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Conformal quantum field theory, subfactors, local extensions.
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\documentclass[12pt]{article}
\usepackage{graphics}
\usepackage{amssymb}
\author{{\sc K.-H. Rehren} \\
Institut f\"ur Theoretische Physik, Universit\"at G\"ottingen
(Germany)}
\title{\vskip-10mm \bf Chiral Observables and Modular Invariants}
%
\def\sig{{\sigma}} \def\eps{\varepsilon} \def\a{\alpha} \def\b{\beta}
\def\g{\gamma} \def\iota{\imath}
\def\sig{\sigma} \def\lam{\lambda}
\def\inv{{^{-1}}} \def\comp{{\scriptstyle\circ}}
\def\arr{\rightarrow} \def\max{^{\rm max}} \def\ext{^{\rm ext}}
\def\stat{^{\rm stat}}
\def\Supp{\hbox{\rm Supp}} \def\Spec{\hbox{\rm Spec}}
\def\frac#1#2{{#1\over#2}} \def\pair#1#2{{\langle#1,#2\rangle}}
\def\eins{\hbox{\rm 1}\mskip-4.4mu\hbox{\rm l}}
\def\Ad{\hbox{\rm Ad}\,} \def\id{{\rm id}}
\def\ZZ{\mathbb{Z}} \def\CC{\mathbb{C}} \def\RR{\mathbb{R}}
\def\NN{\mathbb{N}} \def\MM{\mathbb{M}} \def\CM{\widetilde \MM}
\def\qed{\hspace*{\fill}$\square$} \def\SL{\hbox{SL}} \def\PSL{\hbox{PSL}}
\pagestyle{headings} \setcounter{page}{1}
\parskip2mm plus.4mm minus.5mm \parindent0mm
\textheight223mm \textwidth151mm \topmargin3mm
\evensidemargin2mm \oddsidemargin2mm
\begin{document}
\renewcommand{\today}{}
\maketitle
{\bf Abstract:} {\small Various definitions of chiral observables in a
given M\"obius covariant two-dimensional (2D) theory are shown to be
equivalent. Their representation theory in the vacuum Hilbert space
of the 2D theory is studied. It shares the general characteristics
of modular invariant partition functions, although $\SL(2,\ZZ)$
transformation properties are not assumed. First steps towards
classification are made. }
\vskip8mm
\section{Introduction}
The program of classification of modular invariant partition functions
in 2D conformal quantum field theory (see below for more details) has
seen steady progress since the original A-D-E classification for SU(2)
theories \cite{CIZ}. Apart from explicit classifications for new models
\cite{G}, classification theorems have been established for the
general case \cite{MS,BE}. Yet, the feeling persists that the full
depth of the problem has not yet been sounded.
It is the intention of the present note to show that general
classification theorems of a very similar nature can be derived in a
setting which does not refer to modular transformations of Gibbs
states at all. Our statements are on the decomposition (described by
a ``coupling matrix'') of the vacuum representation of a conformal 2D
quantum field theory upon restriction to its chiral observables. They
can be considered with a different perspective as statements on the
possible 2D extensions of given left and right chiral algebras.
Our mathematical tool is the structure theory of subfactors applied to
the inclusion of local algebras of chiral observables into local algebras
of 2D observables.
Note that a modular invariant partition function is also described by
a coupling matrix which is usually also interpreted as a chiral
decomposition of a 2D vacuum representation. But the classification method
based on arithmetic properties of the representation matrices $S$ and
$T$ of the $\SL(2,\ZZ)$ generators is entirely different and does not
rely on this interpretation. In fact, there seem to be exotic (accidental?)
modular invariants which do not derive from a 2D theory \cite[III]{BE}.
In contrast to the modular invariants program, we make only rather
general structural assumptions on the theory under consideration. We
put the emphasis on the local structure \cite{VK}, rather than the
accidental Lie algebra structure of chiral observables. Thus we avoid
the, somewhat artificial, restriction to chiral algebras which are
related to affine Lie algebras because these are the only ones for
which Gibbs functionals ${\rm Tr}_\pi\,e^{-\b L_0}$ (``characters'')
are known \cite{KP}.
Likewise, the problem that for most $W$-algebras it is not clear on
which suitable set of ``zero mode quantum numbers'' for chiral Gibbs
functionals the modular group should act, does not pose itself in
our approach.
Furthermore, we do not {\em assume} that the left and right chiral
observables are isomorphic, nor that they have isomorphic fusion of
their superselection sectors. Instead, we shall {\em derive} that the
maximal (see below) chiral observables automatically possess
sectors with identical fusion rules.
To be sure, it is not our intention to depreciate the modular point of
view at all. On the contrary, the $\SL(2,\ZZ)$ symmetry between high and low
temperature Gibbs states is one of the most fascinating features of
chiral models which calls for a sound physical understanding. Indeed,
there are general arguments, with reasonable assumptions, in favour of
a modular transformation law which generalizes the one for affine Lie
algebras \cite{KP} as conjectured in \cite{V}. E.g., Cardy \cite{C}
argues with transfer matrix methods and invariance under global
resummations in lattice models before the continuum limit is taken,
and Nahm \cite{N} exploits the operator product algebra of the Schwinger
functions to show that Gibbs states transform into Gibbs states.
None of these, however, provides a completely satisfactory
explanation in terms of the real time local quantum field theory.
On the other hand, the modular group $\SL(2,\ZZ)$ plays a fundamental
role even without any Gibbs functionals to act on by a modular transformation
of the temperature. Namely, the general theory of superselection sectors
collects monodromy data of braid group statistics in numerical
matrices $S\stat$ and $T\stat$, and as a ``maximality'' feature
of braid group statistics, these matrices represent
the modular group \cite{Pal,FG,MM}. In models where both concepts are
defined, one has $S=S\stat$ and $T=T\stat$.
E.g., the Kac-Peterson modular matrices \cite{KP} for affine Lie
algebras can be computed from the statistics of the representations
with positive energy of associated local current algebras.
Furthermore, the matrix entries of $S\stat$ were found \cite[II]{FRS}
to describe the spectrum of the central observables naturally associated
with the nontrivial topology of the space $S^1$. These discoveries are
general structure theorems from local quantum field theory and never
refer to Gibbs functionals (and hardly to conformal invariance).
They show also, however, that a degeneracy ($S\stat$ being not
invertible) can -- and in higher dimensions must -- occur which
obstructs the existence of an $\SL(2,\ZZ)$ representation.
(Algebraic conditions for non-degeneracy are given in \cite{KLM}.)
Thus, even if $\SL(2,\ZZ)$ does not act on chiral characters, it is
likely to be around, with various caveats as in the
discussion above, as a consequence of fundamentals of local quantum
field theory, and an interpretation in terms of Gibbs functionals
would be highly desirable. This issue will not be addressed here.
In the classification program for modular invariant 2D partition
functions, it is assumed that certain chiral observables
$A_L\simeq A_R$ are a priori given along with a collection of
representations (sectors) described by their chiral characters (Gibbs
functionals for the conformal Hamiltonian $L_0$ and suitable other
quantum numbers such as Cartan charges for current algebras).
These characters transform linearly under the group
$\SL(2,\ZZ)$ which is essentially generated by the imaginary unit shift
($T$) and the inversion ($S$) of the inverse temperature parameter
$\b/2\pi$. One then looks for bilinear combinations of chiral
characters with positive integer coefficients $Z_{l,r}$ (the coupling
matrix) which are invariant under the simultaneous $\SL(2,\ZZ)$
transformations for both chiral factors (that is, $Z$ commutes with
$S$ and $T$). The resulting modular invariant partition functions are
considered as Gibbs functionals for two-dimensional energy and
momentum operators in a representation of a 2D conformally invariant
quantum field theory. The latter contains the chiral observables along with
additional local 2D fields which are nonlocal in each light-cone
coordinate separately. In this interpretation, the entries of the
coupling matrix $Z$ clearly are the multiplicities of the sectors of
the chiral algebras within the representation space of the 2D theory.
E.g., one usually imposes the constraint $Z_{0,0}= 1$ on the coupling
matrix which ensures this representation to contain a unique vacuum vector.
One of the most important general classification statements \cite{MS}
asserts that every solution can be turned into a permutation matrix induced
by an ``automorphism of the fusion rules'' with respect to some ``suitably
extended algebra of chiral observables'' $A\ext_L\simeq A\ext_R$.
Furthermore, it was found \cite{BE} that the non-vanishing diagonal
entries of the coupling matrix $Z$ (with respect to the initially given chiral
observables) can be characterized in terms of structure data which refer to
the chiral extension $A\subset A\ext$ only. In the case of SU(2),
these two statements yield the A-D-E classification of \cite{CIZ}.
In this article, we endeavour a somewhat opposite program. We assume a
local 2D conformally invariant quantum field theory, denoted by $B$,
to be given in its vacuum representation $\pi^0$ on a Hilbert space $H$.
Within this theory we identify chiral observables, denoted by $A\max_L$
and $A\max_R$, and show that these are the respective relative commutants
of any initially given chiral observables $A_R$ and $A_L$ within the same
2D theory (Corollary 2.7). We then study the superselection sectors of
the maximal chiral observables which are contained in $H$, that is,
the branching of the irreducible representation $\pi^0$ upon restriction
to the subalgebra $A\max_L \otimes A\max_R$. We show that the coupling
matrix for the chiral observables $A\max$ is described by an isomorphism
between the left and right chiral fusion rules (Corollary 3.5), which
as a side result implies that $A\max$ coincide with $A\ext$ in the
modular classification statement (Lemma 3.4).
We just use the laws controlling local extensions of local algebras,
as established in \cite{LR}. The crucial point is the fact that the
same coupling matrix which describes the vacuum branching (or the
2D partition function), at the same time describes a distinguished DHR
representation of the chiral observables, and an endomorphism of a von
Neumann algebra of the form $A_L\otimes A_R$ canonically associated with a
subfactor $A_L\otimes A_R\subset B$. The constraints on the coupling
matrix arise by the latter endomorphism both being canonical and respecting
the tensor product (these notions are explained in Sect.\ 3).
Unlike locality of the chiral observables, locality of the 2D
net is only implicitly exploited and does not yet enter our (outline
of the) classification itself. It is well known that left and right
chiral sectors (charged fields) cannot be freely composed to yield
local 2D fields \cite{EA,Spt,MS}, and a general algebraic condition in
terms of a statistics operator was given in \cite{LR}. The
incorporation of this condition into our present scheme is still awaiting.
As far as these constraints are concerned, very similar arguments also
apply to ``coset models'' in which a tensor product of two commuting
subtheories is embedded within a given {\em chiral} theory.
Therefore, the same constraints on the coupling matrix
also arise for the branching of the vacuum sector of the ambient
theory upon restriction to the pair of subtheories.
The paper is organized as follows. Section 2 sets the physical stage
with emphasis on the equivalence of various possible definitions
of the chiral observables. In Section 3 the decomposition of the 2D
vacuum representation upon restriction to the chiral observables is
analyzed in the light of the general theory described in \cite{LR}.
The central result is a generalization of the ``automorphism of the
fusion rules'' theorem \cite{MS}. Section 4 discusses the (first)
implications for the classification problem.
The central argument in Section 3 is in fact a theorem on the sector
decomposition of the canonical endomorphism of a von Neumann subfactor.
This theorem, and the associated notion of a normal canonical tensor
product subfactor, is of its own mathematical interest \cite{SF} and
constitutes the common link between various problems in quantum field
theory, such as chiral observables in 2D, and coset models \cite{X2}
and Jones-Wassermann subfactors \cite{X1,KLM} in chiral conformal
quantum field theory. Its mathematical essence seems to be most
appropriately formulated in terms of C* tensor categories.
It furthermore reveals a connection to asymptotic subfactors \cite{O}
and quantum doubles \cite{KLM}. This observation may support the
expected role of quantum double symmetry in 2D conformal quantum
field theory and coset models.
\section{Chiral observables}
We start with the discussion of various alternatives to define chiral
observables within a conformally invariant 2D theory. The reader
mainly interested in modular invariants is invited to skip this
section, and take its results referred to in Sect.\ 3 for granted.
We adopt the algebraic approach to quantum field theory in which the
local algebras are considered rather than the local (Wightman) fields which
possibly generate them. The underlying picture \cite{LQP} is that the
{\em net of algebras}, i.e., the complete collection of inclusion and
intersection relations between algebras associated with smaller and
larger space-time regions, is sufficient in principle to reconstruct
the full physical content of the theory. Specifications of the model,
therefore, have to be formulated as properties of the net of local algebras.
A two-dimensional local conformal quantum field theory is defined on a
covering manifold $\CM$ of Minkowski space-time $\MM=\RR^{(1,1)}$.
This manifold is obtained as follows \cite{LM,BGL}. One first considers
Minkowski space-time as the Cartesian product $\RR \times \RR$ of its
two chiral light-cone directions. On each light-cone,
the M\"obius group $\PSL(2,\RR)$ acts by the rational transformations
$x\mapsto\frac{ax+b}{cx+d}$, thus enforcing the compactification of
$\RR$ to $S^1$ by addition of the point $\infty=-\infty$.
In the quantum field theory, the chiral M\"obius groups are only
projectively represented, leading to a covering of $S^1$ (in which
$\RR$ will be henceforth identified with the interval $(0,2\pi)$).
The covering manifold $\CM$ is the Cartesian product of the coverings
of the two chiral $S^1$, quotiented by the identification
$(x_L,x_R)= (x_L+2\pi,x_R-2\pi)$. Each subset $(a,a+2\pi)\times(b,b+2\pi)$
represents one copy of Minkowski space-time $\MM$ within $\CM$.
The covering manifold
$\CM$ possesses a global causal structure such that the causal complement
of a double cone $O=(a,b)\times(c,d)$\footnote{It is always understood
that $02$, say.
Splitting $f_R$ into two pieces $f_R^\pm$ with supports in $(-2R,a)$ and in
$(-a,2R)$ respectively, yields two Weyl operators $\exp itj(f_R^-)$
and $\exp itj(-f_R^+)$ localized in overlapping left and right halfspaces
whose weak limits as $R\to\infty$ should coincide in the vacuum
representation, and differ in a charged representation by a factor
$\exp itQ$. Nontriviality of these weak limits would invalidate our
conclusion in the proof of (v). A calculation, however, shows that,
due to scale invariance, the cutoff within the fixed interval $(-a,a)$
in comparison to the increase in $R$ behaves like a cutoff in a scaled
interval $(-a/R,a/R)$, and produces an ultraviolet singularity which
causes the weak limits of interest to be zero. Since this ultraviolet
behaviour is a ``universal'' effect of scale invariance, we believe
that the same mechanism protects the validity of our conclusion also
in general models. In any case, (v) will not be needed for the
purposes of this paper.
Since we consider the assumption of the generating property for the
chiral observables as no serious restriction, we reformulate the
statements with this assumption as a default.
{\bf 2.7.\ Corollary: \rm (i) \sl Assume the generating property for some
nets $A_L(I)$ and $A_R(J)$ of subalgebras of $B(I\times J)$ which are
invariant under the respective opposite M\"obius group. Then
$$A\max_L(I)=\bigcap_J B(I\times J)=A_R(J)'\cap B(I\times J),$$
and similarly for $A\max_R$. In particular, the left and right maximal
chiral observables are each other's mutual relative commutants in $B$.
\rm (ii) \sl If the net $B$ is Haag dual, then $A\max_L$ and $A\max_R$
are Haag dual, and
$$A\max_L(I_1)=B(I_2\times J)'\cap B(I\times J)$$
where $I_1$, $I_2$ arise from the interval $I$ by removal of an
interior point, and $J$ is an arbitrary interval.
The corresponding statement holds for $A\max_R$.}
(Again, the assertion of Haag duality for the chiral observables has to
be taken with a little caution.)
We conclude this section with a study of the joint position of the
subalgebras of left and right chiral observables within $B(O)$. We have
{\bf 2.8.\ Proposition: \sl In the vacuum representation of $B$, the left
and right chiral observables are in a tensor product position, i.e.,}
$$A\max_L(I)\vee A\max_R(J)\simeq A\max_L(I)\otimes A\max_R(J).$$
{\it Proof:} The statement follows, by Tomita-Takesaki modular theory
\cite{TT}, from the existence of the conditional expectations $\eps_L$ and
$\eps_R$, cf.\ Lemma 2.2. We want to give a less abstract argument.
Since left and right chiral observables mutually commute,
it is sufficient to consider products $a_La_R$ where $a_L\in A\max_L(I)$
and $a_R\in A\max_R(J)$. Since the vacuum state $\omega$ is conformally
invariant, and since the chiral observables transform under the respective
chiral M\"obius groups only, we have
\begin{eqnarray*} \omega(a_La_R)=\omega(\a_{g_L\times g_R}
(a_La_R))= \omega(\a_{g_L}(a_L)\a_{g_R}(a_R)).\end{eqnarray*}
For suitable elements $g_L$ and $g_R$, the localization of the
transformed observables tends to space-like infinite separation, hence
the cluster property of the vacuum state applies and entails
$$\omega(a_La_R)=\omega(a_L)\omega(a_R).$$
The factorization of the (normal) vacuum state implies the tensor product
factorization of the corresponding algebras. \qed
\section{Representation theory}
A subtheory $A$ of a given theory $B$ is described by a net of subalgebras
(subfactors) $A(O)\subset B(O)$. Conversely, $B$ may be considered as
a (local) extension of a given theory $A$. In the present paper,
$A$ is a net of left and right chiral
observables\footnote{Henceforth, the notation $O=I\times J$
will be understood.}
$O\mapsto A(O)=A_L(I)\otimes A_R(J)$, contained in a two-dimensional
net $O\mapsto B(O)$.
A general analysis of the representation theory in this situation
was initiated in \cite{LR}. As a prerequisite it was required that,
in generalization of an unbroken global gauge symmetry, there is a
consistent family of (normal, faithful) conditional expectations
$\eps_O:B(O)\to A(O)$ which commute with space-time symmetries and
preserve the vacuum state.
In our situation at hand, these expectations are provided by
Takesaki's theorem \cite{TT}, thanks to the fact that Tomita's modular
group for conformal double cone algebras is a subgroup of
$\widetilde G_L\times \widetilde G_R$ and consequently preserves any
M\"obius covariant subtheory of the form $A_L\otimes A_R$. As in
Sect.\ 2, they are coherently implemented by the projection $E_{LR}$ onto
the closure of the
subspace $A_L(I)A_R(J)\Omega$ (not depending on $I\times J$), which
commutes with M\"obius transformations and preserves the vacuum state.
Actually, for the analysis in \cite{LR} nets have to be {\em directed}.
We must therefore pass to the 2D and chiral theories on Minkowski
space $\MM$ and the light-cone axes $\RR$, respectively. As is common
practice, we denote the quasilocal C* algebra generated by a directed
net of von Neumann algebras (say $A(O)$) by the same symbol (say $A$)
as the net itself. We also denote the vacuum representations of $A$
and of $B$ by $\pi_0$ and $\pi^0$, respectively.
In the algebraic approach to quantum field theory, positive energy
representations are conveniently described in terms of DHR
endomorphisms \cite{LQP}, provided Haag duality holds. But the
restriction of $\pi^0$ to the subtheory $A$ is always given by a DHR
endomorphism $\rho$ of $A$
$$\pi^0\vert_A\simeq\pi_0\comp\rho$$
even without assuming Haag duality \cite{LR}. Moreover, $\rho$ is of the
``canonical'' form $\rho=\bar\iota\comp\iota$. Here $\iota:A\to B$
is the embedding homomorphism and $\bar\iota:B\to A$ is a conjugate
homomorphism to $\iota$ in the sense \cite{Dim} that there exist isometric
intertwiners $w\in A, w:\id_A\to\bar\iota\comp\iota\equiv\rho$ and
$v\in B, v:\id_B\to\iota\comp\bar\iota\equiv\gamma$ with
$w^*v=w^*\g(v)=\lambda\inv\cdot\eins$. The number $\lambda \geq 1$ is the
(statistical) dimension of $\rho$ and coincides with the index of the
local subfactor $A(O)\subset B(O)$ which is independent of $O$.
(We assume this index, and hence the dimensions of $\rho$ and all its
subsectors, to be finite throughout.)
The construction given in \cite{LR} starts off from a canonical
endomorphism \cite{L} $\g_O$ of the local von Neumann algebra
$B(O)$ for any fixed double cone $O$ into its subfactor $A(O)$.
$\g_O$ extends to a canonical endomorphism $\g$ of the quasilocal algebra
$B$ into $A$ in such a way that on any $B(\hat O)$, $\hat O\supset O$, it
yields a canonical endomorphism of $B(\hat O)$ into $A(\hat O)$, and
consequently the restriction of $\rho=\g\vert_A$ to $A(\hat O)$ is the
corresponding dual canonical endomorphism. It was shown that $\rho$ is
a DHR endomorphism localized in the fixed double cone $O$, and that
$w\in A(O)$ and $v\in B(O)$ are local operators.
In the present case, $A$ being a tensor product $A_L\otimes A_R$ of C*
algebras, any irreducible representation is also a C* tensor product. As
pointed out by R. Longo, there is a theoretical possibility (in case
the chiral representations are not ``type I'', cf.\ \cite{KLM}), that
the C* tensor products are not spatial. In a large class of models,
including current algebras, this possibility can be ruled out
\cite[Lemma 12]{KLM}, however, and it can presumably never arise when
the statistical dimension is finite. Thus we may assume that the
corresponding subspaces of $H$ are also tensor products.
Let therefore the irreducible decomposition of the restricted vacuum
representation into chiral sectors be given by
$$\pi^0\vert_{A_L\otimes A_R} \simeq \bigoplus_{l,r} Z_{l,r}\;
\pi^L_{l}\otimes \pi^R_{r}$$
with a (possibly rectangular) matrix of nonnegative integers $Z_{l,r}$
where $l,r$ run over the irreducible superselection sectors of the
left and right chiral observables contained in $H$. Equivalently, the
corresponding DHR endomorphism $\rho$ decomposes as
$$\rho\simeq\bigoplus_{l,r} Z_{l,r}\;\rho^L_{l}\otimes\rho^R_{r}$$
with irreducible chiral DHR endomorphisms $\rho^L_{l}$ and $\rho^R_{r}$,
and with the same matrix $Z$. We call $Z$ the {\bf coupling matrix},
and we reserve the labels $l=0$ and $r=0$ for the respective vacuum sectors,
$\rho^L_{0}\simeq\id_{A_L}\equiv\id_L$ and
$\rho^R_{0}\simeq\id_{A_R}\equiv\id_R$.
Making contact with modular invariants, it should be clear that
the coupling matrix also enters the decomposition of the vacuum
partition function of a 2D local theory
$$\hbox{Tr}_{\pi^0} \;e^{-\beta (L^L_0+L^R_0)} =
\sum_{l,r} Z_{l,r}\;\hbox{Tr}_{\pi^L_l} \;e^{-\beta L^L_0}\;
\hbox{Tr}_{\pi^R_r} \;e^{-\beta L^R_0} $$
into chiral characters $\chi_\pi=\hbox{Tr}_{\pi}
\;e^{-\beta L_0}$ of the representations of the chiral observables.
A similar algebraic situation with a tensor product of two nets of
observables embedded into another net also arises in
coset models \cite{X2} in chiral quantum field theory.
These models are given by a net of chiral observables $B(I)$ and a
proper subnet $A(I)$ (e.g., the current algebras associated with a
compact Lie group $G$ and a subgroup $H$). The coset theory is defined
as the net of relative commutants $C(I):=A(I)'\cap B(I)$.
Unless the pair of groups gives rise to a conformal inclusion (in
which case $C(I)$ is trivial), the net
$C$ possesses a stress-energy tensor of its own which
commutes with the stress-energy tensor of $A$. An argument similar
as in Proposition 2.8, making use of the two commuting M\"obius groups
for $A$ and $C$, yields the tensor product position of $A$ and $C$
within $B$. Again, the branching of the vacuum sector of $B$ is
described by a coupling matrix, and our results below can be easily
adapted to coset models.
We are going to study the branching of the vacuum representation
$\pi^0\vert_A$ in terms of the endomorphism $\rho$. It turns
out convenient to do this in a framework of endomorphisms of von
Neumann algebras. For this purpose we use the fact that $\rho$ as a DHR
endomorphism of the quasilocal algebra $A$ has the same decomposition
into irreducibles as its restriction $\rho_O=\rho\vert_{A(O)}$ as a
(dual canonical) endomorphism of a local von Neumann algebra. This
statement is standard if one assumes Haag duality and strong additivity.
But it has also been established without these assumptions in the
chiral case, making use of conformal symmetry and essential duality
instead, provided the statistical dimension is finite \cite{GL}. The
latter argument carries over without difficulty to the 2D case.
We just state this result without repeating its proof.
{\bf 3.1.\ Lemma: \sl Let $A$ be a local net on $\MM$ or $\RR$. Assume
either that $A$ is the restriction of a conformal net on $\CM$
resp.\ $S^1$, or that $A$ satisfies Haag duality and
strong additivity. Let $\sigma$, $\tau$ be two DHR endomorphisms
(in the conformal case: with finite statistical dimension),
localized in some double cone or interval $O$, and $\sigma_O$, $\tau_O$ their restrictions to
$A(O)$. Then the intertwiner spaces $(\sigma,\tau)$ and
$(\sigma_O,\tau_O)$ coincide. In particular, $\sigma$ and $\sigma_O$
have the same decomposition into irreducibles.}
Since our nets $B$ and $A_L$, $A_R$ are conformal, the Lemma applies
to all DHR endomorphisms with finite dimension. It follows that the
decomposition
$$\rho_O\simeq\bigoplus_{l,r} Z_{l,r}\;\rho^L_{l}\otimes\rho^R_{r}$$
of the dual canonical endomorphism for the local subfactor
$A_L(I)\otimes A_R(J)\subset B(O)$ is again described by the same
coupling matrix $Z$, where now $\rho^L_l$ and $\rho^R_r$ are
local restrictions of chiral DHR endomorphisms.
The crucial additional information here is that $\rho$ and hence the
dual canonical endomorphism $\rho_O$ respects the tensor product
structure $A(O)=A_L(I)\otimes A_R(J)$ in the sense that its
irreducible components are equivalent to tensor products of
irreducible endomorphisms of the factor algebras. We call a
von Neumann subfactor
$A\otimes C\subset B$ with this property a {\bf canonical tensor product
subfactor (CTPS)}\footnote{An elementary example of a
subfactor $A\otimes C\subset B$ which is {\em not}
canonical in this sense was suggested to me by H.J.~Borchers:
take $C=A$, and $B$ the crossed product of $A\otimes A$ by the
flip automorphism. Then the dual canonical endomorphism is
the direct sum of the identity and the flip. The latter does not
respect the tensor product.} with associated coupling matrix $Z$.
The subfactors $A_L(I)\otimes A_R(J)\subset B(O)$, or
$A(I)\otimes C(I)\subset B(I)$ for coset models, are examples of
CTPS's. Other examples in conformal quantum field theory are
Jones-Wassermann subfactors arising from partitions of $S^1$ into
four intervals \cite{X1,KLM}.
Since we assume the index to be finite, only finitely many sectors can
contribute which all must have finite dimension, hence the coupling matrix
is a finite matrix. Since we have assumed the defining representation
of $B$ to contain a unique vacuum vector, it follows that its
restriction to the chiral observables contains the joint vacuum
representation exactly once, hence $Z_{0,0} = 1$. This implies that
the multiplicity of $\id_L\otimes \id_R$ in $\rho$ is one, hence
the embedding $A_L\otimes A_R\subset B$ is irreducible (both for the
local von Neumann algebras and for the quasilocal C* algebras).
We summarize the discussion so far:
{\bf 3.2.\ Proposition: \sl The local subfactors $A_L(I)\otimes A_R(J)
\subset B(O)$ are irreducible canonical tensor product subfactors.
The irreducible sector decomposition of their dual canonical
endomorphisms is described by the same finite coupling matrix $Z$ as the
decomposition of the restricted vacuum representation
$\pi^0\vert_{A_L\otimes A_R}$ of $B$.}
We are going to study the constraints on $Z$ being the coupling matrix
of a canonical TPS.
These constraints are then read back as constraints on the representation
$\pi^0\vert_{A_L\otimes A_R}$ or on the 2D partition function.
In the sequel when we write $A_L\otimes A_R\subset B$, we have in
mind the local subfactor $A_L(I)\otimes A_R(J)\subset B(O)$, or with
suitable modifications $A(I)\otimes C(I)\subset B(I)$ in coset
models. But we are actually going to establish general statements on
coupling matrices of CTPS's without reference to quantum field theory.
We shall several times need ``Frobenius reciprocity'', cf.\ \cite{Dim},
which we recall in
{\bf 3.3.\ Lemma: \sl Let $A$, $B$, $C$ be unital C* or von Neumann
algebras and $\a:A\to B$, $\beta:B\to C$, $\g:A\to C$ unital
homomorphisms. Denote by $\langle\g,\a\b\rangle$ the dimension of
the linear space of intertwiners $t\in C$, $t:\g\to\a\b$. Then
$$\langle\bar\a\g,\b\rangle=\langle\g,\a\b\rangle=\langle\g\bar\b,\a\rangle$$
provided the conjugate homomorphisms $\bar\a:B\to A$ or
$\bar\b:C\to B$ exist.}
Here, as before, conjugates are defined in terms of a pair of
intertwiners \cite{Dim}, say $w:\id_A\to\bar\a\a$, $v:\id_B\to\a\bar\a$
which satisfy the relations $\a(w)^*v=\eins_B$, $\bar\a(v)^*w=\eins_A$.
For $X\subset B$ the relative commutant $X'\cap B$ is commonly denoted by
$X^c$. We have
{\bf 3.4.\ Lemma: \sl Let $A_L\otimes A_R\subset B$ be a CTPS with
finite index, and $Z_{l,r}$ its coupling matrix.
Then, $Z_{0,r}\neq 0$ implies $r=0$ if and only if
$\eins\otimes A_R=(A_L\otimes \eins)^c$.
The corresponding statement holds exchanging $A_L$ and $A_R$.}
{\it Proof:} We have to show that $Z_{0,r}\neq 0$ for some $r\neq 0$
(that is, $\rho^R_r\not\simeq\id_R$) if
and only if the inclusion $\eins\otimes A_R\subset (A_L\otimes \eins)^c$
is proper. Note that equality holds if and only if
$X:=(A_L\otimes\eins)\vee (A_L\otimes\eins)^c$ equals $A_L\otimes A_R$
(since $A_L$ is a factor).
Consider now the intermediate subfactor $A_L\otimes A_R\subset X\subset B$.
In terms of the inclusion maps
$\iota_1:A_L\otimes A_R\to X$ and $\iota_2:X\to B$, we have
$$\rho_1\equiv\bar\iota_1\iota_1\prec
\bar\iota_1\bar\iota_2\iota_2\iota_1=\bar\iota\iota\equiv\rho.$$
If $A_L\otimes A_R\subset X$ is proper, then $\iota_1$
is nontrivial and $\rho_1$ contains a nontrivial subsector
$\id_L\otimes\rho^R_{r}$ which is also a subsector of $\rho$ giving rise
to a nonvanishing matrix entry $Z_{0,r}$. Conversely, if
$Z_{0,r}\neq 0$, then $\id_L\otimes\rho^R_r\prec\rho$. By Frobenius
reciprocity (Lemma 3.3), $\iota\prec\iota\comp(\id_L\otimes\rho^R_r)$,
hence there is a nonvanishing intertwiner $\psi\in B$ which satisfies
$$\psi(a_L\otimes a_R)=(a_L\otimes \rho^R_{r}(a_R))\psi.$$
Putting $a_R=\eins$, this implies that $\psi\in (A_L\otimes\eins)^c$,
thus $\psi\in X$, and hence $\iota_1\prec\iota_1\comp(\id_L\otimes\rho^R_r)$.
Again invoking Frobenius reciprocity, $\id_L\otimes\rho^R_{r}\prec\rho_1$.
Thus $A_L\otimes A_R\subset X$ is proper. \qed
The Lemma allows us to characterize the maximal chiral observables by
a normality property of the local subfactors, see Corollary 3.5 below.
We recall that an inclusion $A\subset B$ is called normal if
$(A^c)^c=A$. In general, $A^{cc}\supset A$. It follows that
$(A^{cc})^c\subset A^c$ and $(A^c)^{cc}\supset A^c$, hence
$A^{ccc}=A^c$ which is obviously equivalent to the statement that a
relative commutant is always normal.
We call (with a slight abuse of terminology) a tensor product
subfactor $A\otimes C\subset B$ {\bf normal} if $A\otimes\eins$ and
$\eins\otimes C$ are each other's relative commutants in $B$.
Hence, the local subfactors of chiral observables within 2D conformal
quantum field theories, $A\max_L(I)\otimes A\max_R(J)\subset B(O)$
are examples of normal and canonical TPS's.
Also coset models give rise to local subfactors which are normal CTPS's.
Namely, one obtains normality by extending (if necessary) $A(I)$ by the
relative commutant of $C(I)$.
Normality of the local subfactors is characteristic for the maximal
chiral observables, and a criterium in terms of the coupling matrix is
given in
{\bf 3.5.\ Corollary: \sl The following are equivalent.
\rm (i) \sl $A_L=A\max_L$ and $A_R=A\max_R$.\\
\rm (ii) \sl The local subfactors $A_L(I)\otimes A_R(J)\subset B(O)$
are normal CTPS's.\\
\rm (iii) \sl The coupling matrix satisfies $Z_{0,r}=\delta_{0,r}$ and
$Z_{l,0}=\delta_{l,0}$.\\
\rm (iv) \sl The coupling matrix describes an isomorphism of the left and
right chiral fusion rules (in the sense of Theorem 3.6 below).}
{\it Proof:} (i) and (ii) are equivalent by Corollary 2.7. (ii) and (iii)
are equivalent by Lemma 3.4. (iii) and (iv) are equivalent by the
following Theorem. \qed
(The equivalence (i) $\Leftrightarrow$ (iii) could have been
argued already from Lemma 2.3.)
{\bf 3.6.\ Theorem: \sl Let $A_L\otimes A_R\subset B$ be a CTPS with
finite index, and $Z_{l,r}$ its coupling matrix, that is
$$\rho=\bar\iota\comp\iota \simeq\bigoplus_{l,r} Z_{l,r}\;\rho^L_{l}
\otimes\rho^R_{r}$$
where $\iota: A_L\otimes A_R\to B$ denote the inclusion
map and $\bar\iota$ its conjugate. If the coupling matrix satisfies
$$Z_{0,r}=\delta_{0,r}\quad {\rm and}\quad
Z_{l,0}=\delta_{l,0}$$
(that is, the CTPS is normal and irreducible), then \\
(1) $Z$ is a permutation matrix. It induces a bijection $\,\hat\cdot\,$
with inverse $\,\check\cdot\,$ between the systems of sectors
$\{\rho^L_{l}\}$ and $\{\rho^R_{r}\}$
contributing to the decomposition of $\rho$ such that
$$ Z_{l,r}=\delta_{\hat l,r}=\delta_{l,\check r}.$$
(2) Both systems of sectors $\{\rho^L_{l}\}$ and
$\{\rho^R_{r}\}$ are closed under conjugation and under
decomposition of products (fusion). They satisfy the same fusion rules
$$\rho^L_l\rho^L_k\simeq\bigoplus_m N_{lk}^m\;
\rho^L_m\qquad {\rm and}\qquad
\rho^R_r\rho^R_s\simeq\bigoplus_t \hat N_{rs}^t\; \rho^R_t$$
with
$ N_{lk}^m = \hat N_{\hat l\hat k}^{\hat m}$.
In particular, the bijection $\,\hat\cdot\,$ respects conjugation, and
the dimensions of the corresponding sectors coincide:
$$ d(\rho^R_{\hat l})=d(\rho^L_{l}).$$
(3) The homomorphisms $\iota^L_{l}:=\iota\comp(\rho^L_{l}\otimes\id_R):
A_L\otimes A_R\to B$ are irreducible and mutually inequivalent. The
same holds for $\iota^R_{r}:=\iota\comp(\id_L\otimes\rho^R_{r})$, and
$\iota^R_{r}\simeq\iota^L_{\check r}$. Moreover, }
\begin{eqnarray*}\iota\comp(\rho^L_{l}\otimes\rho^R_{r}) \simeq\bigoplus_{k}
N_{\check{\bar r}l}^{k}\; \iota^L_{k} \simeq\bigoplus_{s}
\hat N_{\hat{\bar l}r}^{s}\; \iota^R_{s}. \end{eqnarray*}
{\it Proof:} The proof adopts and extends methods taken from \cite{M}.
Let the index sets $\{l\}$ and $\{r\}$ label the irreducible sectors
$\rho^L_{l}$ of $A_L$ and $\rho^R_{r}$ of $A_R$, respectively, obtained
by closure under reduction of products of those sectors which contribute to
$\rho$. If among these there are any ``new'' sectors not already
contributing to $\rho$, we extend the coupling matrix $Z$ by zero
columns and rows, but we are eventually going to show that there are
no such new sectors.
Only finitely many columns and rows of $Z$ are non-zero. Since
$\rho=\bar\iota\comp\iota$ is self-conjugate, along with
$\rho^L_{l}\otimes\rho^R_{r}$ also its
conjugate must contribute with the same multiplicity, and hence
$Z_{l,r}=Z_{\bar l,\bar r}$. In particular, both systems $\{\rho^L_{l}\}$
and $\{\rho^R_{r}\}$ are closed under conjugation.
Let the homomorphisms $\iota^L_{l}:A_L\otimes A_R\to B$ be as in (3).
We compute
\begin{eqnarray*} \langle \iota^L_{l},\iota^L_{l'}\rangle
= \langle\iota\comp(\rho^L_{l}\otimes \id_R),
\iota\comp(\rho^L_{l'}\otimes\id_R)\rangle
= \langle \rho^L_{l}\otimes \id_R,
\bar\iota\comp\iota\comp(\rho^L_{l'}\otimes \id_R)\rangle = \qquad \\
= \sum_{k,s} Z_{k,s} \langle \rho^L_{l}\otimes
\id_R,\rho^L_{k}\rho^L_{l'}\otimes \rho^R_{s}\rangle
= \sum_{k,s} Z_{k,s} \langle \rho^L_{l},\rho^L_{k}\rho^L_{l'}\rangle
\langle\id_R,\rho^R_{s}\rangle . \end{eqnarray*}
To this sum contributes only $s=0$ since $\langle \id_R,\rho^R_{s}\rangle
=\delta_{s,0}$, and by the assumed properties of $Z$ also $k=0$ is the only
contribution. Hence
$$\langle \iota^L_{l},\iota^L_{l'}\rangle =
\langle \rho^L_{l},\rho^L_{l'}\rangle = \delta_{l,l'}.$$
Thus the homomorphisms $\iota^L_{l}$ are irreducible and mutually
inequivalent. The symmetric argument applies to $\iota^R_{r}$. Next we compute
\begin{eqnarray*} \langle \iota^L_{l},\iota^R_{\bar r}\rangle = \langle
\rho^L_{l}\otimes \id_R,\bar\iota\comp\iota\comp(\id_L\otimes \bar\rho^R_{r})
\rangle
=\sum_{k,s} Z_{k,s} \langle \rho^L_{l},\rho^L_{k}\rangle \langle
\id_R,\rho^R_{s}\bar\rho^R_{r}\rangle = Z_{l,r}\; . \end{eqnarray*}
As we have seen that both sets of homomorphisms $\{\iota^L_{l}\}$
and $\{\iota^R_{r}\}$ consist of mutually inequivalent irreducibles,
each $\iota^L_{l}$ can be
equivalent to at most one $\iota^R_{\bar r}$. Hence for fixed index $l$,
at most one entry $Z_{l,r}$ can be different from zero and must be
one. It follows also that no $\iota^L_{l}$ associated with a ``new'' sector
$\rho^L_{l}$ can be equivalent to any of the $\iota^R_{r}$, old or
new, and vice versa.
For the ``old'' sectors, we write
$$r=\hat l \quad\hbox{and}\quad l=\check r \qquad\hbox{iff}\quad
Z_{l,r}=1, \quad\hbox{that is, iff}\quad \iota^L_{l}\simeq\iota^R_{\bar r}.$$
That this assignment between old sectors is bijective follows from
transitivity of equivalence of sectors. Since we have already seen that $Z$
is conjugation invariant, this assignment respects conjugation, that is
$$\bar\rho^R_{\hat l}=\rho^R_{\bar{\hat l}}=\rho^R_{\hat{\bar l}}
\qquad {\rm etc.}$$
Next, we consider homomorphisms
$\iota_{l,r}:=\iota\comp(\rho^L_{l}\otimes\rho^R_{r}): A_L\otimes
A_R\to B$ and compute
\begin{eqnarray*} \iota_{l,r} =
\iota^R_{r}\comp(\rho^L_{l}\otimes \id_R)\simeq
\iota^L_{\check{\bar r}}\comp(\rho^L_{l}\otimes \id_R)=
\iota\comp(\bar\rho^L_{\check r}\rho^L_{l}\otimes \id_R) \simeq
\bigoplus_{k} N_{\check{\bar r}l}^{k}\; \iota^L_{k}.
\end{eqnarray*}
The symmetric argument produces also the decomposition
\begin{eqnarray*}\iota_{l,r}=\iota^L_{l}\comp(\id_L\otimes\rho^R_{r})
\simeq \iota\comp(\id_L\otimes\bar\rho^R_{\hat l}\rho^R_{r})
\simeq \bigoplus_{s} \hat N_{\hat{\bar l}r}^{s}\;\iota^R_{s}.
\end{eqnarray*}
In the first of these two decomposition formulae of the same object, no
``new'' label ${k}$ can appear, since we have seen that such a
term $\iota^L_{k}$ is not equivalent to any term $\iota^R_{s}$ in
the second decomposition formula, and vice versa. This shows that the
sets of sectors contributing to the coupling matrix are already closed
under reduction of products. Furthermore, comparison of the two
decomposition formulae shows equality of the multiplicities
$N_{\check{\bar r}l}^{k}$ and $\hat N_{\hat{\bar l}r}^{s}\equiv
\hat N_{\bar r\hat l}^{\bar s}$ if $\bar s=\hat k$. Hence the
bijection $\,\hat\cdot\,$ between the sectors induces an isomorphism
of the fusion rules.
Since finally the fusion rules of a finite system determine the
dimensions uniquely, also the equality of the dimensions follows. \qed
We have thus reproduced a result found previously in the
classification program for modular invariant partition functions with
heavy use of $\SL(2,\ZZ)$ machinery \cite{MS}, reducing every modular
invariant to an ``automorphism of the fusion rules'' for suitably
extended chiral observables. Our analysis is, however, much stronger
since its assumptions are much weaker. Furthermore, it implies that the
``suitably extended'' chiral observables are indeed the maximal chiral
observables defined in 2.1, and coincide with the relative
commutants of the initially given chiral observables (Corollary 2.7(i)).
Second, if possibly the maximal left and right chiral observables are not
isomorphic, then the result still implies an isomorphism of the
respective fusion rules. The corresponding statement is even more interesting
in the case of coset models where typically $A\subset B$ is a theory
with well-known fusion rules, while the coset theory $C=A^c$ is in
general a $W$-algebra whose superselection structure is a priori unknown.
The theorem establishes that the fusion rules of this $W$-algebra are
isomorphic to those of a local extension of the given theory $A$,
namely the relative commutant $A^{cc}$ of $C$, which is in turn
controllable in terms of the representations of $A$ itself. For coset
models based on current algebras, our result seems to be the algebraic
backbone of the modular reasoning as in \cite{SY}.
Finally, we emphasize that the sectors in Theorem 3.6 were never
referred to as being restrictions of DHR sectors. Neither was it
required that their fusion be abelian. The theorem is thus of a
quite more general nature than its specific application to conformal
quantum field theory as treated in this paper.
\section{Towards classification}
Modular invariant partition functions associated with affine Lie
algebras ($A_L \simeq A \simeq A_R$), as far as they have been
classified, exhibit a classification scheme which refers to certain
graphs and their exponents (eigenvalues of the square of the
adjacency matrix) \cite{CIZ,G}. An essential statement is on the
non-vanishing diagonal entries of the coupling matrix $Z$.
A rather general formulation can be found in \cite[II]{BE}.
It entails that $Z_{\lam,\lam}\neq 0$\footnote{In affine models the
DHR sectors of the initially given chiral observables are given in terms of
weights $\lam$ of semisimple Lie algebras. Throughout this section,
we adopt the labels $\lam$ for DHR sectors in order to make the
present generalizations more transparent.}
if and only if the DHR sector $\lam$ of $A$ belongs to a set of
``exponents'' associated with the chiral extensions $A \subset A\ext$.
The set of exponents is a subset of the sectors of $A$.
By modular invariance, the sectors of $A$ label at the
same time also the irreducible representations of their own fusion
algebra, the modular matrix $S$ playing the role of a ``generalized
Fourier transformation'' between the fusion algebra itself and its dual.
On the other hand, modular invariance of the partition function implies
that the coupling matrix coincides with its Fourier transform (up to a
conjugation). Hence, the above statement on the sector $\lam$ being an
exponent can as well be interpreted as a statement on the irreducible
representation $\lam$ of the fusion algebra and on non-vanishing
entries of the Fourier transformed coupling matrix. In the following,
we set out to
formulate a generalization of this version of the statement to the
more general situation we discussed in this paper (without parity
symmetry between left and right chiral algebras, and without
assumption of modular invariance).
Let $A_L\otimes A_R\subset A\max_L\otimes A\max_R\subset B$ denote some
initially given chiral observables embedded into a two-dimensional local
theory $B$ (satisfying the assumptions of section 2) along with their
maximal chiral extensions obtained by passing to the relative
commutants in $B$.
Let $W_L$ and $W_R$ denote the fusion algebras of all irreducible DHR
sectors $\lam_L$, $\lam_R$ of the initially given chiral observables (or
fusion subalgebras containing all sectors which contribute to the
coupling matrix $Z$). Let $W\max_L$ and $W\max_R$ denote the fusion
algebras of the irreducible sectors $\tau_L$, $\tau_R$ of the extended
(= maximal) chiral observables which contribute to the coupling
matrix (i.e., which are contained in the vacuum representation of $B$).
According to Theorem 3.6 and Corollary 3.5, the fusion algebras
$W\max_L$ and $W\max_R$ are isomorphic under the bijection
$\,\hat\cdot\,$. We use this bijection to identify $W\max_L$ with
$W\max_R$, so the coupling matrix with respect to $A\max$ becomes the
unit matrix $\eins$.
To be on safe grounds, we assume that $W_L$ and $W_R$ contain only finitely
many sectors, and that these have finite dimensions. This implies the
same for $W\max$, and ensures that all extensions have finite index.
Restriction and extension prescriptions between DHR sectors
of a theory $B$ and a subtheory $A$ were given in \cite{LR}, and
further analyzed in \cite{BE}. We are going to apply this theory to
the chiral extensions $A\max_L$ of $A_L$, and $A\max_R$ of $A_R$.
The restriction is just the restriction of
representations and coincides with the ``canonical'' prescription in
terms of the inclusion homomorphism $\iota$ and its conjugate, given by
$\tau\mapsto\sig_\tau=\bar\iota\comp\tau\comp\iota$.
It was named $\sig$-restriction in \cite{BE}. In the present situation,
$\sig$-restriction maps $W\max$ into $W$.\footnote{Here and in the
sequel, we often suppress the subscripts $L$ and $R$ when both
chiralities are understood.}
In contrast, the extension prescription $\lam\mapsto\a_\lam$ \cite{LR}
differs from the canonical induction
$\lam\mapsto\iota\comp\lam\comp\bar\iota$; it was named $\a$-induction
in \cite{BE} for distinction.
In particular, unlike canonical induction, $\a$-induction respects sector
composition, and the trivial sector of the subtheory extends to the
trivial sector of the extended theory. Furthermore, $\a$-extensions of
DHR sectors of the subtheory in general are not DHR but only
half-space localized (solitonic) sectors, due to a monodromy
obstruction \cite{LR}. Let $V_L$ and $V_R$ denote the, possibly
non-abelian, fusion algebras of all sectors (labelled $\beta$) generated by
reduction of products of $\a$-extended DHR sectors from $W_L$ and $W_R$.
In \cite{BE}, a reciprocity formula for $\a$-induction and
$\sig$-restriction was found:
$$\langle\a_\lam,\tau\rangle=\langle\lam,\sig_\tau\rangle$$
provided $\lam$ and $\tau$ are DHR sectors of the respective
theories. It entails that $\a_\lam$ and $\iota\comp\lam\comp\bar\iota$,
while otherwise different, contain the same DHR subsectors.
It also entails that, in the present setting, the fusion algebras $V$
contain the abelian subalgebras $W\max$.
Let $B_L$ and $B_R$ denote the rectangular ``branching matrices'', describing
chiral $\sig$-restriction, with non-negative integer entries
$\langle\lam,\sig_\tau\rangle$ which connect the irreducible DHR
sectors $\tau\in W\max$ with $\lam\in W$. Then the (in general
rectangular, $\dim W_L \times \dim W_R$) coupling matrix
with respect to the initially given chiral observables is
$$Z=B_LB_R^t,$$
that is, $Z_{\lam_L,\lam_R}\neq 0$ if and only if the sectors $\lam_L$ and
$\lam_R$ arise by restriction from a pair of sectors of the maximal
chiral observables which are identified by the bijection $\,\hat\cdot\,$ of
Theorem 3.6. This is just the ``block form'' of the coupling matrix
expected by restricting first $\pi^0_B$ to the maximal chiral
observables, and subsequently restricting the sectors so obtained to
the initially given chiral observables.
Each fusion algebra has a ``regular representation'' defined by
representing a sector by its matrix of fusion multiplicities with the other
sectors. $W$ and $W\max$ being abelian, all their irreducible representations
are one-dimensional and contribute with multiplicity one to the
regular representations. The values of the generators of the
fusion algebra in the irreducible representations provide ``character
tables'' which are non-degenerate square matrices. We denote the
one-dimensional representations of $W$ by $\phi\in\widehat W$, and
their character tables by $X$.
The character table defines a ``generalized Fourier transform'' between
any abelian fusion algebra and its representations. The Fourier transformed
coupling matrix is thus defined as
$$\widehat Z = (X_LB_L)(X_RB_R)^t.$$
Its matrix entries are the values of the restriction of the vacuum
sector of the 2D theory $B$, as a DHR sector of $A_L\otimes A_R$,
in the irreducible representations $\phi_L\otimes\phi_R$ of the tensor
product $W_L\otimes W_R$ of the chiral DHR fusion algebras. A priori,
the entries need not to be integers.
Let $\bar\phi\in\widehat W$ denote the conjugate representation of $\phi$.
Since the adjoint in the fusion algebra is given by sector conjugation,
we have $\phi(\bar\lam)=\overline{\phi(\lam)}=\bar\phi(\lam)$. This means
$\widehat C X = XC=\overline{X}$ where $C$ and $\widehat C$ are
the conjugation matrices for the DHR sectors of the initially given chiral
observables $A$ and for the representations of their fusion algebras
$W$, respectively. Furthermore, restriction respects sector conjugation,
hence $BC\max=CB$ where $C\max$ is the conjugation matrix for the
sectors $\tau\in W\max$ of the maximal chiral observables $A\max$.
Thus, since the branching matrices are real, we arrive at
$$\widehat Z\widehat C=(X_LB_L)(X_RB_R)^+\qquad\hbox{or equivalently}\qquad
\widehat Z=(X_LB_L)C(X_RB_R)^+.$$
It follows that a matrix entry of $\widehat Z\widehat C$ to be non-zero
requires that the corresponding complex row vectors of $X_LB_L$ and
$X_RB_R$ are not orthogonal, and a fortiori non-zero. If both the
chiral branching and the chiral fusion algebras are isomorphic, e.g.,
if the theory $B$ is parity symmetric, then a diagonal matrix entry of
$\widehat Z\widehat C$ vanishes if {\em and only if} the corresponding
row vector of $XB$ vanishes.
A modular (transformation) matrix $S$, if it exists, establishes a
natural identification between the generators of a fusion algebra and
its representations, and $X=S$. Since $S^2=C=\widehat C$,
modular $S$-invariance is the statement that the coupling matrix
$Z=SZS^*=\widehat Z\widehat C$ equals its own Fourier transform up to
a conjugation.
This remark implies that the Proposition 4.1 below
reduces to the above-mentioned statement on ``exponents'' in \cite{BE}
in the case with modular invariance.
We have first to adapt definitions made in \cite[II]{BE} to our more
general setting. We introduce certain subsets of $\widehat W$ which reflect
the structure of the chiral extensions.
For a given irreducible
DHR sector $\tau\in W\max$, we define the {\bf $\sig$-supports}
$\Supp_L(\tau)$ and $\Supp_R(\tau)$ as the subsets of those irreducible
representations of $W_L$ and $W_R$ which do not vanish on the
respective restrictions $\sig_\tau$ of $\tau$ to the initially given
left and right chiral observables, that is, those rows of $XB$ which have
non-zero entry in column $\tau$. The notion ``support'' is motivated by
considering the abelian fusion algebra $W$ as an algebra of functions
on the set $\widehat W$ of its one-dimensional representations.
Thus $\Supp(\tau)\subset\widehat W$ is indeed the support of the function
$\sig_\tau\in W$. (The $\sig$-supports were called $\rm Eig(\tau)$
in \cite{BE}.)
As shown in \cite{BE}, $\a$-induction of sectors induces a
homomorphism of fusion algebras $W\to V$. Composing this homomorphism
with the regular representation of $V$ yields another representation,
$\pi_\a$, of $W$. We define the {\bf $\a$-spectra} $\Spec_L$ and $\Spec_R$
as the subsets of those irreducible representations of $W_L$ and $W_R$
which are contained in the $\a$-induced representations $\pi^L_\a$ and
$\pi^R_\a$. (The $\a$-spectra were called $\rm Exp$
in \cite{BE} and are the ``exponents'' mentioned above.)
Now, by virtue of $\a$-$\sig$-reciprocity \cite{BE}, we are going to
derive
{\bf 4.1.\ Proposition: \rm (i) \sl A matrix entry of $\widehat Z\widehat C$
vanishes unless for some sector $\tau\in W\max$, both matrix indices
belong to the respective left and right $\sig$-supports $\Supp(\tau)$.
It also vanishes unless both matrix indices belong to the left and
right $\a$-spectra $\Spec$.
\rm (ii) \sl If (fusion and branching of) the left and right chiral
theories are
isomorphic, then a diagonal matrix entry of $\widehat Z\widehat C$ is
non-zero if and only if the corresponding representation of $W$
belongs to the union $\bigcup_\tau \Supp(\tau)$.}
In fact, there are many interesting cases when $\bigcup_\tau
\Supp(\tau)=\Spec$ (some of them being given below), so the last
statement can be phrased in terms of the $\a$-spectrum $\Spec$.
The Proposition is the desired generalization of the classification statement
\cite{CIZ,G,BE} for modular invariant partition functions.
(The second statement seems not to be
sensible with differing left and right chiral fusion and branching
matrices, since the product of two different row vectors can clearly
vanish without these vectors being zero.)
The Proposition makes assertions about the coupling matrix for the
initially given
chiral observables $A_L\otimes A_R$ embedded into the 2D theory $B$,
in terms of the chiral extensions $A\subset A\max$ to which
$\a$-induction and $\sig$-restriction pertain. Thus the 2D problem
is reduced to a chiral problem. An open issue remains, however, a
model-independent classification of possible $\a$-spectra, and hence
of 2D chiral extensions. The available classifications for affine
Lie and Virasoro algebras (``diagonal or automorphism, orbifold,
exceptional'' \cite{CIZ,G,BE}) refer to the chiral extensions being in turn
trivial, fixpoints under an abelian group, or conformal embeddings,
and are expected to be too coarse in the general case.
{\it Proof of the Proposition:} (i) The first statement
is obvious since by the
representation $\widehat Z\widehat C=(X_LB_L)(X_RB_R)^+$, every matrix
entry is the inner product of row vectors whose components are the
values of the functions $\sig_\tau$, $\tau\in W\max$, evaluated on the
respective left and right
one-dimensional representations. The inner product vanishes whenever
these representations do not belong to the respective $\sig$-supports.
The second statement is a consequence of the first in view of the
Lemma 4.2 below. \\
(ii) For isomorphic left and right chiral fusion and branching,
$X_LB_L=X_RB_R$,
diagonal matrix entries of $\widehat Z\widehat C$ are
norm squares of row vectors of $XB$ which vanish if {\it and only if} all
their entries vanish, hence if {\it and only if} the corresponding
representation of $W$ does not belong to any of the $\sig$-supports
$\Supp(\tau)$, $\tau\in W\max$. \qed
We have used
{\bf 4.2.\ Lemma: \sl $\bigcup_\tau \Supp(\tau)\subset \Spec$. }
{\it Proof:} The one-dimensional representations $\phi$ of an
abelian fusion algebra with generators $\lam$, considered as
vectors with entries $\phi(\lam)$, are pairwise orthogonal
\cite{K}. This property enables us to decide whether a representation
$\phi$ is contained in the $\a$-induced representation $\pi_\a(\lam)$ with
matrix entries $\langle\a_\lam\b_1,\b_2\rangle$, by contracting
the matrix-valued vector $(\pi_\a(\lam))_\lam$ with the vector
$(\overline{\phi(\lam)})_\lam$. Thus $\phi$ belongs to the $\a$-spectrum
$\Spec$ if and only if the resulting matrix
$$(\overline\phi\cdot\pi_\a)_{\b_1\b_2}\equiv
\sum_\lam\overline{\phi(\lam)}\pi_\a(\lam)_{\b_1\b_2}
=\sum_\lam\overline{\phi(\lam)}\langle\a_\lam \b_1,\b_2\rangle$$
is non-zero. But for
$\b_1=\id_{A\max}$, and $\b_2=\tau$ an irreducible sector from
$W\max\subset V$, the matrix entry of the $\a$-induced
representation equals $\langle\lam,\sig_\tau\rangle$ by
$\a$-$\sig$-reciprocity, and the contracted matrix entry equals
$\overline{\phi(\sig_\tau)}$. Hence, if $\phi$ belongs to any of the
$\sig$-supports $\Supp(\tau)$, then $\phi$ belongs to the
$\a$-spectrum $\Spec$. \qed
We list here two ``extremal'', but by no means exhaustive, conditions
to ensure equality in Lemma 4.2, that is, $\bigcup_\tau \Supp(\tau) = \Spec$:
{\bf 4.3.\ Lemma: \sl If $\a$-induction is surjective
(considered as a linear map from $W$ into $V$),
then $\Supp(\id_{A\max})=\bigcup_\tau \Supp(\tau)=\Spec$.
If $\sig$-restriction is surjective (considered as a linear map from $W\max$
into $W$), then $\bigcup_\tau \Supp(\tau) = \Spec = \widehat W$
exhaust all representations of $W$. }
The case of surjective induction was also paid special attention in \cite{BE}.
Indeed, there are many other cases when $\bigcup_\tau \Supp(\tau)=\Spec$,
but we have no satisfactory characterization yet.
{\em Proof:} We want to compute the $\sig$-support $\Supp(\id_{A\max})$.
For this purpose, we multiply $\phi(\sig_{\id_{A\max}})$ with
$\phi(\mu)$, $\mu\in W$. Using in turn $\a$-$\sig$-reciprocity,
the representation condition for $\phi$, Frobenius reciprocity,
the homomorphism property of $\a$-induction, and associativity of
fusion, we arrive at
\begin{eqnarray*}\phi(\sig_{\id_{A\max}})\phi(\mu)=
\sum_\lam\phi(\lam)\phi(\mu)\langle\a_\lam,\id_{A\max}\rangle =
\sum_{\kappa\lam} N_{\kappa\bar\mu}^\lam\phi(\kappa)
\langle\a_\lam,\id_{A\max}\rangle =\\=
\sum_\kappa\phi(\kappa)\langle\a_\kappa\bar\a_\mu,\id_{A\max}\rangle=
\sum_{\kappa,\b}\phi(\kappa)\langle\bar\a_\mu,\b\rangle\langle
\a_\kappa \b,\id_{A\max}\rangle.\end{eqnarray*}
Here the sum over $\b$ extends over all sectors of $V$. The last sum
must vanish for every $\mu$, since the left hand side does, if
$\phi(\sig_{\id_{A\max}})=0$,
i.e., if $\phi\not\in\Supp(\id_{A\max})$. Now, if $\a$-induction
is surjective, then every sector $\b$ arises as a linear combination
of sectors $\a_\mu$, and consequently
$$\sum_\kappa\phi(\kappa)\langle \a_\kappa \b,\id_{A\max}\rangle=
\sum_\lam\overline{\phi(\lam)}\langle \a_\lam,\b\rangle $$
must vanish for all $\b$. These are sufficiently many matrix
entries to ensure the vanishing of the full matrix (since $\langle\a
\b_1,\b_2\rangle=\sum_\b\langle\a,\b\rangle\langle \b\b_1,\b_2\rangle$),
and hence the absence of $\phi$ from the $\a$-spectrum $\Spec$. Hence
$\Spec\subset\Supp(\id_{A\max})$, implying the first claim.
On the other hand, if $\sig$-restriction is surjective, then
$\phi(\sig_\tau)=0$ for all $\tau\in W\max$ implies $\phi(\lam)=0$
for all $\lam\in W$, hence $\phi=0$. Thus the union of the
$\sig$-supports exhausts all representations of $W$, implying the
second claim. \qed
We have thus established some first constraints on the coupling matrix
in terms of representations of fusion algebras.
Further constraints are expected to derive from locality which was
only partially exploited in the form of $\a$-$\sig$-reciprocity in
Proposition 4.1, and in the commutativity of left and right chiral
observables in Theorem 3.6. Notably the condition for locality of the 2D theory
in terms of the local subfactor data and the statistics which was given
in \cite{LR} remains to be transcribed into a condition on
the coupling matrix.
As mentioned in the introduction, chiral locality produces matrices
$S\stat$ and $T\stat$ which represent $\SL(2,\ZZ)$
\cite{Pal,FG}, except for a possible degeneracy of the braiding.
A first implication of the locality condition for the 2D theory is that
$T\stat_LZ=ZT\stat_R$, in accordance with local 2D conformal fields
having integer spin $h_L-h_R$. The companion relation
$S\stat_LZ=ZS\stat_R$, that is, modular invariance of the coupling
matrix with respect to the representation of $\SL(2,\ZZ)$ given by the
statistics, cannot be established for general 2D nets $B$,
however. The surprise is that, as shown here, one can go much of the
way towards classification without knowing these formulae, and that
one can do so whether the involved sectors have a degenerate braiding or
not. (M\"uger's proof \cite{MM} that the degeneracy can always be
removed by an algebraic extension of the chiral observables does not
help here, since this extension is in general not possible within the
given 2D observables.)
\section{Conclusions}
We have shown that in a 2D conformally invariant quantum field theory
with sufficiently many chiral observables to generate the chiral
M\"obius groups, there are maximal algebras of chiral observables
which are, locally, the relative commutants of each other, as well as
of any a priori given chiral observables sharing the generating
property (cf.\ Sect.\ 2).
The representation theory of the chiral observables is governed by a
``canonical tensor product subfactor'' (CTPS) $A_L\otimes A_R\subset B$
given by the respective chiral and 2D local algebras. We have
therefore investigated the general structure of CTPS's and have found
a characterization of the two tensor factors being each other`s relative
commutants (``normality'') in terms of a coupling matrix.
The coupling matrix in this case provides an isomorphism between the
respective fusion rules for the involved sectors of the two tensor factors.
This abstract result, applied to the quantum field theoretical
situation at hand, generalizes a statement on certain ``extended''
chiral observables in the classification program for 2D modular
invariant partition functions, and shows that the latter coincide with
the maximal chiral observables.
Exploiting general properties of $\a$-induction and $\sig$-restriction
between the superselection sectors of the maximal and the a priori
given non-maximal chiral observables, constraints on the coupling
matrix (with respect to the non-maximal chiral observables) are
derived which are the direct counterparts of similar constraints in
the modular classification program.
Yet, modular invariance has not been assumed throughout the analysis.
This supports our conviction that modular invariants are just one aspect
of a deeper and more general mathematical structure (presumably related
to ``asymptotic subfactors'' and ``quantum doubles''). A classification
in terms of graphs still remains to be established in the general
situation. Possibly, additional constraints originating from locality
will play a role here.
\vskip 10mm
{\bf Acknowledgements:} I am indebted to J. B\"ockenhauer for many
helpful and critical comments on an earlier version of this paper, as
well as to A. Recknagel for discussions on modular invariance. I also
thank D. Buchholz, B. Schroer and H.-W. Wiesbrock for useful
suggestions concerning Sect.\ 2, and R. Longo for pointing out a
difficulty in Sect.\ 3.
\small
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\end{document}
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