BCFT: from the boundary to the bulk
72 Tzarigradsko Chaussee
1784 Sofia
School of Computing and Mathematics
NE1 8ST Newcastle upon Tyne
UK
and
SPhT
CEA Saclay
91191 Gif-sur-Yvette
France
Boundary Conformal Field Theory (BCFT) is experiencing a renewal of interest. This is motivated by applications to string and brane theory
and to problems of statistical mechanics and condensed matter. In these situations
it may be important to have an a priori knowledge of the possible boundary conditions compatible with conformal invariance
and to master the algebra of boundary fields etc. There is
however
another reason to be interested in BCFT: as we want to explain in this note
there is much to learn on the general structure and the quantum symmetries of a CFT from the study of its properties in the presence of a boundary. After a brief review of notations and general aspects of rational CFT
we shall discuss how boundary conditions may be systematically classified in terms of non-negative integer valued matrix representations of the fusion algebra
or equivalently in terms of graphs generalising the ADE Dynkin diagrams (see \cite{BPPZ} for more details). We shall see how the ``cells'' -- a concept introduced by Ocneanu and associated with these graphs -- determine many properties of the BCFT or of the associated lattice models and
in particular
are at the heart of the quantum symmetry of the CFT described by an algebraic structure called weak $C^*$-Hopf algebra. A more detailed presentation will appear in \cite{PZ4}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{General set-up} A rational conformal field theory (RCFT) is generally described by data of different nature:
%%%%%%%%%% \noindent $\bullet$ {Chiral data:} %--------------------------- Chiral data specify the chiral algebra ${\goth A}$
e.g.
the Virasoro algebra itself
or a ${\cal W}$ algebra
a current algebra $\widehat{ g}\
$ etc
and its finite set $\calI$ of irreducible representations
$\calV_i\
\ i\in \cal I\
$; notations are such that $i=1$ labels the identity (vacuum) representation and $i^*$ the conjugate of ${\cal V}_i$. The fusion rule multiplicities $N_{ij}{}^k$
$ \calV_i \star \calV_j =\oplus_k\
N_{ij}{}^k\
\calV_k\
$ are assumed to be given by the Verlinde formula
% \be N_{ij}{}^k=\sum_{\ell\in{ I}}{S_{il}\
S_{jl} \left(S_{kl}\right)\!{}^* \over S_{1l}} %\;\in\; {\Bbb Z_{\ge 0}} \
%{\Bbb N}\
\label{Verl} \ee % with $S$ the symmetric
unitary matrix of the modular transformations of the characters $\chi_i(\tau)=\tr e^{2i\pi\tau(L_0-c/24)}$
$\chi_i(\tau)=\sum_{j\in \calI} S_{ij} \chi_j(-1/\tau)$. The nonnegative integers $N_{ij}{}^k$ give the dimensions of linear spaces of chiral vertex operators (CVO) % \be \phi_{ij;t,I}^k(z):\ \calV_i \otimes \calV_j \to \calV_k \
\ \ %t=1
2
\dots N_{ij}{}^k
\ z\in {\Bbb C}\
\ee % with a finite basis label $t=1
\dots N_{ij}{}^k\
$ %with $t=1
$ a basis label and $I$ labelling descendents in $\calV_i$. The chiral data finally include the knowledge of the duality matrices: the genus $0$ fusing $F$ and braiding $B(\pm)$ matrices
and the matrix $S(j)$
which gives the modular transformation of $1$-point conformal block $\bra \phi_{ji}^i \ket$ on the torus. These matrices satisfy a set of consistency relations: pentagon
hexagon and torus identities \cite{MS1}. %In particular
the matrix $F$ satisfies a pentagon %identity expressing the associativity of the fusion of CVO. A typical example is provided by the chiral algebra ${\goth A}=\slh(2)_k$
the affine algebra at level $k$
for which $\calI=\{1,2,\cdots,k+1\}$
$S_{ij}=\sqrt{2/(k+2)}$ $\sin{\pi ij/(k+2)}$
and the $F$ are
up to a gauge transformation
the quantum $6j$ symbols \cite{KiRe}. Those are the basic ingredients of algebraic nature in the construction of a RCFT. In the following
they are supposed to be known. %%%%%%%%%% %\medskip \noindent $\bullet$ {Spectral and OPE data:} %-------------------------------------- The physical spectrum of a RCFT {in the bulk}
i.e. on a closed Riemann surface
is described by irreducible representations of {two} copies of the chiral algebra. Thus the Hilbert space is decomposed according to % \be {\cal H}_P=\oplus_{_{j
\jb\
\in{ \calI}}}\
Z_{j\
\bar{j}}\
\calV_j\otimes \bar{\calV}_{\bar j}\
. \label{Spec} \ee % The integer multiplicities $Z_{j\
\bar{j}}$ (with $Z_{11}=1$
expressing the unicity of the vacuum) are conveniently encoded in the modular invariant torus partition function % \be Z(\tau)= \sum_{j
\in{ \calI}}\
Z_{j\jb}\ \chi_j({\tau})\
\chi_{\jb}(\tau)^* \
. \label{partfn} \ee % The particular case where $Z_{ij}=\delta_{ij}$ will be referred to as the diagonal theory
the others as ``non-diagonal''. On the other hand
the physical fields and their correlators factorise \omit{\be \langle \cdots \Phi_{(i,\ib)}(z
\bar{z}) \cdots \rangle = \dots \sum_{j
\jb
k
\bar{k}
t
\bar{t}}\
d_{(i,\ib) (j,\jb)}^{(k
\bar{k});t
\bar{t}}\cdots \langle \cdots \phi_{i,j;t}^k(z)\
\cdots \rangle\ \otimes \langle \cdots \phi_{\ib
\jb;\bar{t}}^{\bar{k}}(\bar{z})\, \cdots \rangle \ee } \be \Phi_{(i,\ib)}(z
\bar{z})=\!\!\! \sum_{j
\!\!\! d_{(i,\ib) (j,\jb)}^{(k
\bar{t}} \
\phi_{i,j;t}^k(z)\
\otimes \phi_{\ib
\jb;\bar{t}}^{\bar{k}}(\bar{z})\ . \ee % The expansion coefficients $d_{(i,\ib) (j,\jb)}^{(k
\bar{t}}\ $ determine
up to the normalisation of the chiral blocks
the coefficients of the short distance operator product expansion (OPE)
i.e.
these are the {relative} OPE coefficients of the non-diagonal model if $d$ are chosen trivial for the diagonal model of same central charge. These numbers are constrained by the requirement of locality of the physical correlators
which makes use of the braiding matrices $B(\pm)$. The resulting set of coupled quadratic equations has been fully solved only in the $sl(2)$ cases (see \cite{DF,PZ1} and further references therein). \quad Starting with the ADE classification of the $sl(2)$ modular invariants \cite{CIZ}
it has been gradually realised that behind these %spectral and OPE data there are hidden {graphs $G$}. A systematic study of the graphs generalising the $sl(2)$ ADE Dynkin diagrams was initiated in \cite{Kos,DFZ1}. It was empirically observed that these graphs are encoding a non trivial information on some spectral and OPE data
more precisely on those pertaining to the {spin-zero} fields of the theory. % These graphs $G_i$
which share the same set of vertices $\calV$
are labelled by an index $i\in {\cal I}$ --but it is sufficient to restrict $i$ to a generating subset (generating in the sense of fusion)-- and their adjacency matrices are commuting and simultaneously diagonalizable in an orthonormal basis $\{\psi\}$: \noindent $\bullet $ the diagonal part of the physical spectrum
labelled by the so-called {``exponents''}\
: \be {\cal E}=\{j\in \calI| j=\bar j
Z_{jj}\ne 0\}\
\ee counted with the multiplicity $Z_{jj}$
is in one-to-one correspondence with the spectrum of eigenvalues of $G_i$
of the form $S_{ij}/S_{1j}$
$j\in \calE$. \noindent $\bullet$ in the $sl(2)$ (ADE) cases the structure constants $M_{ij}{}^k$ of the so-called Pasquier algebra \cite{Pasq87} can be identified with the (relative) OPE coefficients of the scalar fields \cite{PZ1} \be d_{(i,i) (j,j)}^{(k,k)}=M_{ij}{}^k:=\sum_{a\in \calV}\
{\psi_a^i\
\psi_a^j \
\psi_a^{k\, *}\over \psi_a^1}\
. \label{strc} \ee The summation in (\ref{strc}) runs over the set $\cal V$ of vertices of the Dynkin diagram. The purpose of this note is to show that the study of the CFT on a half-plane or a cylinder %
(i.e. on a manifold with boundary)
\cite{Ca89}
provides an alternative (chiral) approach in which these graphs and the related algebraic structures become manifest. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Graphs and conformal boundary conditions} % We may summarize the results of \cite{BPPZ} as follows: boundary conditions that respect the chiral algebra $\goth A$ are described by a set of commuting
non-negative integer valued matrices $\{n_i= n_{i a}{}^b\}\
i\in \calI\
$ $a
b\in \calV\
$ $|\cal E|=|\cal V|$
s.t. $n_1=I$
$n_i^T=n_{i^*}$
realising a representation of the Verlinde fusion algebra \be n_i\;n_j=\sum_{k\in \calI} N_{ij}{}^k\
n_k \
. \label{nimrep} \ee The matrices $n_i$ thus admit a spectral decomposition \be n_{i a}{}^b =\sum_{j\in \cal E}\
{S_{ij}\over S_{1j}}\ \psi_a^j\ \psi_b^{j*}\
\label{specdec} \ee % where $\psi_a^j$ are unitary matrices of dimension $|\cal E|=|\cal V|$ and the sum runs over the set $\{(j
\alpha)\
j\in \calI\
\alpha=1
\dots
Z_{jj}\}$
for simplicity of notation identified with the set of exponents $\calE$. % This comes about as follows: On the upper half-plane param\-etrised by a coordinate $z\in {\Bbb H}^+$
(resp. on a finite-width strip $w={L\over \pi}\
\log z\
$)
with boundary conditions $b$ and $a$ imposed on the negative and positive real axes
(resp. $\!\!$ on the two sides of the strip)
only one copy of the chiral algebra acts: only real analytic coordinate transformations
$\epsilon(z)=\bar{\epsilon}(\bar{z})$ for real $z=\bar{z}$
are allowed. Thus the Hilbert space of the theory ${\cal H}_{ba}$ splits into a linear sum of representations \be {\cal H}_{ba}=\oplus_{i\in \calI} n_{ib}{}^a \
\calV_i\ . \label{Hab} \ee On a finite segment of the strip of length $T$ with periodic boundary conditions $w\sim w+T$
i.e. on a cylinder
the partition functions $Z_{b|a}$ is thus a linear form in the characters % \be Z_{b|a}(\tau)=\sum_{i\in \calI} n_{ib}{}^a\
\chi_i(\tau)\
\quad \tau= {i T\over 2 L}\
. \label{pfn} \ee % The multiplicities $n_{ia}{}^b$ are constrained by the Cardy consistency condition \cite{Ca89} which expresses that $Z_{b|a}$ can also be evaluated in a dual way: by mapping the cylinder to an annulus region in the full plane
one computes $Z_{b|a}$ as the matrix element of the evolution operator between boundary states $|a\rangle$ and $\langle b|$. The latter are decomposed on a standard basis of complete and orthonormal ``Ishibashi states'' $|j\rangle\!\rangle$ labelled by an exponent $j\in \calE$
according to $|a\rangle =\sum_{j\in \cal E}\
{\psi_a{}^j \over \sqrt{S_{1j}}} \ |j\rrangle $. This gives % \bea Z_{b|a}(\tau) &=& \langle b|e^{{-\pi i\over \tau} (L_0+\overline{L}_0-{c\over 12})} |a\rangle \nonumber
&=& \sum_{j\in \calE}{\psi_a{}^j \psi_b{}^{j*} \over {S_{1j}}} \chi_j\Big({-1\over\tau}\Big)\ . \label{cardy} \eea Comparing the resulting two expressions for %
$Z_{b|a}$ yields the spectral decomposition (\ref{specdec}). See \cite{BPPZ} for a discussion of the assumptions and a more detailed derivation. Each set of matrices $\{n_i\}$ solving (\ref{nimrep})
(\ref{specdec}) may be regarded as the adjacency matrices of a collection of graphs $G_i$
thus explaining the occurence of graphs with the special spectral properties noticed above. % Solving in general the system (\ref{nimrep}) provides a classification of boundary conditions
labelled by the vertices of the graphs $G_i$ and specified by the spectrum (\ref{Hab}). The case $n_i=N_i $ provides the regular representation with spectrum $\cal E=\cal I$ corresponding to the diagonal case; then (\ref{nimrep}) reduces to (\ref{Verl}) with $\psi=S$ and the boundary states are labelled by the same indices $i\in \calI$ as the representations and primary fields \cite{Ca89}. $\bullet$ In $\slh(2)$ RCFT (the WZNW models and the Virasoro minimal models) this reduces to the classification of the symmetric
irreducible
non-negative integer valued matrices of spectrum $|\zg^j|$ $ =|{S_{2 j}\over S_{1 j}}|< 2$. This is well known to lead to an ADE classification
after the spurious ``tadpole'' graphs $A_{2n}/{\Bbb Z}_2$ have been discarded
on the basis that their spectrum does not match any known modular invariant. $\bullet$ A new situation arises in the $sl(3)$ case
where there is no known a priori classification of (oriented) graphs with the required spectral properties. Solutions may be found \cite{DFZ1} and comparison with the complete list of modular invariants \cite{Ga} enables one to discard some spurious solutions and leads to a list of graphs
see \cite{BPPZ}. One also finds that in a few cases
there are isospectral graphs
i.e. more than one solution with a given spectrum
indicating that there are several choices of boundary conditions (and in fact of OPE structure constants
see \cite{BPPZ})
for a given spectrum of spin-zero fields in a RCFT. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Ocneanu quantum graph symmetry} According to Cardy the boundary conditions $(a
b)$ are created by insertions of fields on the boundary ({boundary fields})
$^a\Psi_{j;\beta}^b(x)$
$\beta =1
\dots n_{jb}{}^a\
\ $ $x\in {\Bbb R}$. %
$z\in {\Bbb H}^+$. Exploiting some ideas of Ocneanu \cite{Ocn,Ocn00} one can interpret these fields
extended to the full plane
as generalised CVO. Given a solution of the system of equations (\ref{nimrep})
consider for each $j\in \cal I$ an auxiliary Hilbert space $V^j\cong {\Bbb C}^{m_j}\
$ of dimension $m_j= \sum_{a,b}\
n_{ja}{}^b$ with basis states $|e^{j,\beta}_{ba}\rangle$
% |j % b__|__a $\beta=1
n_{j a}{}^b$
depicted as \hbox{\raise -7mm\hbox{\epsfxsize=3cm\epsfbox{dta11.eps}}}. A scalar product in $\oplus_{j\in \calI}\
V^j$ is defined as % \bea \langle e^{j,\beta}_{ ab} | e^{j',\beta'}_{a'b'}\rangle &=& \delta_{bb'} \delta_{aa'}\
\delta_{jj'}\
\delta_{\beta' \beta}\
\sqrt{P_a\,P_b\over d_j} \
\nonumber
d_j&:=&{S_{j1}\over S_{11}}\
\ P_a:={\psi_a^1\over \psi_1^1}\
. \label{ebasis} \eea % % Restricting to a subspace $V^i \otimes_h\
V^j\
$ of $V^i\otimes V^j\
$ with coinciding intermediate labels
one defines a decomposition $V^i\otimes_h\
V^j\cong \oplus_{k}\
V^k$
or
explicitly
\bea \vert e^{i, \eta}_{ab}\ket \otimes_h \vert e^{j,\zeta}_{b c}\ket &=& \
\sum_{k\in \calI }\ \sum_{\gamma=1}^{n_{kc}{}^a}\
\sum_{t=1}^{N_{ij}{}^k}\ \Fo_{b k}\left[ {i\atop a}\ {j\atop c} \right]_{\eta\, \zeta}^{\gamma\, t}\
% & &\!\!\!\!\! \!\!\!\! \times \sqrt{P_b}\
& &\!\!\!\!\! \Big({d_k \over d_i d_j}\Big)^{{1\over 4}}\ |e^{k\,, \gamma}_{ac}(ij;t)\ket\
. \label{prodh} \eea \omit{Pictorially \hbox{\raise -4mm\hbox{\epsfxsize=4cm\epsfbox{Fone.eps}}} % | | \/ % | | | Fig 2 % ________ = sum ^1F _________ } %%%% The counting of states in both sides is consistent due to (\ref{nimrep}). The Clebsch-Gordan coefficients $\Fo\in {\Bbb C}$ ({``$3j$- symbols''}) satisfy unitarity conditions (completeness and orthogonality) and a pentagon identity
reflecting the associativity of the product (\ref{prodh})
written schematically as % \be F\ \Fo \ \Fo \
= \Fo\ \Fo \
\qquad \label{pentagon} \ee where $F$ are the $6j$-symbols. In the diagonal case $F$ and $\Fo$ are identified
and (\ref{pentagon}) becomes the conventional pentagon identity satisfied by the fusing matrices of the corresponding RCFT. The space $\calA =\oplus_{j\in \calI}\
\hbox{End}\
(V^j)$ is a matrix algebra $\oplus_{j\in \calI}\
M_{m_j}$. A matrix unit basis $\{e_{j;\eta\,,\eta'}^{(ab)\,,(a'b')}
=| e^{j,\eta}_{ab}\ket \bra\
e^{j,\eta'}_{a'b'} | \,, j\in\cal I\}$ in $\calA$ is identified with a product of states in $V^j\otimes V^{j^*}$
and depicted as $4$-point blocks or double triangles
%\medskip % <<< Fig \qquad\hbox{\raise -4mm\hbox{\epsfxsize=5cm\epsfbox{dta12.eps}}}
\noindent Along with the matrix (``vertical'') product
a second ``horizontal'' product (or
alternatively
a coassociative coproduct)
is defined on $\cal A$ via the $3j$-symbols $\Fo\
$
\noindent \hbox{\raise -4mm\hbox{\epsfxsize=7cm\epsfbox{dta2h.eps}}}
\noindent The algebra generated by these objects is the Ocneanu {double triangle algebra} (DTA) \cite{Ocn}
further studied in the subfactor theory
see \cite{BEK,BE} and references therein. Equipped with a counit and an antipode satisfying a weakened version of the Hopf algebra axioms ($\triangle( 1_v)\ne 1_v\otimes 1_v$
etc.)
it was considered in \cite{BoSz} as an example of a {weak $C^*$- Hopf algebra} (WHA). This Ocneanu ``graph quantum algebra''
associated to any solution $\{n_i\,,i\in \calI \}$ of $(\ref{nimrep})$
together with its dual algebra $\hat{\calA}$ structure (see below)
appears as the quantum symmetry of the CFT model
either diagonal or non-diagonal. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Generalised CVO (GCVO)} We define operators $\oplus_{j\in \calI}\
\calV_j\!\otimes\! V_j \!\to\! \oplus_{k\in \calI}\
\calV_k\!\otimes\! V_k$
\noindent associated with the basis states in the tensor product (\ref{prodh})
% \bea ^a\Psi_{i, \beta;I}^c(z) &=&\sum_{j,k, t}\ \phi_{ij;t,I}^k(z) \otimes\
\sum_{ b,\alpha,\gamma}\
\Fo_{ck}\left[{i\atop a} {j\atop b}\right]^{ \alpha\, t}_{ \beta\, \gamma}\
& & \times \sqrt{d_j\over P_c\,P_b} \ | e^{k,\alpha}_{ab}\ket \bra\
e^{j,\gamma}_{cb} | \
\label{gcvo} \eea covariant under the action of $\calA$. Their correlators
obtained by projecting on a state $|0\rangle \otimes |e^1_{aa}\rangle $ in the vacuum space $\calV_1\otimes V^1$
as well as their fusing and braiding properties
are inherited from those of the conventional CVO. In contrast with previous approaches introducing quantum symmetry covariant operators
\cite{MR, MSch}
note that in the diagonal case the $6j$-symbols $F$ of the related quantum group (instead of its $3j$-symbols) appear in the r.h.s. of (\ref{gcvo}). For real $z$ the operators $\Psi$ of (\ref{gcvo}) represent (after an appropriate choice of normalisation) the boundary fields
and their short distance $x_{12}\approx 0$ expansion is recovered
with the $3j$-symbols $\Fo$ serving as {boundary fields OPE coefficients}. The recoupling equation (\ref{pentagon}) expresses then the associativity of the boundary fields OPE \cite{BPPZ} and is equivalent to the Lewellen boundary fields sewing identity \cite{CL}. In general the chiral operators (\ref{gcvo}) have nontrivial braiding with a new braiding matrix $\hat{B}(\epsilon)$
with $4+2$ indices of two types
\bea & &^a\Psi_{j,\alpha}^b(z_1)\
^b\Psi_{k,\gamma}^c(z_2)= \label{newbraid}
& &\sum_{d, \alpha',\gamma'} \hat{B}_{bd}\left[{j \atop a} {k\atop c} \right]_{ \alpha\, \gamma}^{\alpha'\, \gamma'}\!\!\!\!(\epsilon)\ ^a\Psi_{k,\alpha'}^d(z_2)\
^d\Psi_{j,\gamma'}^c(z_1)\
\nonumber \eea % $z_{12}\notin {\Bbb R}_- \
\ \epsilon=\hbox{sign}\
(\hbox{Im}\
z_{12}$). The matrices $\hat{B}$ satisfy various relations: $\bullet$ Inversion (unitarity) relation \be \hat{B}^{12}(\epsilon) \
\hat{B}^{21}(-\epsilon)=1\
. \ee % $\bullet$ The braid group (``Yang-Baxter'') relation %equation % \be \hat{B}^{12}\
\hat{B}^{23}\
\hat{B}^{12}\
= \hat{B}^{23}\
\hat{B}^{23} \
. \label{braidgp} \ee $\bullet$ The braiding--fusing (pentagon) identity \be \hat{B} \ \Fo = \Fo\ \hat{B}\ \hat{B}\
\ee which turns into a recursion relation for $\hat B$
given the fundamental $3j$-symbols
$\bullet$ Intertwining relation \be \Fo\
\Fo\ B= \hat{B}\ \Fo\
\Fo\
\label{intwg} \ee which implies a bilinear representation of $\hat{B}$ in terms of the $3j$-symbols \bea & & \hat{B}_{bd}\left[{j\atop a} {k\atop c} \right]_{ \alpha \gamma}^{\alpha' \gamma'}\!\!\!\!\!\!(\epsilon) = \label{bilin}
& & \sum_{i\,, \delta\,, t }\
{}\Fo_{b i}\left[{j\atop a} {k\atop c}\right]_{\alpha \gamma}^{\delta\, t } e^{-i \pi\epsilon\
\triangle_{jk}^i}\ {}\Fo_{d i}^*\left[{k\atop a} {j\atop c} \right]_{\alpha' \gamma'}^{\delta\, t }\
\nonumber \eea where $\triangle_{jk}^i$ is the combination of conformal weights $\triangle_j+\triangle_k-\triangle_i $. %In the diagonal case $\hat{B}(\epsilon)$ is identified %with $B(\epsilon)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Connection to lattice models} Some of these identities have been already encountered in critical lattice models
the ADE Pas\-quier models and their $\widehat{sl}(n)_{h-n}$ generalisations
in which the degrees of freedom may be regarded as vertices of the previous graphs \cite{Pa1,DFZ1}. In the limit $u\to -i\epsilon\infty$
($\epsilon=\pm1$)
of the spectral parameter $u$
the face Boltzmann weights $W(u)$ satisfy the same equation (\ref{braidgp}) as the braiding matrix $\hat{B}$ of (\ref{newbraid}). Indeed
denoting the representations of $sl(n)$ by their Young tableau
and with $q=e^{2\pi i\over h}\
\bea & & \hat{B}_{bd} \left[{\youngone\atop a}{\youngone\atop c} \right]_{ \za\, \zg}^{\za'\, \zg'}\!\!\!\!(\epsilon)\
= \nonumber
& & 2i \
q^{-\epsilon {1\over 2 n}}\
\lim_{u\to -i\epsilon\infty}\
e^{-i\pi\epsilon u}\
W_{bd}\left({a\atop c} \right)_{ \alpha\, \gamma}^{\alpha'\, \gamma'} (u)\
\nonumber \eea % \be W_{bd}(u)=\sin({\pi\over h}-u) \delta_{bd} +\sin(u)\
[2]_q\
U_{bd} \
\quad U^2=U\
. \ee Using (\ref{bilin})
the Hecke algebra generators $U$ are expressed via the $3j$-symbols
recovering an Ansatz of Ocneanu \cite{Ocn00} \be U_{bd}=\sum_{\zb}\
\Fo_{b\, \youngtwo}\left[{\youngone\atop a} {\youngone\atop c}\right]_{\za\, \zg}^{\zb\ 1}\
\Fo_{d\, \youngtwo}^*\left[{\youngone\atop a} {\youngone\atop c} \right]_{\za'\, \zg'}^{\zb\ 1 }\
. \ee Moreover
equation (\ref{intwg}) shows that we can identify the $3j$- symbols $\Fo$ with the {``intertwining cells}'' studied in these models
see e.g. \cite{DFZ1}; then lattice results provide solutions for particular $\Fo$
namely
those for which one of the labels in $\cal I$ is equal to $\youngone$. For $sl(2)$: $\youngtwo=1\
\ $ $ [2]_q\ U_{bd} = \delta_{ac}\
{\sqrt{P_b\,P_d}\over P_a }$
while for $sl(3)$: $\youngtwo=\youngone^*\
$ and these cells $\Fo_{b\
\youngone {}^*}$ exist for all graphs listed in \cite{BPPZ} but one
according to the recent work of Ocneanu
\cite{Ocn00}. % completing previous results of {Jimbo et al, Fendley, DiFZ, Sochen}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Bulk fields} For $(j
\bar{j})$ in the physical spectrum (\ref{Spec})
$Z_{j\bar{j}}\not =0$
and $z\in {\Bbb H}^+\
$ define (upper) half-plane bulk fields by two-point compositions of %generalised CVO (\ref{gcvo}) GCVO (\ref{gcvo}) \be \Phi_{(j
\bar{j})}^H(z
\bar z) =\sum_{a
b\
\atop{\beta'\,,\beta}}\ C^{a,b,a;\beta,\beta'}_{(j
\bar{j})}\ {}^a\Psi_{j, \beta}^b(z)\ ^b\Psi_{\bar{j}^*, \beta'}^a(\bar{z}) \
. \label{bulkf} \ee % {}From this operator representation
which can be also rewritten %
using (\ref{gcvo})
in terms of the conventional CVO
one recovers the bulk field small distance $z-\bar{z}=2i\
y\approx 0$ expansion
in the leading order % \bea & & %\bra e^1_{aa}| \Phi_{(j
\bar z) = \label{bulkbdy}
& & %|e^1_{aa}\ket %= \!\!\!\sum_{p, a, \za, t } R_{(a;\za)}^{(j
\bar{j};t)}(p)\
%\label{bulkbdy}
& & \bra p| \phi_{j\bar{j}^*; t}^p(2 i y)|\bar{j}^*\ket\
%\bra e^1_{aa}| {}^{a}\Psi_{p, \za}^a(x) %|e^1_{aa}\ket + \dots \
\nonumber \eea % with $R=\sum \
\Fo\ C $ -- the bulk-boundary reflection coefficients of Cardy--Lewellen \cite{CL}. Given the chiral representation (\ref{bulkf}) all the bulk-boun\-da\-ry sewing relations of \cite{CL} are derived from the duality identitities of the conventional CVO. $\bullet$ Diagonal case: the two basic bulk-boundary Lewellen equations are not independent and are equivalent to the torus duality identity of Moore-Seiberg with $R(p) \sim S(p)$ \cite{BPPZ}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The dual DTA and the physical spectrum} The dual (in the algebraic sense) of Ocneanu DTA leads to the consideration of new graphs $\tilde{G}$
\cite{Ocn,Ocn00}
(see also \cite{BE, BEK})
with a set of vertices $ x \in \tilde{\calV}=\{1
\sum_{i,j}\
(Z_{ij})^2\}$. %
These graphs are described by a set of nonnegative integer valued matrices $\big(\tilde{V}_{ij}\big)_x^{\ y}$
forming a representation of the product of two Verlinde algebras
thus generalising (\ref{nimrep}) \be \tilde{V}_{i_1j_1} \tilde{V}_{i_2j_2}= \sum_{i_3,j_3} N_{i_1i_2}{}^{i_3} N_{j_1j_2}{}^{j_3}\ \tilde{V}_{i_3j_3} \ee with the additional condition that $\big(\tilde{V}_{ij^*}\big)_1^{\ 1}= Z_{ij}$
the matrix of (\ref{partfn}). The eigenvalues of $\tilde{V}_{ij}$ are described by this matrix $Z$
are labelled by the set $\tilde{\calE} = \{ (\ell
\bar\ell)\
\ell
\bar\ell\in \calI\
$ taken with multiplicity $(Z_{\ell\bar\ell})^2\} $
$|\tilde{\calE}| = |\tilde{\calV}|$
and are of the form $S_{i\ell}S_{j\bar\ell}/S_{1\ell}S_{1\bar\ell}$. % For example in the $E_6$ case
this graph has 12 vertices that we label by a pair $x=(a
b)$
$a=1
\cdots
6$
$b=1
2$ of two vertices of the ordinary $E_6$ diagram; the new graph is generated by ${\tilde V}_{2,1}$ and ${\tilde V}_{1,2}$ and the corresponding edges are depicted in red and blue respectively on the following figure due to Ocneanu
\qquad{\epsfxsize=5cm\epsfbox{ocne6.eps}} \noindent On that figure $ (P_{ab})_{ij}:=\sum_{c=1,5,6} n_{ia}{}^c n_{jb}{}^c$
expressed in terms of $n_i$ of (\ref{nimrep})
give the matrices $({\tilde V}_{ij})_1{}^x$
generalising a well known formula for $Z=P_{11}$ \cite{DFZ2,PZ2}. This Ocneanu graph $\tilde{G}$ gives rise to a (noncommutative in general) fusion algebra with structure constants $\tilde{N}_{xy}{}^z$; the nodes of the graph can be equivalently associated with the matrices ${\tilde N}_x$. This algebra also admits a representation by matrices $(\tilde{n}_x)_a^b= \tilde{n}_{xa}{}^{b}$ of size $|\calV|$ with nonnegative integer entries \be \sum_{b\in \calV}\
\tilde{n}_{xa}{}^b \
\tilde{n}_{yb}{}^c= \sum_{z \in \tilde{\calV}}\
\tilde{N}_{xy}{}^z\
\tilde{n}_{za}{}^c \
. \ee These integers describe the dimensions $\tilde{m}_x=\sum_{a,b}\
\tilde{n}_{xa}{}^b$ of the dual spaces $\hat{V}_x$
spanned by states (dual triangles) $|E_{a x;\za}^b\ket\
$ $\za =1
\tilde{n}_{ax}^{b}$. The ``dual double triangles'' span a basis of $\hat{\calA}$
with vertical and horizontal products reversed. The two basis sets are related by a ``fusing'' matrix
$\Fos$
satisfying the pentagon identity \be \Fos \ \Fos = \Fos \ \Fo\ \Fo^*\
. \ee More pentagon relations involve the dual $3j$- and $6j$-symbols; the full set of such relations is named the ``Big Pentagon identity'' in \cite{BoSz}. In the diagonal cases
all these $F$ matrices coincide. The problem is to extend this general construction beyond the diagonal case. {Example:} $sl(2)$ $D_r$ series
$r={h\over 2}+1$ odd. Here $\tilde{\calV}=\{x\equiv k =1,2,\dots, h-1\}\
$ $\tilde{\calE}=\{(l, \zeta(l))\, |\,
l=1,2,\dots, h-1\}\
$ %fusion rules preserving automorphism where $\zeta(l)=h-l$ for $l$ even
$\zeta(l)=l$ for $l$ odd. Then $({\tV_{ij}})_1{}^k=N_{i \zeta(j)}{}^k$ is diagonalised by $S$ and $\tilde{G}= A_{h-1}\
$\tilde{N}_k=N_k\
$ $\tilde{n}_k= n_k$. Hence $\cal A$ is self-dual
with both the $3j$- and $6j$-symbols being self-dual. The $3j$-symbols $\Fo$ were found in \cite{Ru} and it remains to determine the $ \Fos$ to have a non-diagonal example of WHA. \medskip While $\calA$ gives meaning to the generalised CVO
the CFT interpretation of the dual $\hat{\calA}$ is less clear. The representations of the $\tilde{N}$-algebra carry the labels $(j
\jb)$ of the physical fields with $Z_{(j,\jb)}\ne 0$. In the cases $Z_{ij}=0
1$
when this algebra is commutative with only 1-dimensional representations
one can consider its dual algebra
a generalised Pasquier algebra
with structure constants \be \tilde{M}_{(i
\bar{i}) (j
\bar{j})}^{(k
\bar{k})} = \sum_{x\in \tilde{\calV}}\
{\Psi_x^{(i
\bar{i})}\over \Psi_x^{(1,1)}}\
\Psi_x^{(j,\jb)}\
\Psi_x^{(k
\bar{k})*}\
\ee where $\Psi_x^{(i
\bar{i})}$ is a unitary matrix diagonalising $\tilde{V}_{ij}$ and $\tilde{N}_x$. In all $sl(2)$ cases of this type
$A$
$D_{odd}$
$E_6\
E_7\
E_8\
$ one finds \cite{PZ4} a relation generalising (1.4)
\be \quad \tilde{M}_{(i
\bar{k})}= \Big (d_{(i
\bar{k})} \Big)^2\
\label{gP} \ee involving the relative OPE coefficients of fields of non-zero spin. For $E_6
$ one has $ \tilde{M}_{(i
\bar{k})}= M_{ij}^k\ M_{\bar{i} \bar{j}}^{\bar{k}}\
$ \noindent in agreement with the known factorisation of the OPE coefficients in these cases. The problem remains to recover (\ref{gP}) directly in the field theory framework as it has been achieved in the boundary CFT \cite{PSS, BPPZ} for its scalar counterpart (\ref{strc}). \bigskip %%%%%%%%%%%%%%%%% \noindent{Acknowledgements:~}
\noindent Our warmest thanks to Adrian Ocneanu for his generous explanations of his recent work prior to publication. We also thank Paul Pearce and Peter Vecsernyes for useful discussions. V.B.~P. acknowledges partial support of the Bulgarian National Research Foundation (Contract $\Phi$-643). J.-B.~Z. acknowledges partial support of the EU Training and Mobility of Researchers Program (Contract FMRX-CT96-0012). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %\begin{thebibliography}{99} %%%%%%%%%%%%% \begin{thebibliography}{99}\label{bib} \nc{\e}[1]{{\em #1\/}} \nc{\art}[7]{\bibitem{#1}#2 {\sl#3} {#4} (#5) #6 \e{#7}} \nc{\book}[5]{\bibitem{#1}#2 {\sl#3} (#4, #5)} \nc{\proc}[8]{\bibitem{#1}#2 \e{#3} {\sl in} #4 {\sl#5} (#6, #7, #8)} \nc{\cm}[4]{\bibitem{#1}#2 {cond-mat preprint} #3
\e{#4}} \nc{\pth}[4]{\bibitem{#1}#2 {hep-th preprint} #3
\e{#4}} \nc{\qa}[4]{\bibitem{#1}#2 {q-alg preprint} #3
\e{#4}} \nc{\CMP}[6]{\art{#1}{#2, }{Comm.~Math.~Phys.}{#3}{#4}{#5, }{#6}} \nc{\IJMPA}[6]{\art{#1}{#2, }{Int.~J.~Mod.~Phys.~A}{#3}{#4}{#5, }{#6}} \nc{\IJMPB}[6]{\art{#1}{#2, }{Int.~J.~Mod.~Phys.~B}{#3}{#4}{#5,}{#6}} \nc{\JPA}[6]{\art{#1}{#2, }{J.~Phys.~A}{#3}{#4}{#5, }{#6}} \nc{\JSP}[6]{\art{#1}{#2, }{J.~Stat.~Phys.}{#3}{#4}{#5, }{#6}} \nc{\LMP}[6]{\art{#1}{#2, }{Lett.~Math.~Phys.}{#3}{#4}{#5, }{#6}} \nc{\MPLA}[6]{\art{#1}{#2, }{~Mod.~Phys.~Lett.~A}{#3}{#4}{#5, }{#6}} \nc{\NPB}[6]{\art{#1}{#2, }{Nucl.~Phys.~B}{#3}{#4}{#5, }{#6}} \nc{\NPP}[6]{\art{#1}{#2, }{Nucl.~Phys.~Proc.~Suppl.}{#3}{#4}{#5, }{#6}} \nc{\PLA}[6]{\art{#1}{#2, }{Phys.~Lett.~A}{#3}{#4}{#5, }{#6}} \nc{\PLB}[6]{\art{#1}{#2, }{Phys.~Lett.~B}{#3}{#4}{#5, }{#6}} \nc{\PRL}[6]{\art{#1}{#2, }{Phys.~Rev.~Lett.}{#3}{#4}{#5, }{#6}} \nc{\PA}[6]{\art{#1}{#2, }{Physica~A}{#3}{#4}{#5, }{#6}} \nc{\FP}[6]{\art{#1}{#2, }{Fortschr.~Phys.}{#3}{#4}{#5, }{#6}} %%%%%%%%%%%%%%%%%%%%% \newcommand\jpha[3] {{J. Phys.\ }{A #1} (#2) #3} \newcommand{\mathQA}[1]{\href{http://xxx.lanl.gov/abs/math.QA/#1}{q-alg/#1}} \newcommand{\mathOA}[1]{\href{http://xxx.lanl.gov/abs/math.OA/#1}{math.OA/#1}} \bibitem{Ca89} J.L.\ Cardy
\npb{324}{1989}{581-596}. %{Boundary Conditions, Fusion Rules and the Verlinde Formula} % \bibitem{BPPZ} R.E. Behrend
P.A. Pearce
V.B. Petkova and J.-B. Zuber \plb{444}{1998}{163-166} %[{hep-th/9809097}]; [\hepth{9809097}]; %{On the classification of bulk and boundary conformal field theories}; \npb{570}{2000}{525-589} %[{hep-th/9908036}]. [\hepth{9908036}]. % \bibitem{PZ4} V.B. Petkova and J.-B. Zuber
{\sl The many faces of Ocneanu triangular cells}
in preparation. % \bibitem{MS1} G. Moore and N. Seiberg
\cmp{123}{1989}{177-254}. %{Classical and Quantum Conformal Field Theory} %; {\sl Lectures on RCFT, }{Physics, Geometry and Topology, %}{Plenum Press, New York, 1990}. % \bibitem{KiRe} A.N. Kirillov and N.Yu. Reshetikhin
% preprint LOMI E-9-88
in {New developments of the theory of knots}
{T. Kohno ed., }{World Sc. 1990.}{} % \bibitem{DF} Vl.S. Dotsenko and V.A. Fateev
\npb{251[FS13]}{1985}{691-734}. %{Four-point correlation functions and the operator algebra in 2d %conformal invariant theories with central charge $c\le 1$} % \bibitem{PZ1} V.B. Petkova and J.-B. Zuber
\npb{438}{1995}{347-372} %[{hep-th/9410209}]. [\hepth{9410209}]. %{On structure constants of $sl(2)$ theories} % \bibitem{CIZ} A. Cappelli
C. Itzykson and J.-B. Zuber
\npb{280}{1987}{445-465}; %{Modular Invariant Partition Functions in Two Dimensions} \cmp{113}{1987}{1-26}.
%{The A-D-E Classification of Minimal and $A_1^{(1)}$ %Conformal Invariant Theories}; A. Kato
\mpla{2}{1987}{585-600}. % %{Classification of Modular Invariant Partition Functions in %Two Dimensions } % \bibitem{Kos} I. Kostov \npb{300}{1988}{559-587}. %{Free field presentation of the $A_n$ coset models on the torus } % \bibitem{DFZ1} P. Di Francesco and J.-B. Zuber
\npb{338}{1990}{602-646}. %{$SU(N)$ Lattice Integrable Models Associated with Graphs } % \bibitem{Pasq87} V. Pasquier
\jpha{20}{1987}{5707-5717}; %{Operator Content of the $ADE$ lattice models} and the preprint version of that article
reprinted in V. Pasquier
{\sl Mod\`eles Exacts Invariants Conformes}
{Th\`ese d'Etat, Orsay, 1988.} % \bibitem{Ga} T. Gannon
\cmp{161}{1994}{233-263} %[{hep-th/9212060}]; [\hepth{9212060}]; %{The Classification of Affine SU(3) Modular Invariant Partition Functions}; {\sl Ann. Inst. H. Poincar\'e}
Phys.Theor. {65} (1996) 15-56
[\hepth{9404185}]. %{The classification of SU(3) modular invariant revisited} % % \bibitem{Ocn} A. Ocneanu
{}{Lectures at Bariloche Summer School, } {Argentina, Jan 2000}. % \bibitem{BEK} J. B\"ockenhauer
D. E. Evans and Y. Kawahigashi
\cmp{208}{1999}{429-487} [\mathOA{9904109}]; %{On $\alpha$-induction, chiral generators and modular invariants %for subfactors} %\bibitem{BEK2} J. B\"ockenhauer
D. E. Evans and Y. Kawahigashi \cmp{210}{2000}{733-784} [\mathOA{9907149}]. %{Chiral structure of modular invariants for subfactors} % \bibitem{BE} J. B\"ockenhauer and D. E. Evans
%\cmp{200}{1999}{57-103}
[ \hepth{9805023}]. %Modular invariants
graphs and $\alpha$-induction for nets of subfactors
The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory ``in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden ``quantum symmetry''. The Ocneanu triangular cells -- the $3j$-symbols of these symmetries, admit various interpretations and make a link between different problems.
petzub.tex
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% eqns (6.1) and (6.2) = version sent to hep-th

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\def\youngone{{ \hbox{\epsfxsize=1.5mm\epsfbox{onebox.eps}} }}
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\conference{Nonperturbative Quantum Effects 2000}

\title{ BCFT: from the boundary to the bulk}

\author{V.B. Petkova 
$\!\!$\thanks{
permanent address:
Institute for Nuclear Research and Nuclear Energy,
72 Tzarigradsko Chaussee,
1784 Sofia, Bulgaria;
%E-mail: 
%\email{
petkova@inrne.bas.bg}
\ and J.-B. Zuber\\
School of Computing and Mathematics\\
University of Northumbria\\
NE1 8ST Newcastle upon Tyne, UK\\
E-mail: \email{valentina.petkova@unn.ac.uk}\\
and\\
SPhT, CEA Saclay\\
91191 Gif-sur-Yvette, France\\
E-mail:  \email{zuber@spht.saclay.cea.fr}}

\abstract{
The study of boundary conditions in rational conformal field
theories is not only physically important. It also reveals a lot
on the structure of the theory ``in the bulk''. The same graphs
classify both the torus and the cylinder partition functions and
provide data on their hidden ``quantum symmetry''.  The Ocneanu
triangular cells -- the $3j$-symbols of these symmetries, admit
various interpretations and make a link between different
problems.


}
%
\keywords{conformal field theory, boundary, graphs, weak Hopf algebra}
 


\begin{document}
%
\section{Introduction}

Ten years after the work of Cardy \cite{Ca89}, Boundary Conformal 
Field Theory (BCFT) is experiencing a  renewal of interest. This 
is motivated by applications to string and brane theory, and to 
problems of statistical mechanics and condensed matter. In these 
situations, it may be important to have an a priori knowledge of
the possible boundary conditions compatible with conformal 
invariance, and to master the algebra of boundary fields etc. 
There is, however, another reason to be interested in BCFT: 
as we want to explain in this note, there is much to learn 
on the general structure and the quantum symmetries of a CFT from 
the study of its properties in the presence of a boundary.
After a brief review of notations and general aspects of
rational CFT, we shall discuss how boundary conditions may be
systematically classified in terms of 
non-negative integer valued matrix representations 
of the fusion algebra, or equivalently in terms of graphs generalising 
the ADE Dynkin diagrams (see \cite{BPPZ} for more details). We shall see 
 how the ``cells'' -- a concept introduced by
Ocneanu and  associated with these graphs --
 determine many properties
of the BCFT or of the associated lattice models 
 and, in particular,  are at the heart of the quantum symmetry of
 the 
 CFT described by an algebraic structure called weak  $C^*$-Hopf
 algebra.  A more detailed presentation will appear in \cite{PZ4}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{General set-up}
 A rational conformal field theory (RCFT) is generally described
by data of different nature:\\
%%%%%%%%%%
\noindent
$\bullet$  {\bf Chiral data:} 
%--------------------------- 
Chiral data  specify the  chiral
algebra ${\goth A}$, e.g., the   Virasoro algebra itself, or a
${\cal  W}$ algebra, a current algebra $\widehat{ g}\,$ etc, and
its finite set $\calI$ of irreducible representations,
$\calV_i\,, \ i\in \cal  I\,$; 
notations are such that $i=1$ labels
the identity (vacuum)  representation and $i^*$ the conjugate of
${\cal V}_i$.
  The  fusion rule multiplicities  $N_{ij}{}^k$, $
\calV_i \star \calV_j =\oplus_k\, N_{ij}{}^k\,  \calV_k\,, $ are
assumed to be given by the Verlinde formula,
%
\be 
N_{ij}{}^k=\sum_{\ell\in{ I}}{S_{il}\, S_{jl}
\left(S_{kl}\right)\!{}^*
\over S_{1l}} %\;\in\; {\Bbb Z_{\ge 0}}
\, ,
%{\Bbb N}\,,
\label{Verl} 
\ee 
%
with $S$ the symmetric, unitary matrix of the modular
transformations of the characters $\chi_i(\tau)=\tr
e^{2i\pi\tau(L_0-c/24)}$, $\chi_i(\tau)=\sum_{j\in \calI} S_{ij}
\chi_j(-1/\tau)$.
The nonnegative integers $N_{ij}{}^k$ give the dimensions 
of linear spaces of chiral vertex operators (CVO)
%
\be 
\phi_{ij;t,I}^k(z):\ \calV_i \otimes \calV_j \to \calV_k \,, \ \
%t=1,2,\dots N_{ij}{}^k,\ 
z\in {\Bbb C}\,,
\ee 
%
with a finite basis label  $t=1,2,\dots N_{ij}{}^k\,,$
%with $t=1,2,\dots N_{ij}{}^k\,$ a basis label
and  $I$ labelling descendents in $\calV_i$.
The chiral data finally include the knowledge of the duality
matrices: the genus $0$ fusing $F$ and braiding $B(\pm)$
matrices, and the matrix $S(j)$, which gives the modular
transformation of $1$-point conformal block $\bra \phi_{ji}^i
\ket$ on the  torus. These matrices satisfy a set of
consistency relations: pentagon, hexagon and  torus identities
\cite{MS1}. 
%In particular, the matrix $F$ satisfies a pentagon
%identity expressing the associativity of the fusion of CVO.

A typical example is provided by the chiral algebra ${\goth
A}=\slh(2)_k$, the affine algebra at level $k$, for which
$\calI=\{1,2,\cdots,k+1\}$, $S_{ij}=\sqrt{2/(k+2)}$ $\sin{\pi
ij/(k+2)}$,  and the $F$ are, up to a gauge transformation, the
quantum $6j$ symbols \cite{KiRe}.

Those are the basic ingredients of algebraic nature in the
construction of a RCFT. In the following, they are supposed to be
known.

%%%%%%%%%% 
%\medskip
\noindent
$\bullet$  {\bf Spectral and OPE data:}
%--------------------------------------
The physical spectrum of a RCFT {\it in the bulk}, i.e. on a
closed Riemann surface, is described  by irreducible
representations of {\it two} copies of  the chiral algebra. Thus
the Hilbert space is 
decomposed according to
%
\be 
{\cal H}_P=\oplus_{_{j,\jb\,\in{ \calI}}}\,
 Z_{j\, \bar{j}}\, \calV_j\otimes \bar{\calV}_{\bar j}\,.
 \label{Spec}
\ee 
%
The integer multiplicities $Z_{j\,\bar{j}}$ (with $Z_{11}=1$,
expressing the unicity of the vacuum) are conveniently encoded in
the modular invariant torus partition function
%
\be 
Z(\tau)=
\sum_{j,\jb\,\in{ \calI}}\,  Z_{j\jb}\ \chi_j({\tau})\,
\chi_{\jb}(\tau)^* \,.
\label{partfn}  
\ee 
%
 The particular case where $Z_{ij}=\delta_{ij}$ will be referred
to as the diagonal theory, the others as ``non-diagonal''.

On the other hand, 
the  physical fields and their correlators factorise
\omit{\be 
\langle \cdots \Phi_{(i,\ib)}(z,\bar{z}) \cdots \rangle
 =
 \dots \sum_{j, \jb, k, \bar{k}, t,\bar{t}}\,  
d_{(i,\ib) (j,\jb)}^{(k,\bar{k});t, \bar{t}}\cdots
\langle \cdots \phi_{i,j;t}^k(z)\,  \cdots \rangle\ \otimes
\langle \cdots \phi_{\ib,\jb;\bar{t}}^{\bar{k}}(\bar{z})\, \cdots \rangle
\ee 
}
\be
\Phi_{(i,\ib)}(z,\bar{z})=\!\!\! \sum_{j, \jb, k, \bar{k}, t,\bar{t}}\,
\!\!\! d_{(i,\ib) (j,\jb)}^{(k,\bar{k});t, \bar{t}} \,
\phi_{i,j;t}^k(z)\, \otimes
\phi_{\ib,\jb;\bar{t}}^{\bar{k}}(\bar{z})\ .
\ee
%
The expansion coefficients
$d_{(i,\ib) (j,\jb)}^{(k,\bar{k});t, \bar{t}}\ $
determine, up to the  normalisation of the chiral blocks,
the  coefficients of the short distance operator product expansion (OPE),
 i.e., these are the {\it relative} OPE coefficients of
 the non-diagonal model if $d$ are chosen trivial for the  diagonal
 model of  same central charge.
   These numbers are constrained by the requirement of locality 
of the physical correlators, which makes use 
of the braiding matrices $B(\pm)$. 
The resulting set of coupled quadratic equations
has been fully solved only  in the $sl(2)$ cases 
(see \cite{DF,PZ1} and further references therein).
\quad 


Starting with the ADE  classification of the $sl(2)$ modular
invariants \cite{CIZ}, it has been gradually realised that behind
these 
%spectral and OPE 
data there are hidden {\bf graphs $G$}.  A
systematic study of the graphs generalising the $sl(2)$ ADE
Dynkin diagrams was initiated in \cite{Kos,DFZ1}.  
It was empirically observed that these graphs are encoding a non
trivial information on some spectral and OPE data, more precisely
on those pertaining to the {\it spin-zero} fields of the theory.
%
These graphs $G_i$, which share the same set of vertices $\calV$,
are labelled by an index $i\in {\cal I}$ --but it is sufficient
to restrict $i$ to a generating subset (generating in the sense
of fusion)-- and their adjacency matrices are commuting and
simultaneously diagonalizable in an orthonormal basis $\{\psi\}$:


\noindent $\bullet $
the  diagonal part of the physical spectrum, 
labelled by the so-called  {\it ``exponents''}\,:
\be 
{\cal  E}=\{j\in  \calI| j=\bar j, Z_{jj}\ne 0\}\ ,
\ee
counted with the multiplicity $Z_{jj}$, is in 
one-to-one correspondence with the spectrum of eigenvalues of $G_i$,
of the form $S_{ij}/S_{1j}$, $j\in \calE$.

\noindent $\bullet$ in the $sl(2)$ (ADE) cases
the structure constants $M_{ij}{}^k$ of the so-called 
Pasquier algebra \cite{Pasq87}
can be identified with the (relative) OPE coefficients
of the scalar fields \cite{PZ1}
\be 
d_{(i,i) (j,j)}^{(k,k)}=M_{ij}{}^k:=\sum_{a\in \calV}\, {\psi_a^i\,
\psi_a^j \,\psi_a^{k\, *}\over \psi_a^1}\,.
\label{strc} 
\ee 
The summation in (\ref{strc}) runs over the set $\cal V$ of vertices
of the Dynkin diagram.

The purpose of this note is to show that the study  
of the CFT on a half-plane or a cylinder %, 
(i.e. on a manifold with boundary), \cite{Ca89},
provides an alternative (chiral) approach in
which  these graphs and the related algebraic structures become
manifest. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Graphs and conformal boundary conditions} 
% 
We may summarize the results of \cite{BPPZ} as follows: boundary
conditions that respect the chiral algebra $\goth A$ are
described by a set of commuting, non-negative integer valued
matrices $\{n_i= n_{i a}{}^b\}\,,i\in \calI\,,$ $a,b\in \calV\,,$
$|\cal E|=|\cal V|$, s.t. $n_1=I$, $n_i^T=n_{i^*}$, realising a
representation of the Verlinde fusion algebra
\be 
n_i\;n_j=\sum_{k\in \calI} N_{ij}{}^k\,n_k \,. 
\label{nimrep} 
\ee 
 The matrices $n_i$ thus   admit a spectral decomposition
\be 
n_{i a}{}^b =\sum_{j\in \cal  E}\, {S_{ij}\over S_{1j}}\
\psi_a^j\ \psi_b^{j*}\,,
\label{specdec}   
\ee 
% 
where $\psi_a^j$ are unitary matrices of dimension $|\cal E|=|\cal V|$
and the sum runs over the 
set $\{(j,\alpha)\,, j\in \calI\,,
\alpha=1,2,\dots, Z_{jj}\}$, 
for simplicity of notation identified with
 the set  of exponents $\calE$.
%
This comes about as follows: On the upper half-plane 
 param\-etrised
by a coordinate $z\in {\Bbb H}^+$,  (resp. on a finite-width
strip $w={L\over \pi}\, \log z\,$), with boundary conditions $b$
and $a$ imposed on the negative and positive real axes, (resp. $\!\!$ on
the two sides of the strip), only one copy of the chiral algebra
acts: only real analytic coordinate transformations,
$\epsilon(z)=\bar{\epsilon}(\bar{z})$ for real $z=\bar{z}$, are
allowed. Thus the Hilbert space of the theory ${\cal H}_{ba}$
splits into a linear sum of representations
\be 
{\cal H}_{ba}=\oplus_{i\in \calI} n_{ib}{}^a \,\calV_i\ .
\label{Hab}
\ee 
On a finite segment of the strip of length $T$ with periodic boundary 
conditions $w\sim w+T$, i.e. on a cylinder, 
the partition functions $Z_{b|a}$ is thus a linear form
in the characters
%
\be 
Z_{b|a}(\tau)=\sum_{i\in \calI} n_{ib}{}^a\, \chi_i(\tau)\,, \quad
\tau= {i T\over 2 L}\,.
\label{pfn} 
\ee 
%
The multiplicities $n_{ia}{}^b$ are constrained  by
the Cardy consistency condition  \cite{Ca89}  
which expresses that $Z_{b|a}$ can also be evaluated in a dual way:
by  mapping the cylinder to an annulus region in the full plane, 
one computes $Z_{b|a}$ as the matrix element of the
evolution operator between boundary states 
$|a\rangle$ and $\langle b|$. The latter are decomposed on a standard basis 
of complete and orthonormal ``Ishibashi states'' $|j\rangle\!\rangle$ 
labelled by an exponent $j\in \calE$, 
according to $|a\rangle =\sum_{j\in \cal E}\,  
{\psi_a{}^j \over \sqrt{S_{1j}}}
\ |j\rrangle $. This gives
%
\bea
Z_{b|a}(\tau)
&=&
\langle b|e^{{-\pi i\over \tau} (L_0+\overline{L}_0-{c\over 12})}
|a\rangle
\nonumber\\ 
&=&
 \sum_{j\in \calE}{\psi_a{}^j \psi_b{}^{j*} \over {S_{1j}}}
\chi_j\Big({-1\over\tau}\Big)\ . \label{cardy}  
\eea 
Comparing the resulting two expressions for %\\
  $Z_{b|a}$ 
yields the
spectral decomposition (\ref{specdec}). See \cite{BPPZ} for a discussion of
the assumptions and a more detailed derivation. 

Each set of matrices  $\{n_i\}$ solving (\ref{nimrep}), (\ref{specdec})  
may be regarded as the adjacency matrices of a collection of graphs
$G_i$, thus explaining the occurence of graphs with the special 
spectral properties noticed above.
% 
Solving in general  the system  (\ref{nimrep}) 
provides   a classification of boundary conditions,   labelled by
the vertices of the graphs $G_i$ 
and specified by the spectrum (\ref{Hab}).  
The  case $n_i=N_i $ provides the regular
 representation with spectrum $\cal  E=\cal  I$ corresponding to the
 diagonal case;
then (\ref{nimrep}) reduces to (\ref{Verl}) with $\psi=S$
and the boundary states are labelled by the same indices $i\in \calI$ as 
the representations and primary fields \cite{Ca89}.


$\bullet$
 In $\slh(2)$ RCFT  
(the WZNW models and the Virasoro minimal models)
this reduces to the classification of the symmetric, irreducible, non-negative
integer valued matrices of spectrum $|\zg^j|$ 
$ =|{S_{2 j}\over S_{1 j}}|< 2$.
This is well known to lead to an ADE classification, after
the  spurious ``tadpole'' graphs  $A_{2n}/{\Bbb Z}_2$ have been discarded, 
on the basis that their spectrum does  not match any known modular
invariant.


$\bullet$
A new situation arises in the $sl(3)$ case, where 
there is no known a priori classification of (oriented) graphs 
with the required spectral properties. Solutions may be found
\cite{DFZ1}
and comparison with the complete list of modular invariants \cite{Ga}
enables one to discard some spurious solutions and leads to 
a list of graphs, see \cite{BPPZ}. 
One also finds that in a few cases, there are isospectral graphs, i.e.
 more than one solution with a given spectrum, indicating that 
there are several choices of boundary conditions
 (and in fact of OPE  structure constants, see \cite{BPPZ}), 
for a given spectrum of  spin-zero fields in a RCFT.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{ Ocneanu quantum graph symmetry} 

According to Cardy the boundary conditions $(a,b)$ are
created by insertions of 
 fields on the boundary ({\it boundary fields}),
  $^a\Psi_{j;\beta}^b(x)$,
$\beta =1,2,\dots n_{jb}{}^a\,,\ $ $x\in {\Bbb R}$.
%, $z\in {\Bbb H}^+$.
Exploiting some ideas of Ocneanu \cite{Ocn,Ocn00}
one can interpret these fields, extended to the full plane, as 
generalised CVO.

  Given a solution of the system of equations
   (\ref{nimrep}), consider 
for each $j\in \cal I$ an auxiliary  Hilbert space 
$V^j\cong {\Bbb C}^{m_j}\,$ of dimension $m_j= \sum_{a,b}\,
n_{ja}{}^b$ with basis states $|e^{j,\beta}_{ba}\rangle$, 
%					       |j
%					    b__|__a
$\beta=1,2,\dots,n_{j a}{}^b$, depicted as  
\hbox{\raise -7mm\hbox{\epsfxsize=3cm\epsfbox{dta11.eps}}}.
A scalar product in $\oplus_{j\in \calI}\, V^j$ is defined as
%
\bea 
 \langle e^{j,\beta}_{ ab} |
e^{j',\beta'}_{a'b'}\rangle 
&=& \delta_{bb'} \delta_{aa'}\,
\delta_{jj'}\,\delta_{\beta' \beta}\,\sqrt{P_a\,P_b\over  d_j}
\,, \nonumber \\
d_j&:=&{S_{j1}\over S_{11}}\,,\ P_a:={\psi_a^1\over \psi_1^1}\,. \label{ebasis}
\eea 
%
%
Restricting to a subspace $V^i \otimes_h\, V^j\,$ of $V^i\otimes V^j\,,$
with coinciding intermediate labels, one defines a decomposition
$V^i\otimes_h\, V^j\cong \oplus_{k}\, N_{ij}{}^k\,V^k$,
  or, explicitly,
\bea 
\vert e^{i, \eta}_{ab}\ket \otimes_h \vert e^{j,\zeta}_{b c}\ket &=&
\, \sum_{k\in \calI }\ \sum_{\gamma=1}^{n_{kc}{}^a}\,
\sum_{t=1}^{N_{ij}{}^k}\  
\Fo_{b k}\left[ {i\atop a}\ {j\atop c} \right]_{\eta\, \zeta}^{\gamma\, t}\,
\nonumber \\
% & &\!\!\!\!\! \!\!\!\!
\times \sqrt{P_b}\, 
& &\!\!\!\!\! \Big({d_k \over d_i d_j}\Big)^{{1\over 4}}\
|e^{k\,, \gamma}_{ac}(ij;t)\ket\,.
\label{prodh} 
\eea 
\omit{Pictorially \hbox{\raise -4mm\hbox{\epsfxsize=4cm\epsfbox{Fone.eps}}} 
%	|    |	               \/
%	|    |		        |			Fig 2
%      ________  = sum ^1F  _________
}
%%%%
The counting of states in both sides is consistent
 due to (\ref{nimrep}).
 The  Clebsch-Gordan coefficients $\Fo\in {\Bbb C}$
({\bf ``$3j$- symbols''})  satisfy
 unitarity conditions (completeness and orthogonality)
and a pentagon identity,  reflecting the  associativity of the
product (\ref{prodh}), written schematically as
%
\be 
 F\  \Fo \ \Fo \, =   \Fo\ \Fo \,,\qquad 
\label{pentagon} 
\ee 
where $F$ are the $6j$-symbols. In the diagonal case $F$ and $\Fo$ 
are identified, and (\ref{pentagon})  becomes the conventional pentagon 
identity satisfied by the fusing matrices of the corresponding RCFT.


The space  $\calA =\oplus_{j\in \calI}\, \hbox{End}\,(V^j)$
is a matrix algebra $\oplus_{j\in \calI}\, M_{m_j}$.
A  matrix unit basis
$\{e_{j;\eta\,,\eta'}^{(ab)\,,(a'b')}\\
=| e^{j,\eta}_{ab}\ket \bra\,e^{j,\eta'}_{a'b'} |
\,, j\in\cal I\}$ in $\calA$ is identified
with a product of states in $V^j\otimes V^{j^*}$,
and depicted as $4$-point blocks or double triangles  \\
%\medskip
% <<<  Fig
\qquad\hbox{\raise -4mm\hbox{\epsfxsize=5cm\epsfbox{dta12.eps}}}\\ 
\noindent
 Along with the matrix (``vertical'') product, 
 a second ``horizontal'' product
 (or, alternatively, a coassociative coproduct), is defined   on $\cal A$
 via the $3j$-symbols $\Fo\,$
 \\ \noindent
\hbox{\raise -4mm\hbox{\epsfxsize=7cm\epsfbox{dta2h.eps}}}\\ 
\noindent
The algebra generated by these objects
 is the Ocneanu {\bf double triangle algebra} (DTA) \cite{Ocn},
further studied in the subfactor theory, see \cite{BEK,BE} and 
references therein.
Equipped with a counit and an antipode satisfying a weakened version
of the Hopf algebra axioms ($\triangle( 1_v)\ne 1_v\otimes 1_v$, etc.),
  it was considered in  \cite{BoSz} as an example of   a
{\it weak $C^*$- Hopf algebra} (WHA). This  Ocneanu ``graph
quantum algebra'',   
associated to any solution $\{n_i\,,i\in \calI \}$ of $(\ref{nimrep})$,  
together with its dual algebra $\hat{\calA}$ structure (see below), 
appears as the quantum symmetry of the 
 CFT model,  either diagonal or non-diagonal. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Generalised CVO (GCVO)}

We define operators  
$\oplus_{j\in \calI}\, \calV_j\!\otimes\! V_j \!\to\!
\oplus_{k\in \calI}\, \calV_k\!\otimes\! V_k$\\
\noindent associated with the basis states in the tensor product
(\ref{prodh}), 
%
\bea 
^a\Psi_{i, \beta;I}^c(z) 
&=&\sum_{j,k, t}\ \phi_{ij;t,I}^k(z)  \otimes\,
 \sum_{ b,\alpha,\gamma}\,
\Fo_{ck}\left[{i\atop a} {j\atop b}\right]^{ \alpha\, t}_{
\beta\, \gamma}\, \nonumber \\
& &  \times \sqrt{d_j\over P_c\,P_b} \
| e^{k,\alpha}_{ab}\ket \bra\,e^{j,\gamma}_{cb} | \,,
\label{gcvo}  
\eea 
covariant under the action of $\calA$.
Their correlators, obtained by projecting on a state
$|0\rangle \otimes |e^1_{aa}\rangle  $ in the vacuum 
space $\calV_1\otimes V^1$,   as well as 
 their fusing  and braiding properties, are inherited 
 from those of the conventional CVO.
 In contrast with previous approaches  introducing quantum 
 symmetry covariant operators, e.g.,  \cite{MR, MSch}, note that in
the diagonal case the $6j$-symbols $F$  of the
related quantum group (instead of its $3j$-symbols)
 appear in the r.h.s. of (\ref{gcvo}).
 For real $z$ the operators  $\Psi$ of (\ref{gcvo})
represent (after an appropriate choice of normalisation)
the boundary fields, and their short  distance 
$x_{12}\approx 0$  expansion is recovered, with the $3j$-symbols 
 $\Fo$ serving as {\it boundary fields OPE coefficients}.  
The recoupling equation (\ref{pentagon}) expresses then the
associativity of the boundary fields OPE  \cite{BPPZ} and  is
equivalent to the  Lewellen boundary fields sewing identity
\cite{CL}. 
 
 In general the chiral operators (\ref{gcvo}) have nontrivial
braiding with a new braiding matrix $\hat{B}(\epsilon)$, with
$4+2$ indices of two types,
\bea  
& &^a\Psi_{j,\alpha}^b(z_1)\, ^b\Psi_{k,\gamma}^c(z_2)= 
\label{newbraid} \\
& &\sum_{d,
\alpha',\gamma'}
\hat{B}_{bd}\left[{j \atop a} {k\atop c} \right]_{
\alpha\, \gamma}^{\alpha'\, \gamma'}\!\!\!\!(\epsilon)\
^a\Psi_{k,\alpha'}^d(z_2)\, ^d\Psi_{j,\gamma'}^c(z_1)\,,
\nonumber 
\eea  
%
$z_{12}\notin {\Bbb R}_- \,, \
\epsilon=\hbox{sign}\,(\hbox{Im}\,z_{12}$). 
The matrices $\hat{B}$ satisfy various relations:
 
$\bullet$
 Inversion (unitarity) relation
\be  
\hat{B}^{12}(\epsilon) \, \hat{B}^{21}(-\epsilon)=1\,.
\ee  
%

 $\bullet$
 The braid group (``Yang-Baxter'') relation %equation 
%
\be  
\hat{B}^{12}\, \hat{B}^{23}\,  \hat{B}^{12}\,
=  \hat{B}^{23}\, \hat{B}^{12}\,  \hat{B}^{23} \,.
\label{braidgp}
\ee  

 $\bullet$  The braiding--fusing (pentagon) identity
\be  
 \hat{B} \ \Fo =  \Fo\  \hat{B}\  \hat{B}\,,
\ee  
which turns into a recursion relation for $\hat B$,
 given  the fundamental  $3j$-symbols,

  $\bullet$
Intertwining relation
 \be  
\Fo\, \Fo\ B= \hat{B}\ \Fo\, \Fo\,, \label{intwg}
\ee  
which implies a bilinear representation of $\hat{B}$ 
in terms of the $3j$-symbols  
\bea 
& &  \hat{B}_{bd}\left[{j\atop a} {k\atop c} \right]_{
\alpha \gamma}^{\alpha' \gamma'}\!\!\!\!\!\!(\epsilon)
 =
  \label{bilin}\\
 & & \sum_{i\,, \delta\,, t }\,
{}\Fo_{b i}\left[{j\atop a} {k\atop c}\right]_{\alpha
\gamma}^{\delta\,  t }
 e^{-i \pi\epsilon\, \triangle_{jk}^i}\
{}\Fo_{d i}^*\left[{k\atop a} {j\atop c} \right]_{\alpha'
\gamma'}^{\delta\,  t }\,,
\nonumber 
\eea  
where $\triangle_{jk}^i$  is the combination of
  conformal weights $\triangle_j+\triangle_k-\triangle_i $.
%In the diagonal case  $\hat{B}(\epsilon)$ is identified
%with $B(\epsilon)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Connection to lattice models}
Some of these identities have been already encountered in
critical lattice models, the  ADE Pas\-quier models and their
$\widehat{sl}(n)_{h-n}$ generalisations, in which the degrees of
freedom may be regarded as vertices of the previous graphs
\cite{Pa1,DFZ1}.


In the limit $u\to -i\epsilon\infty$, ($\epsilon=\pm1$), of the
spectral parameter $u$,  the face  Boltzmann weights  $W(u)$
satisfy the same equation (\ref{braidgp}) as the braiding matrix
$\hat{B}$ of (\ref{newbraid}).  Indeed, denoting the
representations of $sl(n)$ by their Young tableau, and with
$q=e^{2\pi i\over h}\,$,
\bea  
& & \hat{B}_{bd}  
\left[{\youngone\atop a}{\youngone\atop c} \right]_{
\za\, \zg}^{\za'\, \zg'}\!\!\!\!(\epsilon)\, 
= \nonumber \\
& &  2i \, q^{-\epsilon {1\over 2 n}}\, \lim_{u\to -i\epsilon\infty}\,
e^{-i\pi\epsilon u}\, W_{bd}\left({a\atop c} \right)_{
\alpha\, \gamma}^{\alpha'\, \gamma'} (u)\,,
 \nonumber 
\eea  
%
\be  
W_{bd}(u)=\sin({\pi\over h}-u) \delta_{bd} +\sin(u)\, [2]_q\, U_{bd}
\,,\quad U^2=U\,.
 \ee 
Using (\ref{bilin}), 
the Hecke algebra generators $U$ are expressed via the $3j$-symbols,
recovering an Ansatz of Ocneanu \cite{Ocn00} 
\be  
U_{bd}=\sum_{\zb}\, 
\Fo_{b\, \youngtwo}\left[{\youngone\atop a} {\youngone\atop
c}\right]_{\za\, \zg}^{\zb\ 1}\, 
\Fo_{d\, \youngtwo}^*\left[{\youngone\atop a}
{\youngone\atop c} 
\right]_{\za'\, \zg'}^{\zb\ 1 }\,.
\ee  

Moreover, equation (\ref{intwg}) shows that we can identify  
the $3j$- symbols  $\Fo$  with the 
{\it ``intertwining
 cells}''   studied in these models, see e.g. \cite{DFZ1}; then
 lattice results provide solutions for particular $\Fo$, namely, those
for which one of the labels in $\cal I$ is equal to $\youngone$.

For $sl(2)$: $\youngtwo=1\,,\ $
 $ [2]_q\ U_{bd}  = \delta_{ac}\,{\sqrt{P_b\,P_d}\over P_a }$, while 
for $sl(3)$: $\youngtwo=\youngone^*\,,$ and  these cells $\Fo_{b\,\youngone
{}^*}$
exist for all graphs listed in \cite{BPPZ} but one, according to   
the recent work of  Ocneanu, \cite{Ocn00}.
% completing previous results of {\bf Jimbo et al, Fendley, DiFZ, Sochen}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Bulk fields}

For $(j,\bar{j})$ in the physical spectrum (\ref{Spec}), 
i.e., $Z_{j\bar{j}}\not =0$, and
$z\in {\Bbb H}^+\,,$ define  (upper) half-plane bulk fields 
by two-point compositions of 
%generalised CVO (\ref{gcvo})
GCVO (\ref{gcvo})
\be 
\Phi_{(j,\bar{j})}^H(z, \bar z)
=\sum_{a,b\,,\atop{\beta'\,,\beta}}\
C^{a,b,a;\beta,\beta'}_{(j,\bar{j})}\
{}^a\Psi_{j, \beta}^b(z)\ ^b\Psi_{\bar{j}^*,
\beta'}^a(\bar{z}) \,.
\label{bulkf}  
\ee 
%
{}From this operator representation, which
can be also rewritten
%, using (\ref{gcvo}),
 in terms of the conventional CVO, one  
recovers the bulk field
small distance $z-\bar{z}=2i\,y\approx 0$ expansion,
in the leading order 
%
\bea
& & 
%\bra e^1_{aa}|
\Phi_{(j,\bar{j})}^H(z, \bar z) =
\label{bulkbdy} \\ & & 
%|e^1_{aa}\ket
%=
\!\!\!\sum_{p, a, \za, t } R_{(a;\za)}^{(j,\bar{j};t)}(p)\,
%\label{bulkbdy} \\ & & 
\bra  p| \phi_{j\bar{j}^*; t}^p(2 i y)|\bar{j}^*\ket\, 
%\bra e^1_{aa}|
{}^{a}\Psi_{p, \za}^a(x)
%|e^1_{aa}\ket
 +   \dots \,,
\nonumber 
\eea
%
with $R=\sum \, \Fo\ C $ -- the bulk-boundary  reflection
coefficients  of Cardy--Lewellen \cite{CL}.  Given the chiral
representation (\ref{bulkf}) all the bulk-boun\-da\-ry  sewing
relations of \cite{CL} are derived from the duality
identitities of the conventional CVO.

$\bullet$
Diagonal case: the two basic bulk-boundary Lewellen equations 
are not independent and are 
equivalent to the torus duality identity of
Moore-Seiberg with  $R(p) \sim S(p)$ \cite{BPPZ}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{The dual DTA and the physical spectrum}

The dual (in the algebraic sense)
of Ocneanu DTA leads to the consideration of new  graphs
$\tilde{G}$, \cite{Ocn,Ocn00},
(see also  \cite{BE, BEK}),
with a set of vertices
$ x \in \tilde{\calV}=\{1,2,\dots, \sum_{i,j}\, (Z_{ij})^2\}$. %, 
These graphs are described by  
a set of nonnegative integer
valued matrices    $\big(\tilde{V}_{ij}\big)_x^{\ y}$, 
forming a representation of the  product of two  Verlinde
algebras, thus generalising (\ref{nimrep})
\be
 \tilde{V}_{i_1j_1} \tilde{V}_{i_2j_2}=
\sum_{i_3,j_3} N_{i_1i_2}{}^{i_3} N_{j_1j_2}{}^{j_3}\
\tilde{V}_{i_3j_3} 
\ee
with the additional condition that
$\big(\tilde{V}_{ij^*}\big)_1^{\ 1}= Z_{ij}$, the matrix of 
(\ref{partfn}).  The eigenvalues of
$\tilde{V}_{ij}$ are described by this matrix $Z$,
i.e., are labelled by the set $\tilde{\calE} =
\{ (\ell,\bar\ell)\,, \ell,\bar\ell\in \calI\,,$ taken
with multiplicity $(Z_{\ell\bar\ell})^2\} $,
$|\tilde{\calE}| = |\tilde{\calV}|$  , 
and are of the form 
 $S_{i\ell}S_{j\bar\ell}/S_{1\ell}S_{1\bar\ell}$. 
%
For example in the $E_6$ case, this graph has 12 vertices
that we label by a pair $x=(a,b)$, $a=1,2,\cdots,6$, $b=1,2$
of two vertices of the ordinary $E_6$ diagram; the new graph is
generated by ${\tilde V}_{2,1}$ and    ${\tilde V}_{1,2}$
and the corresponding edges are depicted in red and blue 
respectively on the following figure 
due to Ocneanu \\
\qquad{\epsfxsize=5cm\epsfbox{ocne6.eps}}

\noindent
On that figure $ (P_{ab})_{ij}:=\sum_{c=1,5,6} n_{ia}{}^c
n_{jb}{}^c$, expressed in terms of $n_i$ of (\ref{nimrep}), give
the  matrices $({\tilde V}_{ij})_1{}^x$, generalising a well
known formula for $Z=P_{11}$
\cite{DFZ2,PZ2}.  

 
This Ocneanu graph $\tilde{G}$ gives rise to a (noncommutative in
general) fusion algebra with structure constants
$\tilde{N}_{xy}{}^z$; the nodes of the graph can be equivalently
associated with the matrices ${\tilde N}_x$. This algebra also
admits a representation by matrices $(\tilde{n}_x)_a^b=
\tilde{n}_{xa}{}^{b}$ of size $|\calV|$ with nonnegative integer
entries
\be  
\sum_{b\in \calV}\, \tilde{n}_{xa}{}^b \, \tilde{n}_{yb}{}^c=
\sum_{z \in \tilde{\calV}}\,
\tilde{N}_{xy}{}^z\,
\tilde{n}_{za}{}^c \,.
\ee  
These integers describe the dimensions $\tilde{m}_x=\sum_{a,b}\,
\tilde{n}_{xa}{}^b$ of the dual spaces $\hat{V}_x$, spanned by
states (dual triangles) $|E_{a x;\za}^b\ket\,,$ $\za =1,2,\dots,
\tilde{n}_{ax}^{b}$.  The ``dual double triangles'' span a basis
of  $\hat{\calA}$, with vertical and horizontal products
reversed. The two basis sets are related by a ``fusing'' matrix,
$\Fos$, satisfying the  pentagon identity
\be  
\Fos \ \Fos = \Fos \ \Fo\ \Fo^*\,.
\ee  
More pentagon relations involve the dual  $3j$- and $6j$-symbols;
the full set of  such relations is  named the  ``Big Pentagon
identity'' in \cite{BoSz}.  In the diagonal cases, all these $F$
matrices coincide.  The problem is to extend this general
construction beyond the diagonal case.

{\bf Example:} $sl(2)$ $D_r$ series, $r={h\over 2}+1$ odd.   
Here
$\tilde{\calV}=\{x\equiv  k =1,2,\dots, h-1\}\,,$
$\tilde{\calE}=\{(l, \zeta(l))\,  |\,\\ l=1,2,\dots, h-1\}\,,$
%fusion rules preserving automorphism  
where $\zeta(l)=h-l$ for $l$ even, $\zeta(l)=l$ for $l$ odd.
Then
$({\tV_{ij}})_1{}^k=N_{i \zeta(j)}{}^k$
is diagonalised by  $S$ and $\tilde{G}= A_{h-1}\,$,
$\tilde{N}_k=N_k\,,$
  $\tilde{n}_k= n_k$. Hence
$\cal A$ is  self-dual,  with both the $3j$- and $6j$-symbols
being self-dual. The $3j$-symbols $\Fo$ were found in \cite{Ru}
and it remains to determine the $ \Fos$ 
to  have a non-diagonal example of WHA.
\medskip

While $\calA$ gives meaning to the generalised CVO, the CFT
interpretation of the  dual $\hat{\calA}$ is less clear.  The
representations of the  $\tilde{N}$-algebra carry the labels
$(j,\jb)$ of  the physical fields with $Z_{(j,\jb)}\ne 0$.  In
the cases $Z_{ij}=0,1$, when this algebra is commutative with
only 1-dimensional representations,  one can consider its dual
algebra, a generalised Pasquier algebra, with structure constants
\be
\tilde{M}_{(i,\bar{i}) (j,\bar{j})}^{(k,\bar{k})} =
\sum_{x\in \tilde{\calV}}\,
{\Psi_x^{(i,\bar{i})}\over \Psi_x^{(1,1)}}\,
\Psi_x^{(j,\jb)}\,
\Psi_x^{(k,\bar{k})*}\,,
\ee
where  $\Psi_x^{(i,\bar{i})}$ is a unitary matrix diagonalising
$\tilde{V}_{ij}$ and $\tilde{N}_x$.
In all
$sl(2)$ cases of this type, i.e., $A$, $D_{odd}$,
$E_6\,,E_7\,,E_8\,,$ 
one finds \cite{PZ4}  a relation generalising (1.4),
\be
\quad
\tilde{M}_{(i,\bar{i}) (j,\bar{j})}^{(k,\bar{k})}=
\Big (d_{(i,\bar{i}) (j,\bar{j})}^{(k,\bar{k})} \Big)^2\,,
\label{gP}
\ee
involving
the relative OPE  coefficients  of  fields of
non-zero spin. For $E_6, E_8\,,  $ one has
$
\tilde{M}_{(i,\bar{i}) (j,\bar{j})}^{(k,\bar{k})}=
M_{ij}^k\ M_{\bar{i} \bar{j}}^{\bar{k}}\,,
$
\noindent
in agreement with the known factorisation of the OPE coefficients
in these cases.  The problem remains to recover  (\ref{gP})
directly in the field theory framework  as it has been achieved
in the boundary CFT \cite{PSS, BPPZ} for its scalar counterpart
(\ref{strc}).

\bigskip


%%%%%%%%%%%%%%%%%


\noindent{\bf Acknowledgements:~} \\ \noindent
Our warmest thanks to Adrian Ocneanu for his generous
explanations of his recent work prior to publication.
We also thank Paul Pearce and Peter Vecsernyes for useful
discussions.
V.B.~P. acknowledges partial support of the Bulgarian National
Research Foundation (Contract $\Phi$-643). J.-B.~Z.  acknowledges
partial support of the EU Training and Mobility of Researchers
Program (Contract FMRX-CT96-0012).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\begin{thebibliography}{99}

%%%%%%%%%%%%%



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\newcommand\jpha[3]   
                {{\it J. Phys.\ }{\bf A #1} (#2) #3} 
\newcommand{\mathQA}[1]{\href{http://xxx.lanl.gov/abs/math.QA/#1}{\tt
q-alg/#1}}
\newcommand{\mathOA}[1]{\href{http://xxx.lanl.gov/abs/math.OA/#1}{\tt
math.OA/#1}}


\bibitem{Ca89} J.L.\ Cardy, \npb{324}{1989}{581-596}.
%{Boundary Conditions, Fusion Rules and the Verlinde Formula}
%
\bibitem{BPPZ} R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B.
Zuber \plb{444}{1998}{163-166}
%[{\tt hep-th/9809097}];
[\hepth{9809097}];
%{On the classification of bulk and boundary conformal field theories};
\npb{570}{2000}{525-589}
%[{\tt hep-th/9908036}].
[\hepth{9908036}].
%
\bibitem{PZ4} V.B. Petkova and J.-B. Zuber,
{\sl The many faces of Ocneanu triangular cells}, in preparation.
%
\bibitem{MS1} G. Moore and N. Seiberg, \cmp{123}{1989}{177-254}.
%{Classical and Quantum Conformal Field Theory}
%; {\sl Lectures on RCFT, }{Physics, Geometry and Topology,
%}{Plenum Press, New York, 1990}.
%
\bibitem{KiRe} A.N. Kirillov and N.Yu. Reshetikhin,
% preprint LOMI E-9-88,
in {\it New developments of the theory of knots}, {T. Kohno
ed., }{World Sc. 1990.}{}
%
\bibitem{DF} Vl.S. Dotsenko and  V.A. Fateev, \npb{251[FS13]}{1985}{691-734}.
%{Four-point correlation functions and the operator algebra in 2d
%conformal invariant theories with central charge $c\le 1$}
%
\bibitem{PZ1} V.B. Petkova and J.-B. Zuber, \npb{438}{1995}{347-372}
%[{\tt hep-th/9410209}].
[\hepth{9410209}].
%{On structure constants of $sl(2)$ theories}
%
\bibitem{CIZ} A. Cappelli, C. Itzykson and J.-B. Zuber,
\npb{280}{1987}{445-465};
%{Modular Invariant Partition Functions in Two Dimensions}
 \cmp{113}{1987}{1-26}.\\
%{\it The A-D-E Classification of Minimal and $A_1^{(1)}$
%Conformal Invariant Theories};
A. Kato, \mpla{2}{1987}{585-600}. %
%{\it Classification of Modular Invariant Partition Functions in
%Two Dimensions }
%
\bibitem{Kos} I. Kostov \npb{300}{1988}{559-587}.
%{Free field presentation of the $A_n$ coset models on the torus }
%
\bibitem{DFZ1} P. Di Francesco and J.-B. Zuber,  \npb{338}{1990}{602-646}.
%{$SU(N)$ Lattice Integrable Models Associated with Graphs }
%
\bibitem{Pasq87} V. Pasquier, \jpha{20}{1987}{5707-5717};
%{Operator Content of the $ADE$ lattice models}
 and the preprint version of that article, reprinted in
V. Pasquier, {\sl Mod\`eles Exacts Invariants Conformes},
{\rm Th\`ese d'Etat, Orsay, 1988.}
%
\bibitem{Ga} T. Gannon, \cmp{161}{1994}{233-263}
%[{\tt hep-th/9212060}];
[\hepth{9212060}];
%{The Classification of Affine SU(3) Modular Invariant Partition Functions};
{\sl Ann. Inst. H. Poincar\'e}, Phys.Theor. {\bf 65} (1996) 15-56,
[\hepth{9404185}].
%{\it The classification of SU(3) modular invariant revisited}
%
%
\bibitem{Ocn} A. Ocneanu, 
%{Paths on Coxeter Diagrams: From Platonic Solids 
%and Singularities to Minimal Models and Subactors }
%{Low Dimensional Topology, Statistical Mechanics and Quantum Field Theory}
in {\sl Lectures on Operator Theory, }
 {Fields Institute, Waterloo, Ontario, April 26--30, 1995, }{(Notes taken by
S. Goto)} 
{Fields Institute Monographies, AMS 1999,}{ Rajarama Bhat et al, eds.}
%
\bibitem{Ocn00} A. Ocneanu, {}{Lectures at Bariloche Summer School, }
{Argentina, Jan 2000}.
%
\bibitem{BEK} J. B\"ockenhauer , D. E. Evans and Y. Kawahigashi, 
\cmp{208}{1999}{429-487}
[\mathOA{9904109}];
%{On $\alpha$-induction, chiral generators and modular invariants
%for subfactors} 
%\bibitem{BEK2} J. B\"ockenhauer, D. E. Evans and Y. Kawahigashi
\cmp{210}{2000}{733-784}
[\mathOA{9907149}].
%{Chiral structure of modular invariants for subfactors}
%
\bibitem{BE} J. B\"ockenhauer and  D. E. Evans,
%\cmp{200}{1999}{57-103}, [ \hepth{9805023}].
%Modular invariants, graphs and $\alpha$-induction for nets of subfactors};
\cmp{205}{1999}{183-228}
[\hepth{9812110}].
%
\bibitem{BoSz} G. B\"ohm and K. Szlachanyi, \lmp{200}{1996}{437-456}
[\qalg{9509008}]. %{A coassociative $C^*$- quantum group with
%non-integral dimensions} 
%
\bibitem{MR} G. Moore and N.Yu. Reshetikhin, \npb{328}{1989}{557-574}.
%
\bibitem{MSch} G. Mack and V. Schomerus, \npb{370}{1992}{185-230}.
%
\bibitem{CL} J.L. Cardy and D.C. Lewellen,  \plb{259}{1991}
{274-278};
D.C. Lewellen,  \npb{372}{1992}{654-682}.
%
\bibitem{Pa1} V. Pasquier, \npb{285}{1987}{162--172}.
%{Two-Dimensional Critical Systems Labelled by Dynkin Diagrams}
%
\bibitem{DFZ2} P. Di Francesco and J.-B.  Zuber, 
%{$SU(N)$ Lattice Integrable Models and Modular Invariance}
in {  Recent Developments in Conformal Field Theories, }{Trieste
Conference, }{1989, S. Randjbar-Daemi}{ E. Sezgin and J.-B. Zuber eds.}
{World Scientific 1990); P. Di Francesco,  \ijmpa{7}{1992}{407-500}.
%{\it Integrable Lattice Models , Graphs and Modular Invariant
%Conformal Field Theories}} 
%
\bibitem{PZ2} V.B. Petkova and J.-B.  Zuber, \npb{463}{1996}{161-193}
[{\tt hep-th/9510175}];
%{ From CFT to Graphs}
%\bibitem{PZ3} 
%V.B. Petkova and J.-B.  Zuber, 
{\sl Conformal Field Theory and Graphs} [\hepth{ 9701103}].
%%
\bibitem{Ru} I. Runkel, \npb{579}{2000}{561-589} [\hepth{9908046}].
%{ \it Structure constants for the D-series Virasoro minimal models}
%
\bibitem{PSS} G. Pradisi, A. Sagnotti and Ya.S. Stanev, 
\plb{381}{1996}{97-104}
[\hepth{9603097}].
%{Completeness conditions for boundary operators in 2D conformal field theory}
%


}
\end{thebibliography}

\end{document}


hep-th0009219.extracted_bib
\begin{thebibliography}{99}

%%%%%%%%%%%%%



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%%%%%%%%%%%%%%%%%%%%%

\newcommand\jpha[3]   
                {{\it J. Phys.\ }{\bf A #1} (#2) #3} 
\newcommand{\mathQA}[1]{\href{http://xxx.lanl.gov/abs/math.QA/#1}{\tt
q-alg/#1}}
\newcommand{\mathOA}[1]{\href{http://xxx.lanl.gov/abs/math.OA/#1}{\tt
math.OA/#1}}


\bibitem{Ca89} J.L.\ Cardy, \npb{324}{1989}{581-596}.
%{Boundary Conditions, Fusion Rules and the Verlinde Formula}
%
\bibitem{BPPZ} R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B.
Zuber \plb{444}{1998}{163-166}
%[{\tt hep-th/9809097}];
[\hepth{9809097}];
%{On the classification of bulk and boundary conformal field theories};
\npb{570}{2000}{525-589}
%[{\tt hep-th/9908036}].
[\hepth{9908036}].
%
\bibitem{PZ4} V.B. Petkova and J.-B. Zuber,
{\sl The many faces of Ocneanu triangular cells}, in preparation.
%
\bibitem{MS1} G. Moore and N. Seiberg, \cmp{123}{1989}{177-254}.
%{Classical and Quantum Conformal Field Theory}
%; {\sl Lectures on RCFT, }{Physics, Geometry and Topology,
%}{Plenum Press, New York, 1990}.
%
\bibitem{KiRe} A.N. Kirillov and N.Yu. Reshetikhin,
% preprint LOMI E-9-88,
in {\it New developments of the theory of knots}, {T. Kohno
ed., }{World Sc. 1990.}{}
%
\bibitem{DF} Vl.S. Dotsenko and  V.A. Fateev, \npb{251[FS13]}{1985}{691-734}.
%{Four-point correlation functions and the operator algebra in 2d
%conformal invariant theories with central charge $c\le 1$}
%
\bibitem{PZ1} V.B. Petkova and J.-B. Zuber, \npb{438}{1995}{347-372}
%[{\tt hep-th/9410209}].
[\hepth{9410209}].
%{On structure constants of $sl(2)$ theories}
%
\bibitem{CIZ} A. Cappelli, C. Itzykson and J.-B. Zuber,
\npb{280}{1987}{445-465};
%{Modular Invariant Partition Functions in Two Dimensions}
 \cmp{113}{1987}{1-26}.\\
%{\it The A-D-E Classification of Minimal and $A_1^{(1)}$
%Conformal Invariant Theories};
A. Kato, \mpla{2}{1987}{585-600}. %
%{\it Classification of Modular Invariant Partition Functions in
%Two Dimensions }
%
\bibitem{Kos} I. Kostov \npb{300}{1988}{559-587}.
%{Free field presentation of the $A_n$ coset models on the torus }
%
\bibitem{DFZ1} P. Di Francesco and J.-B. Zuber,  \npb{338}{1990}{602-646}.
%{$SU(N)$ Lattice Integrable Models Associated with Graphs }
%
\bibitem{Pasq87} V. Pasquier, \jpha{20}{1987}{5707-5717};
%{Operator Content of the $ADE$ lattice models}
 and the preprint version of that article, reprinted in
V. Pasquier, {\sl Mod\`eles Exacts Invariants Conformes},
{\rm Th\`ese d'Etat, Orsay, 1988.}
%
\bibitem{Ga} T. Gannon, \cmp{161}{1994}{233-263}
%[{\tt hep-th/9212060}];
[\hepth{9212060}];
%{The Classification of Affine SU(3) Modular Invariant Partition Functions};
{\sl Ann. Inst. H. Poincar\'e}, Phys.Theor. {\bf 65} (1996) 15-56,
[\hepth{9404185}].
%{\it The classification of SU(3) modular invariant revisited}
%
%
\bibitem{Ocn} A. Ocneanu, 
%{Paths on Coxeter Diagrams: From Platonic Solids 
%and Singularities to Minimal Models and Subactors }
%{Low Dimensional Topology, Statistical Mechanics and Quantum Field Theory}
in {\sl Lectures on Operator Theory, }
 {Fields Institute, Waterloo, Ontario, April 26--30, 1995, }{(Notes taken by
S. Goto)} 
{Fields Institute Monographies, AMS 1999,}{ Rajarama Bhat et al, eds.}
%
\bibitem{Ocn00} A. Ocneanu, {}{Lectures at Bariloche Summer School, }
{Argentina, Jan 2000}.
%
\bibitem{BEK} J. B\"ockenhauer , D. E. Evans and Y. Kawahigashi, 
\cmp{208}{1999}{429-487}
[\mathOA{9904109}];
%{On $\alpha$-induction, chiral generators and modular invariants
%for subfactors} 
%\bibitem{BEK2} J. B\"ockenhauer, D. E. Evans and Y. Kawahigashi
\cmp{210}{2000}{733-784}
[\mathOA{9907149}].
%{Chiral structure of modular invariants for subfactors}
%
\bibitem{BE} J. B\"ockenhauer and  D. E. Evans,
%\cmp{200}{1999}{57-103}, [ \hepth{9805023}].
%Modular invariants, graphs and $\alpha$-induction for nets of subfactors};
\cmp{205}{1999}{183-228}
[\hepth{9812110}].
%
\bibitem{BoSz} G. B\"ohm and K. Szlachanyi, \lmp{200}{1996}{437-456}
[\qalg{9509008}]. %{A coassociative $C^*$- quantum group with
%non-integral dimensions} 
%
\bibitem{MR} G. Moore and N.Yu. Reshetikhin, \npb{328}{1989}{557-574}.
%
\bibitem{MSch} G. Mack and V. Schomerus, \npb{370}{1992}{185-230}.
%
\bibitem{CL} J.L. Cardy and D.C. Lewellen,  \plb{259}{1991}
{274-278};
D.C. Lewellen,  \npb{372}{1992}{654-682}.
%
\bibitem{Pa1} V. Pasquier, \npb{285}{1987}{162--172}.
%{Two-Dimensional Critical Systems Labelled by Dynkin Diagrams}
%
\bibitem{DFZ2} P. Di Francesco and J.-B.  Zuber, 
%{$SU(N)$ Lattice Integrable Models and Modular Invariance}
in {  Recent Developments in Conformal Field Theories, }{Trieste
Conference, }{1989, S. Randjbar-Daemi}{ E. Sezgin and J.-B. Zuber eds.}
{World Scientific 1990); P. Di Francesco,  \ijmpa{7}{1992}{407-500}.
%{\it Integrable Lattice Models , Graphs and Modular Invariant
%Conformal Field Theories}} 
%
\bibitem{PZ2} V.B. Petkova and J.-B.  Zuber, \npb{463}{1996}{161-193}
[{\tt hep-th/9510175}];
%{ From CFT to Graphs}
%\bibitem{PZ3} 
%V.B. Petkova and J.-B.  Zuber, 
{\sl Conformal Field Theory and Graphs} [\hepth{ 9701103}].
%%
\bibitem{Ru} I. Runkel, \npb{579}{2000}{561-589} [\hepth{9908046}].
%{ \it Structure constants for the D-series Virasoro minimal models}
%
\bibitem{PSS} G. Pradisi, A. Sagnotti and Ya.S. Stanev, 
\plb{381}{1996}{97-104}
[\hepth{9603097}].
%{Completeness conditions for boundary operators in 2D conformal field theory}
%


}
\end{thebibliography}