\ Phys.\ {\bf B285}, 162 (1987).
\bibitem{BehrP96}R. E. Behrend, P. A. Pearce and D. L. O'Brien, J. Stat.\
Phys.\ {\bf 84}, 1 (1996);
R. E. Behrend and P. A. Pearce, J. Phys.\ {\bf A29}, 7827 (1996);
R. E. Behrend and P. A. Pearce, Int.\ J. Mod.\ Phys.\ {\bf B11}, 2833 (1997).
\bibitem{ZhouP94}Y.-K. Zhou and P. A. Pearce, Int.\ J. Mod.\ Phys.\ {\bf
B8}, 3531 (1994).
\bibitem{DFZLZZ}P. Di Francesco, and J.-B. Zuber, Nucl.\ Phys.\ {\bf B338},
602 (1990);
S. Loesch, Y.-K. Zhou and J.-B. Zuber, Int.\ J. Mod.\ Phys.\ {\bf A12}, 4425
(1997).

\end{references}




\end{document}



