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496 YASUYUKI KAWAHIGASHI AND ROBERTO LONGO
using positive and negative braidings, respectively. Then we have a nonneg-
ative integer Z
λµ
= dim Hom(α
+
λ
,α
−
µ
). Recall that a completely rational net
produces a unitary representation of SL(2, Z) by [54] and [40] in general. Then
[5, Cor. 5.8] says that this matrix Z with nonnegative integer entries and
normalization Z
00
= 1 is in the commutant of this unitary representation, re-
gardless of whether the extension is local or not, and this gives a very strong
constraint on possible extensions of the Virasoro net. Such a matrix Z is called
a modular invariant in general and has been extensively studied in conformal
field theory. (See [14, Ch. 10] for example.) For a given unitary representation
of SL(2, Z), the number of modular invariants is always finite and often very
small, such as 1, 2, or 3, in concrete examples. The complete classification
of modular invariants for a given representation of SL(2, Z) was first given in
[11] for the case of the SU(2)
k
WZW-models and the minimal models, and
several more classification results have been obtained by Gannon. (See [22]
and references there.)
Our approach to the classification problem of local extensions of a given
net makes use of the classification of the modular invariants. For any local
extension, we have indeed a modular invariant coming from the theory of α-
induction as explained above. For each modular invariant in the classification
list, we check the existence and uniqueness of corresponding extensions. In
complete generality, we expect neither existence nor uniqueness, but this ap-
proach is often powerful enough to get a complete classification in concrete
examples. This is the case of SU(2)
k
. (Such a classification is implicit in [6],
though not explicitly stated there in this way. See Theorem 2.4 below.) Also
along this approach, we obtain a complete classification of the local extensions
of the Virasoro nets with central charge less than 1 in Theorem 4.1. By the
stated canonical appearance of the Virasoro nets as subnets, we derive our final
classification in Theorem 5.1. That is, our labeling of a conformal net in terms
of pairs of Dynkin diagrams is given as follows. For a given conformal net
with central charge c<1, we have a Virasoro subnet. Then the α-induction
applied to this extension of the Virasoro net produces a modular invariant Z
λµ
as above and such a matrix is labeled with a pair of Dynkin diagrams as in
[11]. This labeling gives a complete classification of such conformal nets.
Some extensions of the Virasoro nets in our list have been studied or
conjectured by other authors [3], [69] (they are related to the notion of W -
algebra in the physical literature). Since our classification is complete, it is
not difficult to identify them in our list. This will be done in Section 6.
Before closing this introduction we indicate possible background references
to aid the readers; some have been already mentioned. Expositions of the basic
structure of conformal nets on S
1
and subnets are contained in [26] and [46],
respectively. Jones index theory [34] is discussed in [43] in connection with
quantum field theory. Concerning modular invariants and α-induction one can
CLASSIFICATION OF LOCAL CONFORMAL NETS
497
look at references [3], [5], [6]. The books [14], [29], [17], [35] deal respectively
with conformal field theory from the physical viewpoint, algebraic quantum
field theory, subfactors and connections with mathematical physics and infinite
dimensional Lie algebras.
2. Preliminaries
In this section, we recall and prepare necessary results on extensions of
completely rational nets in connection with extensions of the Virasoro nets.
2.1. Conformal nets on S
1
. We denote by I the family of proper intervals
of S
1
.Anet A of von Neumann algebras on S
1
is a map
I ∈I→A(I) ⊂ B(H)
from I to von Neumann algebras on a fixed Hilbert space H that satisfies:
A. Isotony.IfI
1
⊂ I
2
belong to I, then
A(I
1
) ⊂A(I
2
).
The net A is called local if it satisfies:
B. Locality.IfI
1
,I
2
∈Iand I
1
∩ I
2
= ∅ then
[A(I
1
), A(I
2
)] = {0},
where brackets denote the commutator.
The net A is called M¨obius covariant if in addition satisfies the following prop-
erties C, D, E:
C. M¨obius covariance.
1
There exists a strongly continuous unitary repre-
sentation U of PSL(2, R)onH such that
U(g)A(I)U(g)
∗
= A(gI),g∈ PSL(2, R),I∈I.
Here PSL(2, R) acts on S
1
by M¨obius transformations.
D. Positivity of the energy. The generator of the one-parameter rotation
subgroup of U (conformal Hamiltonian) is positive.
E. Existence of the vacuum. There exists a unit U-invariant vector Ω ∈H
(vacuum vector), and Ω is cyclic for the von Neumann algebra

I∈I
A(I).
(Here the lattice symbol

denotes the von Neumann algebra generated.)
1
M¨obius covariant nets are often called conformal nets. In this paper however we shall
reserve the term ‘conformal’ to indicate diffeomorphism covariant nets.
498 YASUYUKI KAWAHIGASHI AND ROBERTO LONGO
Let A be an irreducible M¨obius covariant net. By the Reeh-Schlieder
theorem the vacuum vector Ω is cyclic and separating for each A(I). The
Bisognano-Wichmann property then holds [8], [21]: the Tomita-Takesaki mod-
ular operator ∆
I
and conjugation J
I
associated with (A(I), Ω), I ∈I, are
given by
U(Λ
I
(2πt))=∆
it
I
,t∈ R, U(r
I
)=J
I
,(1)
where Λ
I
is the one-parameter subgroup of PSL(2, R) of special conformal
transformations preserving I and U(r
I
) implements a geometric action on A
corresponding to the M¨obius reflection r
I
on S
1
mapping I onto I

, i.e. fixing
the boundary points of I, see [8].
This immediately implies Haag duality (see [28], [10]):
A(I)

= A(I

),I∈I,
where I

≡ S
1
 I.
We shall say that a M¨obius covariant net A is irreducible if

I∈I
A(I)=
B(H). Indeed A is irreducible if and only if Ω is the unique U-invariant vector
(up to scalar multiples), and if and only if the local von Neumann algebras A(I)
are factors. In this case they are III
1
-factors (unless A(I)=C identically);
see [26].
Because of Lemma 2.1 below, we may always consider irreducible nets.
Hence, from now on, we shall make the assumption:
F. Irreducibility. The net A is irreducible.
Let Diff(S
1
) be the group of orientation-preserving smooth diffeomor-
phisms of S
1
. As is well known Diff(S
1
) is an infinite dimensional Lie group
whose Lie algebra is the Virasoro algebra (see [53], [35]).
By a conformal net (or diffeomorphism covariant net) A we shall mean a
M¨obius covariant net such that the following holds:
G. Conformal covariance. There exists a projective unitary representation
U of Diff(S
1
)onH extending the unitary representation of PSL(2, R)
such that for all I ∈Iwe have
U(g)A(I)U(g)
∗
= A(gI),g∈ Diff(S
1
),
U(g)AU(g)
∗
= A, A ∈A(I),g∈ Diff(I

),
where Diff(I) denotes the group of smooth diffeomorphisms g of S
1
such that
g(t)=t for all t ∈ I

.
If A is a local conformal net on S
1
then, by Haag duality,
U(Diff(I)) ⊂A(I).
Notice that, in general, U(g)Ω =Ω,g ∈ Diff(S
1
). Otherwise the Reeh-
Schlieder theorem would be violated.
CLASSIFICATION OF LOCAL CONFORMAL NETS
499
Lemma 2.1. Let A bealocalM¨obius (resp. diffeomorphism) covariant
net. The center Z of A(I) does not depend on the interval I and A has a
decomposition
A(I)=

⊕
X
A
λ
(I)dµ(λ)
where the nets A
λ
are M¨obius (resp. diffeomorphism) covariant and irre-
ducible. The decomposition is unique (up to a set of measure 0). Here Z =
L
∞
(X, µ).
2
Proof. Assume A to be M¨obius covariant. Given a vector ξ ∈H,
U(Λ
I
(t))ξ = ξ, ∀t ∈ R, if and only if U(g)ξ = ξ, ∀g ∈ PSL(2, R); see [26].
Hence if I ⊂
˜
I are intervals and A ∈A(
˜
I), the vector AΩ is fixed by U(Λ
I
(·))
if and only if it is fixed by U(Λ
˜
I
(·)). Thus A is fixed by the modular group
of (A(I), Ω) if and only if it is fixed by the modular group of (A(
˜
I), Ω). In
other words the centralizer Z
ω
of A(I) is independent of I; hence, by locality,
it is contained in the center of any A(I). Since the center is always contained
in the centralizer, it follows that Z
ω
must be the common center of all the
A(I)’s. The statement is now an immediate consequence of the uniqueness of
the direct integral decomposition of a von Neumann algebra into factors.
Furthermore, if A is diffeomorphism covariant, then the fiber A
λ
in the
decomposition is diffeomorphism covariant too. Indeed Diff(I) ⊂A(I) de-
composes through the space X and so does Diff(S
1
), which is generated by
{Diff(I),I ∈I}(cf. e.g. [42]).
Before concluding this subsection, we explicitly say that two conformal
nets A
1
and A
2
are isomorphic if there is a unitary V from the Hilbert space
of A
1
to the Hilbert space of A
2
, mapping the vacuum vector of A
1
to the
vacuum vector of A
2
, such that V A
1
(I)V
∗
= A
2
(I) for all I ∈I. Then V also
intertwines the M¨obius covariance representations of A
1
and A
2
[8], because of
the uniqueness of these representations due to eq. (1). Our classification will
be up to isomorphism. Yet, as a consequence of these results, our classification
will indeed be up to the a priori weaker notion of isomorphism where V is not
assumed to preserve the vacuum vector.
Note also that, by Haag duality, two fields generate isomorphic nets if and
only if they are relatively local, that is, belong to the same Borchers class (see
[29]).
2.1.1. Representations. Let A be an irreducible local M¨obius covariant
(resp. conformal) net. A representation π of A is a map
I ∈I→π
I
,
2
If H is nonseparable the decomposition should be stated in a more general form.
500 YASUYUKI KAWAHIGASHI AND ROBERTO LONGO
where π
I
is a representation of A(I) on a fixed Hilbert space H
π
such that
π
˜
I

A(I)
= π
I
,I⊂
˜
I.
We shall always implicitly assume that π is locally normal, namely π
I
is normal
for all I ∈I, which is automatic if H
π
is separable [60].
We shall say that π is M¨obius (resp. conformal) covariant if there exists a
positive energy representation U
π
of PSL(2, R)
˜
(resp. of Diff(S
1
)
˜
) such that
U
π
(g)A(I)U
π
(g)
−1
= A(gI),g∈ PSL(2, R)
˜
(resp. g ∈ Diff(S
1
)
˜
).
(Here PSL(2, R)
˜
denotes the universal central cover of PSL(2, R) and
Diff(S
1
)
˜
the corresponding central extension of Diff(S
1
).) The identity rep-
resentation of A is called the vacuum representation; if convenient, it will be
denoted by π
0
.
A representation ρ is localized in an interval I
0
if H
ρ
= H and ρ
I

0
= id.
Given an interval I
0
and a representation π on a separable Hilbert space, there
is a representation ρ unitarily equivalent to π and localized in I
0
. This is due
to the type III factor property. If ρ is a representation localized in I
0
, then by
Haag duality ρ
I
is an endomorphism of A(I)ifI ⊃ I
0
. The endomorphism ρ is
called a DHR endomorphism [15] localized in I
0
. The index of a representation
ρ is the Jones index [ρ
I

(A(I

))

: ρ
I
(A(I))] for any interval I or, equivalently,
the Jones index [A(I):ρ
I
(A(I))] of ρ
I
,ifI ⊃ I
0
. The (statistical) dimension
d(ρ)ofρ is the square root of the index.
The unitary equivalence [ρ] class of a representation ρ of A is called a
sector of A.
2.1.2. Subnets. Let A beaM¨obius covariant (resp. conformal) net on
S
1
and U the unitary covariance representation of the M¨obius group (resp. of
Diff(S
1
)).
AM¨obius covariant (resp. conformal) subnet B of A is an isotonic map
I ∈I→B(I) that associates to each interval I a von Neumann subalgebra
B(I)ofA(I) with U(g)B(I)U(g)
∗
= B(gI) for all g in the M¨obius group (resp.
in Diff(S
1
)).
If Ais local and irreducible, then the modular group of (A(I), Ω) is ergodic
and so is its restriction to B(I); thus each B(I) is a factor. By the Reeh-
Schlieder theorem the Hilbert space H
0
≡ B(I)Ω is independent of I. The
restriction of B to H
0
is then an irreducible local M¨obius covariant net on H
0
and we denote it here by B
0
. The vector Ω is separating for B(I); therefore
the map B ∈B(I) → B|
H
0
∈B
0
(I) is an isomorphism. Its inverse thus
defines a representation of B
0
that we shall call the restriction to B of the
vacuum representation of A (as a sector this is given by the dual canonical
endomorphism of A in B). Indeed we shall sometimes identify B(I) and B
0
(I)
although, properly speaking, B is not a M¨obius covariant net because Ω is
not cyclic. Note that if A is conformal and U(Diff(I)) ⊂B(I) then B
0
is a
conformal net (compare with Prop. 6.2).
CLASSIFICATION OF LOCAL CONFORMAL NETS
501
If B is a subnet of A we shall denote here B

the von Neumann algebra
generated by all the algebras B(I)asI varies in the intervals I. The subnet
B of A is said to be irreducible if B

∩A(I)=C (if B is strongly additive this
is equivalent to B(I)

∩A(I)=C). If [A : B] < ∞ then B is automatically
irreducible.
The following lemma will be used in the paper.
Lemma 2.2. Let A beaM¨obius covariant net on S
1
and B aM¨obius
covariant subnet. Then B

∩A(I)=B(I) for any given I ∈I.
Proof . By equation (1), B(I) is globally invariant under the modu-
lar group of (A(I), Ω); thus by Takesaki’s theorem there exists a vacuum-
preserving conditional expectation from A(I)toB(I) and an operator A ∈
A(I) belongs to B(I) if and only if AΩ ∈
B(I)Ω. By the Reeh-Schlieder theo-
rem
B

Ω=B(I)Ω and this immediately entails the statement.
2.2. Virasoro algebra and Virasoro nets. The Virasoro algebra is the
infinite dimensional Lie algebra generated by elements {L
n
| n ∈ Z} and c
with relations
[L
m
,L
n
]=(m − n)L
m+n
+
c
12
(m
3
− m)δ
m,−n
(2)
and [L
n
,c] = 0. It is the (complexification of) the unique, nontrivial one-
dimensional central extension of the Lie algebra of Diff(S
1
).
We shall only consider unitary representations of the Virasoro algebra
(i.e. L
∗
n
= L
−n
in the representation space) with positive energy (i.e. L
0
> 0in
the representation space), indeed the ones associated with a projective unitary
representation of Diff(S
1
).
In any irreducible representation the central charge c is a scalar, indeed
c =1− 6/m(m + 1), (m =2, 3, 4, )orc ≥ 1 [20] and all these values are
allowed [23].
For every admissible value of c there is exactly one irreducible (unitary,
positive energy) representation U of the Virasoro algebra (i.e. projective uni-
tary representation of Diff(S
1
)) such that the lowest eigenvalue of the confor-
mal Hamiltonian L
0
(i.e. the spin) is 0; this is the vacuum representation with
central charge c. One can then define the Virasoro net
Vir
c
(I) ≡ U(Diff(I))

.
Any other projective unitary irreducible representation of Diff(S
1
) with a given
central charge c is uniquely determined by its spin. Indeed, as we shall see,
these representations with central charge c correspond bijectively to the ir-
reducible representations (in the sense of Subsection 2.1.1) of the Vir
c
net;
namely, their equivalence classes correspond to the irreducible sectors of the
Vir
c
net.
502 YASUYUKI KAWAHIGASHI AND ROBERTO LONGO
In conformal field theory, the Vir
c
net for c<1 are studied under the
name of minimal models (see [14, Ch. 7, 8], for example). Notice that they are
indeed minimal in the sense they contain no nontrivial subnet [12].
For the central charge c =1− 6/m(m + 1), (m =2, 3, 4, ), we have
m(m − 1)/2 characters χ
(p,q)
of the minimal model labeled with (p, q),
1 ≤ p ≤ m − 1, 1 ≤ q ≤ m with the identification χ
(p,q)
= χ
(m−p,m+1−q)
,
as in [14, Subsec. 7.3.4]. They have fusion rules as in [14, Subsec. 7.3.3] and
they are given as follows.
χ
(p,q)
χ
(p

,q

)
=
min(p+p

−1,2m−p−p

−1)

r=|p−p

|+1,r+p+p

:odd
min(q+q

−1,2(m+1)−q−q

−1)

s=|q−q

|+1,s+q+q

:odd
χ
(r,s)
.(3)
Note that here the product χ
(p,q)
χ
(p

,q

)
denotes the fusion of characters and
not their pointwise product as functions.
For the character χ
(p,q)
, we have a spin
h
p,q
=
((m +1)p − mq)
2
− 1
4m(m +1)
(4)
by [23]. (Also see [14, Subsec. 7.3.3].) The characters {χ
(p,q)
}
p,q
have the S,
T -matrices of Kac-Petersen as in [14, Sec. 10.6].
2.3. Virasoro nets and classification of the modular invariants. Cappelli-
Itzykson-Zuber [11] and Kato [36] have made an A-D-E classification of the
modular invariant matrices for SU(2)
k
. That is, for the unitary representation
of the group SL(2, Z) arising from SU(2)
k
as in [14, Subsec. 17.1.1], they clas-
sified matrices Z with nonnegative integer entries in the commutant of this
unitary representation, up to the normalization Z
00
= 1. Such matrices are
called modular invariants of SU(2)
k
and labeled with Dynkin diagrams A
n
, D
n
,
E
6,7,8
by looking at the diagonal entries of the matrices as in the table (17.114)
in [14]. Based on this classification, Cappelli-Itzykson-Zuber [11] also gave a
classification of the modular invariant matrices for the above minimal mod-
els and the unitary representations of SL(2, Z) arising from the S, T -matrices
mentioned at the end of the previous subsection. From our viewpoint, we will
regard this as a classification of matrices with nonnegative integer entries in
the commutant of the unitary representations of SL(2, Z) arising from the Vi-
rasoro net Vir
c
with c<1. Such modular invariants of the minimal models are
labeled with pairs of Dynkin diagrams of A-D-E type such that the difference
of their Coxeter numbers is 1. The classification tables are given in Table 1
for so-called type I (block-diagonal) modular invariants, where each modular
invariant (Z
(p,q),(p

,q

)
)
(p,q),(p

,q

)
is listed in the form

Z
(p,q),(p

,q

)
χ
(p,q)
χ
(p

,q

)
,
and we refer to [14, Table 10.4] for the type II modular invariants, since we
are mainly concerned with type I modular invariants in this paper. (Note
that the coefficient 1/2 in the table arises from a double counting due to the
CLASSIFICATION OF LOCAL CONFORMAL NETS
503
Label

Z
(p,q ),(p

,q

)
χ
(p,q )
χ
(p

,q

)
(A
n−1
,A
n
)

p,q
|χ
(p,q )
|
2
/2
(A
4n
,D
2n+2
)

q:odd
|χ
(p,q )
+ χ
(p,4n+2−q )
|
2
/2
(D
2n+2
,A
4n+2
)

p:odd
|χ
(p,q )
+ χ
(4n+2−p,q )
|
2
/2
(A
10
,E
6
)
10

p=1

|χ
(p,1)
+ χ
(p,7)
|
2
+ |χ
(p,4)
+ χ
(p,8)
|
2
+ |χ
(p,5)
+ χ
(p,11)
|
2

/2
(E
6
,A
12
)
12

q=1

|χ
(1,q )
+ χ
(7,q )
|
2
+ |χ
(4,q )
+ χ
(8,q )
|
2
+ |χ
(5,q )
+ χ
(11,q )
|
2

/2
(A
28
,E
8
)
28

p=1

|χ
(p,1)
+ χ
(p,11)
+ χ
(p,19)
+ χ
(p,29)
|
2
+ |χ
(p,7)
+ χ
(p,13)
+ χ
(p,17)
+ χ
(p,23)
|
2

/2
(E
8
,A
30
)
30

q=1

|χ
(1,q )
+ χ
(11,q )
+ χ
(19,q )
+ χ
(29,q )
|
2
+ |χ
(7,q )
+ χ
(13,q )
+ χ
(17,q )
+ χ
(23,q )
|
2

/2
Table 1: Type I modular invariants of the minimal models
identification χ
(p,q)
= χ
(m−p,m+1−q)
.) Here the labels come from the diagonal
entries of the matrices again, but we will give our subfactor interpretation of
this labeling later.
2.4. Q-systems and classification. Let M be an infinite factor. A Q-
system (ρ, V, W ) in [44] is a triple of an endomorphism of M and isometries
V ∈ Hom(id,ρ), W ∈ Hom(ρ, ρ
2
) satisfying the following identities:
V
∗
W = ρ(V
∗
)W ∈ R
+
,
ρ(W )W = W
2
.
The abstract notion of a Q-system for tensor categories is contained in [47].
(We had another identity in addition to the above in [44] as the definition of
a Q-system, but it was proved to be redundant in [47].)
If N ⊂ M is a finite-index subfactor, the associated canonical endomor-
phism gives rise to a Q-system. Conversely any Q-system determines a sub-
factor N of M such that ρ is the canonical endomorphism for N ⊂ M : N is
given by
N = {x ∈ M | Wx= ρ(x)W }.
We say (ρ, V, W) is irreducible when dim Hom(id,ρ) = 1. We say that
two Q-systems (ρ, V
1
,W
1
) and (ρ, V
2
,W
2
) are equivalent if we have a unitary
u ∈ Hom(ρ, ρ) satisfying
V
2
= uV
1
,W
2
= uρ(u)W
1
u
∗
.
This equivalence of Q-systems is equivalent to inner conjugacy of the corre-
sponding subfactors.
504 YASUYUKI KAWAHIGASHI AND ROBERTO LONGO
Subfactors N ⊂ M and extensions
˜
M ⊃ M of M are naturally related by
Jones basic construction (or by the canonical endomorphism). The problem
we are interested in is a classification of Q-systems up to equivalence when a
system of endomorphisms is given and ρ is a direct sum of endomorphisms in
the system.
2.5. Classification of local extensions of the SU(2)
k
net. As a preliminary
to our main classification theorem, we first deal with local extensions of the
SU(2)
k
net. The SU(n)
k
net was constructed in [63] using a representation of
the loop group [53]. By the results on the fusion rules in [63] and the spin-
statistics theorem [26], we know that the usual S- and T -matrices of SU(n)
k
as in [14, Sec. 17.1.1] and those arising from the braiding on the SU(n)
k
net
as in [54] coincide.
We start with the following result.
Proposition 2.3. Let A beaM¨obius covariant net on the circle. Suppose
that A admits only finitely many irreducible DHR sectors and each sector is
sum of sectors with finite statistical dimension. If B is an irreducible local
extension of A, then the index [B : A] is finite.
Proof. As in [45, Lemma 13], we have a vacuum-preserving conditional ex-
pectation B(I) →A(I). The dual canonical endomorphism θ for A(I) ⊂B(I)
decomposes into DHR endomorphisms of the net A, but we have only finitely
many such endomorphisms of finite statistical dimensions by assumption. Then
the result in [33, p. 39] shows that multiplicity of each such DHR endomor-
phism in θ is finite; thus the index (= d(θ)) is also finite.
We are interested in the classification problem of irreducible local exten-
sions B when A is given. (Note that if we have finite index [B : A], then the
irreducibility holds automatically by [3, I, Corollary 3.6], [13].) The basic case
of this problem is the one where A(I) is given from SU(2)
k
as in [63]. In this
case, the following classification result is implicit in [6], but for the sake of
completeness, we state and give a proof to it here as follows. Note that G
2
in
Table 2 means the exceptional Lie group G
2
.
Theorem 2.4. The irreducible local extensions of the SU(2)
k
net are in
a bijective correspondence to the Dynkin diagrams of type A
n
, D
2n
, E
6
, E
8
as
in Table 2.
Proof. The SU(2)
k
net A is completely rational by [66]; thus any
local extension B is of finite index by [40, Cor. 39] and Proposition 2.3.
For a fixed interval I, we have a subfactor A(I) ⊂B(I) and can apply
the α-induction for the system ∆ of DHR endomorphisms of A. Then the
matrix Z given by Z
λµ
= α
+
λ
,α
−
µ
 is a modular invariant for SU(2)
k
by
CLASSIFICATION OF LOCAL CONFORMAL NETS
505
level k Dynkin diagram Description
n − 1, (n ≥ 1) A
n
SU(2)
k
itself
4n − 4, (n ≥ 2) D
2n
Simple current extension of index 2
10 E
6
Conformal inclusion SU(2)
10
⊂ SO(5)
1
28 E
8
Conformal inclusion SU(2)
28
⊂ (G
2
)
1
Table 2: Local extensions of the SU(2)
k
net
[5, Cor. 5.8] and thus one of the matrices listed in [11]. Now we have the
locality of B, so that Z
λ,0
= α
+
λ
, id = λ, θ, where θ is the dual canonical
endomorphism for A(I) ⊂B(I) by [64], and the modular invariant matrix Z
must be block-diagonal, which is said to be of type I as in Table 1. Considering
the classification of [11], we have only the following possibilities for θ.
θ =id, for the type A
k+1
modular invariant at level k,
θ = λ
0
⊕ λ
4n−4
, for the type D
2n
modular invariant at level k =4n −4,
θ = λ
0
⊕ λ
6
, for the type E
6
modular invariant at level k =12,
θ = λ
0
⊕ λ
10
⊕ λ
18
⊕ λ
28
, for the type E
8
modular invariant at level k =28.
By [64], [3, II, Sec. 3], we know that all these cases indeed occur, and
we have the unique Q-system for each case by [41, Sec. 6]. (In [41, Def. 1.1],
Conditions 1 and 3 correspond to the axioms of the Q-system in Subsection 2.4,
Condition 4 corresponds to irreducibility, and Condition 3 corresponds to chiral
locality in [46, Th. 4.9] in the sense of [5, p. 454].) By [46, Th. 4.9], we conclude
that the local extensions are classified as desired.
Remark 2.5. The proof of uniqueness for the E
8
case in [41, Sec. 6] uses
vertex operator algebras. Izumi has recently given a direct proof of uniquenss
of the Q-system using an intermediate extension. We later obtained another
proof based on 2-cohomology vanishing for the tensor category SU(2)
k
in [39].
An outline of the arguments is as follows.
Suppose there are two Q-systems for this dual canonical endomorphism of
an injective type III
1
factor M. We need to prove that the two corresponding
subfactors N
1
⊂ M and N
2
⊂ M are inner conjugate. First, it is easy to
prove that the paragroups of these two subfactors are isomorphic to that of
the Goodman-de la Harpe-Jones subfactor [24, Sec. 4.5] arising from E
8
.Thus
we may assume that these two subfactors are conjugate. From this, one shows
that the two Q-systems differ only by a “2-cocycle” of the even part of the
tensor category SU(2)
28
. Using the facts that the fusion rules of SU(2)
k
have
no multiplicities and that all the 6j-symbols are nonzero, one proves that any
such 2-cocycle is trivial. This implies that the two Q-systems are equivalent.