Conformal Subnets and Intermediate Subfactors

Roberto Longo^{∗} Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica, I-00133 Roma, Italy

           e-mail: longo@mat.uniroma2.it

Abstract

Given an irreducible local conformal net A of von Neumann algebras on S^{1} and a finite-index conformal subnet B ⊂ A, we show that A is completely rational iff B is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [14] hold. Our proofs are based on an analysis of the net inclusion B ⊂ A; among other things we show that, for a fixed interval I, every von Neumann algebra R intermediate between B(I) and A(I) comes from an intermediate conformal net L between B and A with L(I) = R. We make use of Watatani’s result, extended for the purpose from the type II case to arbitrary factors, on the finiteness of the set ℑ(N , M) of intermediate subfactors in an irreducible inclusion of factors N ⊂ M with finite Jones index [M : N]. We provide an explicit bound for the cardinality of ℑ(N , M) which depends only on [M : N].

∗Supported in part by MURST and INDAM-GNAMPA.

1 Introduction

Operator algebraic methods have been used to good effect in Conformal Quantum Field Theory, in particular in understanding general model independent structure (e.g. [6, 14, 17, 23, 24, 38]), in the analysis of concrete models (e.g. [5, 42, 43, 45]) and for applications in different contexts (e.g. [33]). In most cases it seems to be impossible to proceed by different methods.

Because of their relevance in different areas, among others Topological QFT and 3-manifold invariants, conformal models with a rational and modular representation theory have been the subject of much attention, also in the physical literature (cf. [15]).

In [14] intrinsic, model independent conditions selecting a class of (local, irreducible) conformal nets A of von Neumann algebras on S^{1} with the right rationality/modularity properties were given. A is completely rational if

1. A is split,

2. A is strongly additive,

3. the 2-interval inclusion of factors A(E) ⊂ A(E′)′ has finite Jones index μ_{A}.

Here both E ⊂ S^{1} and E′ ≡ S^{1} ∖ E are the union of two proper intervals. The split and strongly additivity properties are well-studied basic properties, see Section 3.5 for their definitions, and we do not dwell on them here, cf. [13, 7, 11] and [8, 24]. If A is completely rational, then A(E) ⊂ A(E′)′ is obtained by a quantum double construction in [34], in particular

μ_{A} = ∑_{i} d(ρ_{i})^{2} ,

where the sum is taken over all the irreducible sectors of A. Every representation of A (on a separable Hilbert space) is Möbius covariant and decomposes into the direct sum of irreducible representations with finite statistical dimension. There are only finitely many inequivalent irreducible representations, i.e. A is rational, and the associated braiding is non-degenerate, i.e. the representation tensor category is modular.

At this point the problem of verifying the complete rationality of known models arises. Certain examples were discussed in [14]. As an illustration from [14], consider the case of a non-trivial finite group G acting on a completely rational A; if the fixed-point orbifold subnet A^{G} is also completely rational, then μ_{A^{G}} ≥ |G|^{2}, while ∑_{π∈Ĝ} d(ρ_{π})^{2} = |G|, where the the ρ_{π}’s are the untwisted DHR sectors of A^{G} [12], and this shows that twisted sectors must appear. As A is the initial data, one would infer the complete rationality of A^{G} from that of A. By [14] A^{G} inherits from A the split property and the finiteness of the μ-index. F. Xu [45] has then shown that A^{G} also inherits the strong additivity property and this has inspired our paper.

We shall now show that if B is any conformal subnet of A with finite index, then B is completely rational iff A is completely rational.

As a consequence, if B is a cofinite subnet of A, namely [A : B ∨ B^{c}] < ∞, where B ∨ B^{c} is the subnet generated by B and its relative commutant B^{c} in A, then A is completely rational iff both B and B^{c} are completely rational.

The subnet B^{c} is called the coset subnet associated with B ⊂ A, as it generalizes a coset construction that plays an important rôle in the theory of Kac-Moody Lie algebras, allowing one to construct of the minimal series representations of the Virasoro algebra [20].

Coset models have been intensively studied by Xu in [42, 43] by operator algebraic methods. In one approach he makes use of [14] too. Thanks to his work, coset models associated with many loop group inclusions are cofinite, rational and modular, see the list in Section 3.5.1. Property 3 holds, but the validity of strong additivity was left open.

By our work in all these examples B^{c} is strongly additive, thus B^{c} turns out to be completely rational and this completes the above discussion an explains the rationality/modularity structure better.

We now comment on our proof that the complete rationality property (and also the ‘split & strongly additivity’ property) for finite-index inclusions of conformal nets B ⊂ A are hereditary. That ‘Property 1 & 3’ passes from A to B and viceversa is shown in [14]. The remaining more difficult point we have to prove is that B is strongly additive if A is split and strongly additive, see Sect. 3.5.

To this end, we have analyzed a finite-index inclusion of conformal nets B ⊂ A by considering the relative superselection structure. In particular we show that, for a fixed interval I, every von Neumann algebra R intermediate between B(I) and A(I) comes from an intermediate conformal net L between A and B with L(I) = R.

Here we make use of a result of Watatani [41], following previous work by Popa [36], to the effect that the set ℑ(N , M) of intermediate subfactors in an irreducible inclusion of II_{1}-factors N ⊂ M with finite Jones index [M : N] is finite. We extend this result to the case of arbitrary factors as we need it when N and M are type III factors. In the process, we provide an explicit bound for the cardinality |ℑ(N , M)| of ℑ(N , M) which depends only on [M : N] and implies that

|ℑ(N , M)| ≤ ℓ^{ℓ} ,

where ℓ = [M : N]^{2}. There may be better bounds taking account of further structure associated with intermediate subfactors [3, 4], see the comments at the end of Section 2.2.

We conclude this introduction and include references to the books [1, 15, 25, 39] for basic facts on Operator Algebras and Quantum Field Theory, see also [28] for subfactors and sectors.

2 On subfactors and intermediate factors

The first part of this paper is devoted to an analysis of subfactors and intermediate factors, that will be used later on.

2.1 Some basic structure

Let N ⊂ M be an irreducible inclusion of infinite factors with finite index [M : N]. We denote by γ the canonical endomorphism of M into N and by θ the dual canonical endomorphism θ ≡ γ↾_{N}. The Q-system associated with γ is denoted by (γ, T, S), namely T ∈ M and S ∈ N are the unique (up to a phase) isometries in (ι, γ) and (ι, θ), where ι always denotes the identity automorphism.

Let {[ρ_{i}], i = 0, … N} be the family of the irreducible sectors in the decomposition of [θ], namely

[θ] = ⨁_{i=0}^{N} N_{i}[ρ_{i}] .  (1)

By Frobenious reciprocity for each i the Hilbert space of isometries in M (not necessarily with right support 1)

K_{i} ≡ {R ∈ M : Rx = ρ_{i}(x)R ∀x ∈ N }  (2)

has dimension N_{i}, indeed the map

v ∈ (ρ, θ) → v^{∗}T ∈ K_{i}  (3)

is an anti-linear isomorphism of (ρ, θ) with K_{i}, whose inverse is given by

X ∈ K_{i} → [M : N]ε(T X^{∗}) ∈ (ρ, θ) ,

where ε is the expectation of M onto N. See [34, 26] for the following.

Lemma 1. Let {R_{i,k}}_{k=1}^{N_{i}} be an orthogonal basis of K_{i} with the normalization R_{i,k}^{∗}R_{i,k} = d(ρ_{i}). Then every X ∈ M has a unique Fourier expansion

X = ∑_{i=0}^{N} ∑_{k=1}^{N_{i}} x_{i,k}R_{i,k}  (4)

where the coefficients x_{i,k} belong to N, indeed x_{i,k} = d(ρ_{i})ε(XR_{i,k}^{∗}).

Proof It is immediate that ε(R_{i,k}R_{j,h}^{∗}) ∈ (ρ_{i}, ρ_{j}), thus ε(R_{i,k}R_{j,h}^{∗}) = 0 if i ≠ j. Let \{v_{1}, … v_{N_{i}}\} be an orthonormal basis of isometries for K_{i}, then R_{i,k} ≡ [M : N]^{\frac{1}{2}} v_{k}^{∗}T satisfy

R_{i,k}^{∗}R_{i,h} = T^{∗}v_{k}v_{h}^{∗}T = δ_{hk}d(ρ_{i})

because v_{k}v_{h}^{∗} ∈ (θ, θ) and T T^{∗} ∈ M is the Jones projection for implementing the expectation ε_{1} : N → θ(M) so that ε_{1}↾_{(θ,θ)} is the associated trace, see [30]. Therefore

ε(R_{i,k}R_{i,h}^{∗}) = [M : N]ε(v_{k}^{∗}T T^{∗}v_{h}) = ε(v_{k}^{∗}v_{h}) = δ_{hk}.  (5)

Now M = N T, thus M is generated by M = ∑_{i,k} NR_{i,k} because ∑ K_{i} has right support one. By the orthogonality relations (5) every X ∈ M has the expansion given by formula (4). □

Denote by ℑ(N , M) the set of intermediate von Neumann algebras between N and M. Clearly if R ∈ ℑ(N , M), then R is a factor and indeed both N ⊂ R and R ⊂ M are irreducible finite-index inclusions of factors.

Let R ∈ ℑ(N , M) and set K′_{i} ≡ K_{i} ∩ R. Then K′_{i} is a Hilbert subspace of K_{i}, so we may choose the R_{i,k} so that {R_{i,k}}_{k=1}^{N′_{i}} is a basis for K′_{i}, where N′_{i} = dimK′_{i}. We may also re-order the ρ_{i}’s so that N′_{i} > 0 iff i ≤ N′ where N′ ≤ N is an integer.

Proposition 2. With the above notations, X ∈ M belongs to R iff the in the expansion (4) x_{i,k} = 0 for all k > N′_{i}, namely

X = ∑_{i=0}^{N′} ∑_{k=1}^{N′_{i}} x_{i,k}R_{i,k} .  (6)

In particular R is generated by N and the K′_{i}’s.

Proof Recall that [M : R] < ∞, so there exists an expectation ε_{R} : M → R. From the definition (2) ε_{R}(K_{i}) ⊂ K_{i}, thus ε_{R}↾_{K_{i}} ∈ B(K_{i}) is a norm one projection. Clearly ε_{R}(K_{i}) = K′_{i} thus if X ∈ M has the expansion (4) we have

ε_{R}(X) = ∑_{i=0}^{N′} ∑_{k=1}^{N′_{i}} x_{i,k}R_{i,k} ,

which implies the statement in the proposition. □

The following theorem is due to Watatani [41] in the II_{1}-case, related results are contained in [36]. A proof for factors of arbitrary type will be given in next section, where we shall obtain a bound for the cardinality |ℑ(N , M)| of ℑ(N , M).

Theorem 3. ℑ(N , M) is a finite set.

We note here a simple proof in the special case where N_{i} = 1, i.e. each irreducible sub-sector of θ is contained in θ with multiplicity one, and M is infinite. In this case the expansion (4) takes the form X = ∑_{i=0}^{N} x_{i}R_{i}. By re-ordering the R_{i}’s if necessary we may assume that there is an integer 0 < N_{1} ≤ N such that R_{i} ∈ N iff 0 ≤ i ≤ N_{1}.

With ε_{L} the expectation from M onto L, ε_{L}(R_{i}) belongs to K_{i}, hence it must be a multiple of R_{i} by assumption. Therefore ε_{L}(R_{i}) = R_{i} if 0 ≤ i ≤ N_{1} and ε_{L}(R_{i}) = 0 otherwise. Thus X = ∑_{i=0}^{N} x_{i}R_{i} belongs to N iff x_{i} = 0, N_{1} < i ≤ N. Therefore an intermediate subfactor is uniquely associated with a certain subset of {1, 2, …, N}, hence there are at most 2^{N} intermediate subfactors.

Corollary 4. Let L be an intermediate subfactor between N and M and β : 𝔊 → Aut(M) a (pointwise weakly continuous) action of a connected topological group 𝔊 with β_{g}(N) = N, g ∈ 𝔊. Then β_{g}(L) = L, g ∈ 𝔊.

Proof We consider on ℑ(M, N) the topology of pointwise weak convergence of the associated conditional expectations (L_{i} → L iff ε_{L_{i}}(x) → ε_{L}(x) weakly for all x ∈ M). Then β implements a continuous action of 𝔊 on ℑ(M, N). The corollary is thus immediate because any continuous action of a connected group on a discrete set is trivial. □

2.2 A bound for the number of intermediate subfactors

Let N ⊂ M be an irreducible inclusion with finite index and denote by ε the conditional expectation from M to N. We shall now determine a bound for |ℑ(M, N)|. Our proof is inspired by the papers [10, 41].

We assume that there exists a faithful normal state ω on N (otherwise replacing it by a weight). By considering the GNS representation of M associated with ω̃ ≡ ω ·ε, we may assume that M acts on a Hilbert space H with cyclic and separating vector Ω so that ω̃ = (Ω, ·Ω). Then e ≡ [N Ω], equal to JeJ, is the Jones projection for N ⊂ M and M_{1} ≡ ⟨M, e⟩ = JN′J is the Jones extension, where J is the modular conjugation of M associated with Ω. The projection e ∈ N′ ∩ M_{1} and

ε(e) = λ ≡ [M : N]^{−1}.

Let R, S be intermediate factors between N and M and denote by p ≡ [RΩ] and q ≡ [SΩ] their associated Jones projections. Note that JpJ = p, JqJ = q and R_{1} ≡ ⟨R, p⟩ = JR′J and S_{1} ≡ ⟨S, q⟩ = JS′J are the corresponding Jones extensions and so there is a chain of inclusions

N ⊂ R, S ⊂ M ⊂ R_{1}, S_{1} ⊂ M_{1} .

Clearly p ∈ R′ ∩ R_{1} and q ∈ S′ ∩ S_{1}, thus p and q both belong to N′ ∩ M_{1}.

Proposition 5. If ||p − q|| < λ/2 then R = S.

Proof We may assume R ≠ N as otherwise p = e, thus q = e because q ≥ e and λ ≤ 1.

As we have [M : N] = [M : R][R : N], it follows that

[M : N] ≥ 2[M : R]  (7)

because [R : N] ≥ 2 [27]. In particular λ ≤ 1/2. Let ε_{R_{1}} be the expectation from M_{1} onto R_{1} and set q′ ≡ ε_{R_{1}}(q). Then q′ ∈ S′ ∩ R_{1} since obviously q′ ∈ R_{1} and if x ∈ S

xq′ = xε_{R_{1}}(q) = ε_{R_{1}}(xq) = ε_{R_{1}}(qx) = q′x

because S ⊂ R_{1}. Moreover 0 ≤ q′ ≤ 1 and q′ ≠ 0 because ε_{R_{1}} is positive and faithful. Setting δ = λ/2 we have

||p − q′|| = ||p − ε_{R_{1}}(q)|| = ||ε_{R_{1}}(p − q)|| ≤ ||p − q|| < δ .  (8)

Therefore the spectrum sp(q′) ⊂ [0, δ) ∪ (1 − δ, 1], see Lemma 8.

Thus the spectral projection q′′ ≡ χ_{(1−δ,1]}(q′) is a projection in S′ ∩ R_{1} and ||q′′ − q′|| < δ, thus

||p − q′′|| ≤ ||p − q′|| + ||q′ − q′′|| < 2δ ≤ 1 ,

and this implies that p and q′′ are equivalent projections of N′ ∩ R_{1}. Indeed the phase v in the polar decomposition of t ≡ pq′′ is a partial isometry in N′ ∩ R_{1} with v^{∗}v = q′′, vv^{∗} = p (see Lemma 9 below). Then we can define an isomorphism Φ of S into R by

Φ(x)p ≡ vxv^{∗} , x ∈ S ,

as pR_{1}p = Rp [27]. Moreover Φ(x) = x for all x ∈ N because v ∈ N′.

We have the intertwining relation

Φ(x)v = vx , x ∈ S .

With ε′ the conditional expectation from M_{1} onto M we then have

Φ(x)ε′(v) = ε′(v)x , x ∈ S ,

where ε′(v) ∈ N′ ∩ M = ℂ, thus ε′(v) ≠ 0 would imply that Φ is the identity on S and S ⊂ R. Reversing the rôle of R and S also R ⊂ S, so R = S.

To show that indeed ε′(v) ≠ 0 set λ_{0} ≡ ε′(p) = [M : R]^{−1} and notice that, by using Lemma 9, we have

|ε′(v) − λ_{0}| ≤ ||ε′(v) − ε′(t)|| + ||ε′(t) − ε′(p)|| ≤ ||v − t|| + ||t − p|| = ||v − t|| + ||p(q′′ − p)|| ≤ 2δ + ||q′′ − p|| < 4δ = 2λ ,

thus ε′(v) ≠ 0 because 2λ ≤ λ_{0} by eq. (7). □

Corollary 6. Let N ⊂ M be an irreducible inclusion of factors. The cardinality of the set of intermediate factors between N and M is bounded by

|ℑ(N , M)| ≤ (4(n + 2)\sqrt{n} + 1)^{n^{2}} ,  (9)

where n is the largest integer such that n + 1 ≤ [M : N].

Proof By the above proposition |ℑ(N , M)| is dominated by the maximum number of projections ≥ e in N′ ∩ M_{1} whose mutual distance is ≥ λ/2. As e is a minimal central projection of N′ ∩ M_{1}, we can naturally embed N′ ∩ M_{1} into ℂ ⊕ Mat_{m}(ℂ), where m an integer with m + 1 ≤ [M : N] (m = ∑_{1}^{N} N_{i} in eq. (1)). Indeed, as J(N′ ∩ M_{1})J = N′ ∩ M_{1}, and AdJ implements an anti-automorphism of N′ ∩ M_{1}, we can assume this anti-automorphism to extend to an anti-automorphism of ℂ ⊕ Mat_{m}(ℂ) preserving the two components, in other words we may assume that the AdJ-invariant part of N′ ∩ M_{1} is contained in ℝ ⊕ Mat_{m}(ℝ).

Thus |ℑ(N , M)| is dominated by the maximum number of projections in Mat_{m}(ℂ) whose mutual distance is larger than \frac{1}{2(n+2)} where n is the largest interger such that n + 1 ≤ [M : N] (so λ > \frac{1}{n+2}). Moreover, as Jp_{i}J = p_{i}, we may regard the p_{i}’s as elements of Mat_{m}(ℝ).

The following Lemma 7 with ϵ = \frac{1}{2(n+2)} then gives

|ℑ(N , M)| ≤ (4(n + 2)\sqrt{m} + 1)^{m^{2}} ≤ (4(n + 2)\sqrt{n} + 1)^{n^{2}} .
□

The following lemma slightly improves [7, Lemma 2.6].

Lemma 7. Let ϵ > 0 and {p_{1}, p_{2}, …, p_{k}} be elements in the unit ball of Mat_{n}(ℝ) such that ||p_{i} − p_{j}|| ≥ ϵ if i ≠ j. Then k ≤ (\frac{2\sqrt{n}}{ϵ} + 1)^{n^{2}}.

Proof As the uniform and Hilbert-Schmidt norms are related by ||X|| ≤ ||X||_{HS} ≤ \sqrt{n}||X||, the p_{i}’s give vectors of norm less \sqrt{n} in the Euclidean space ℝ^{n^{2}} (identified with Mat_{n}(ℝ) with the Hilbert-Schmidt norm) with mutual distance larger than ϵ. Denoting by B(r) the open ball of radius r in ℝ^{n^{2}} we then have

k < \frac{Vol(B(\sqrt{n} + ϵ/2))}{Vol(B(ϵ/2))} = (\frac{2\sqrt{n}}{ϵ} + 1)^{n^{2}} .

□

The following lemmata are variations of known facts (cf. e.g. [39]) and are included for convenience.

Lemma 8. Let x be a positive linear operator, 0 ≤ x ≤ 1, and p a selfadjoint projection with ||x − p|| ≤ δ < 1/2. Then sp(x) ⊂ [0, δ] ∪ [1 − δ, 1].

Proof With ℓ ∈ (δ, 1 − δ) we have x − ℓ = (p − ℓ)(1 + (p − ℓ)^{−1}(x − p)), thus x − ℓ is invertible if ||(p − ℓ)^{−1}(x − p)|| < 1, which is the case if δ||(p − ℓ)^{−1}|| < 1. This holds because ||(p − ℓ)^{−1}|| = max{ℓ^{−1}, (1 − ℓ)^{−1}} < δ^{−1}. □

Lemma 9. Let p and q be selfadjoint projections on a Hilbert space H and t = vh be the polar decomposition of t ≡ pq. If ||p − q|| ≤ δ < 1, then v^{∗}v = q, vv^{∗} = p and ||v − t|| ≤ δ.

Proof As ||(p − q)^{2}|| ≤ δ^{2} < 1, the operator 1 − (p − q)^{2} is invertible. Thus s ≡ pq + (1 − p)(1 − q) is also invertible, indeed s^{−1} = (1 − (p − q)^{2})^{−1}s^{∗}, and this implies that v is a partial isometry from q to p. Then we have

||v − t|| = ||v − vh|| = ||v(1 − h)|| ≤ ||(1 − h)↾_{qH}|| = ||(1 − \sqrt{qpq})↾_{qH}|| ≤ ||(1 − qpq)↾_{qH}|| = ||q(q − p)q|| ≤ ||q − p|| ≤ δ .

□

The bound (9) implies that

|ℑ(N , M)| ≤ ℓ^{ℓ} ,

where ℓ = [M : N]^{2}. The arguments in this section can be improved, in particular taking into account that the p_{i}’s are projections in Lemma 7, leading to a better bound |ℑ(N , M)| ≤ ℓ_{1}^{ℓ_{2}} where however ℓ_{2} is still quadratic in the index. It would be interesting to see if a bound |ℑ(N , M)| ≤ [M : N]^{[M:N]} holds. This is the case of the example N = M^{G} with G is a finite group where, because of the Galois correspondence (see e.g. [26]), |ℑ(N , M)| ≤ |G|! = |ℑ(N , M)|! . We note that we have not made use of the specific form of the projections associated with intermediate subfactors and the canonical algebra they generate [3, 4].

3 Conformal nets and subnets

We now begin our study of conformal nets. Their subnets will be analyzed through the relative superselection structure.

3.1 Conformal nets

Let I denote the family of proper intervals of S^{1}, namely connected subsets of S^{1} of positive measure (length) strictly less than 2π. The subnet structure is rather simple for net on the 4-dimensional Minkowski spacetime (see [9]), but this does not extend to the low-dimensional case.

A net (or precosheaf) A of von Neumann algebras on S^{1} is a map

I ∈ I → A(I) ⊂ B(H)

from I to von Neumann algebras on fixed a Hilbert space H that satisfies:

A. Isotony. If I_{1} ⊂ I_{2} belong to I, then

A(I_{1}) ⊂ A(I_{2}).

The net A is called a (local) conformal net if in addition it satisfies the following properties:

B. Locality. If I_{1}, I_{2} ∈ I and I_{1} ∩ I_{2} = ∅ then

[A(I_{1}), A(I_{2})] = {0},

where brackets denote the commutator^{1}.

C. Conformal invariance. There exists a strongly continuous unitary representation U of PSL(2, ℝ) on H 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 Möbius transformations. We shall denote also by α_{g} = AdU(g) the adjoint action on B(H).

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).

We shall say that a conformal net is irreducible if ∨_{I∈I}A(I) = B(H). Here the lattice symbol ∨ denotes the von Neumann algebra generated. We recall the following Lemma whose proof can be found in [23].

Lemma 10. Let A be a conformal net. The following are equivalent:

(i) A is irreducible;

(ii) Ω is cyclic for ∨_{I∈I}A(I) and unique U-invariant;

(iii) Ω is cyclic for ∨_{I∈I}A(I) and the local von Neumann algebras A(I) are factors. In this case they are III_{1}-factors (unless A(I) = ℂ identically).

Let A be an irreducible conformal net. By the Reeh-Schlieder theorem [18] the vacuum vector Ω is cyclic and separating for each A(I). The Bisognano-Wichmann property then holds [6, 19]: the Tomita-Takesaki modular operator Δ_{I} and conjugation J_{I} associated with (A(I), Ω), I ∈ I, are given by

U(Λ_{I}(2πt)) = Δ_{I}^{it}, t ∈ ℝ,  (10)
U(r_{I}) = J_{I},  (11)

1The locality condition will be always assumed in this paper, with the exception of Subsection 3.5.3

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, the Möbius reflection on S^{1} mapping I onto I′, i.e. fixing the boundary points of I, see [6].

This immediately implies Haag duality:

A(I)′ = A(I′), I ∈ I ,

where I′ ≡ S^{1} ∖ I.

3.2 Representations

Let A be an irreducible local conformal net. A representation π of A is a map

I ∈ I → π_{I} ,

where π_{I} is a representation of A(I) on a fixed Hilbert space H_{π} such that

πĨ↾_{A(I)} = π_{I}, I ⊂ Ĩ;

we shall always assume that π is locally normal, namely π_{I} is normal for all I ∈ I, which is automatic if H_{π} is separable [39].

We shall say that a representation ρ is localized in a 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 the type III factor property.

Let ρ be a representation of A localized in a given interval I_{0}. By Haag duality ρ satisfies the following properties

(a) If I ∈ I and I ⊃ I_{0} then ρ_{I} is an endomorphism of A(I), and ρĨ↾_{A(I)} = ρ_{I} for all Ĩ ∈ I, Ĩ ⊃ I;

(b) If I_{1} ∈ I and I_{1} ∩ I_{0} = ∅, then ρ_{I_{1}} is the identity on A(I_{1});

(c) If I, I_{1} ∈ I and I ⊃ I_{0} ∪ I_{1}, there exists a unitary u ∈ A(I) such that the representation I → ρ′_{I} ≡ uρ_{I}(·)u^{∗} is localized in I_{1} (that is to say ρ′_{I_{2}} acts identically on A(I_{2}) if I_{1} ∩ I_{2} = ∅ for all I_{2} ∈ I).

We now make a stereographic identification ℝ = S^{1} ∖ {∞} and denote by I_{0} ⊂ I the family of bounded intervals of ℝ, namely of the intervals of S^{1} whose closure do not contain the point ∞ of S^{1}.

We denote by A_{0} the restriction of A to ℝ (i.e. to I_{0}) and by 𝔄_{0} the associated quasi-local C^{∗}-algebra 𝔄_{0} ≡ \overline{∪_{I∈I_{0}}A(I)} (norm closure). For a characterization of the so obtained net on ℝ, see [24].

Given I_{0} ∈ I_{0} a DHR endomorphism ρ of A_{0} localized in I_{0} is a map

I ⊃ I_{0} → ρ_{I}

that associates to each I ∈ I_{0} a representation ρ_{I} of A(I) on H such that the above conditions (a), (b), (c) hold true with I replaced by I_{0}.

Clearly a DHR endomorphism determines an endomorphism of 𝔄_{0}, still denoted by ρ, such that ρ_{I} = ρ↾_{A(I)}, I ⊂ I_{0}. The above properties (a), (b), (c) are immediately expressed in terms of such endomorphism of 𝔄_{0}; we shall use the two descriptions interchangeably without further specifications.

Proposition 11. Let ρ be a DHR endomorphism on A_{0} localized in an interval I_{0} ∈ I_{0}. There exists a unique representation ρ̃ of A extending ρ and localized I_{0}.

Proof Our aim is to define consistently a representation ρ_{I} of A(I) for every I ∈ I. To this end, given I ∈ I, choose I_{1} ∈ I_{0}, I_{1} ⊂ I′ and L ∈ I_{0} let be an interval with L ⊃ I_{0} ∪ I_{1}. Take then a DHR endomorphism of A_{0} equivalent to ρ and localized in I_{1}, ρ′ = Adu · ρ for some unitary u ∈ A(L). We set

ρ̃_{I}(a) = u^{∗}au, a ∈ A(I) ,

namely ρ̃_{I} = Adu^{∗} · ρ′_{I}. Clearly ρ̃_{I} = ρ_{I} if I ∈ I_{0} and a routine checking shows that I ∈ I → ρ̃_{I} is indeed a representation of A. □

A representation π of A on a Hilbert space H_{π} is covariant if there exists a unitary representation U_{π} of the universal covering group PSL(2, ℝ)^{}˜ of PSL(2, ℝ) on H_{π} such that

AdU_{π}(g) · π_{I} = π_{gI} · AdU(g), g ∈ PSL(2, ℝ)^{}˜ , I ∈ I_{0}.

Here U has been lifted to PSL(2, ℝ)^{}˜ . π is said to have positive energy if the generator of the rotation unitary subgroup of U_{π} is positive.

Let ρ be a representation of A localized in I_{0} ∈ I. By a local cocycle (w.r.t. to ρ) we shall mean the assignement of an interval I ⊃ I_{0}, a symmetric neighborhood U of the identity of PSL(2, ℝ)^{}˜ such that I_{0} ∪ gI_{0} ⊂ I, ∀g ∈ U and a strongly continuous unitary valued map z : g ∈ U → z_{ρ}(g) ∈ A(I) such that

z_{ρ}(g) ∈ A(I)  (12)
z_{ρ}(gh) = z_{ρ}(g)α_{g}(z_{ρ}(h)),  (13)
Adz_{ρ}(g)^{∗} · ρĨ(a) = α_{g} · ρ_{g^{−1}Ĩ} · α_{g^{−1}}(a), a ∈ A(Ĩ),  (14)

for some open interval Ĩ with Ī ⊂ Ĩ and all g, h ∈ U such that I ∪ gI ⊂ Ĩ. We shall then say that z is localized in I.

If this holds, then eq. (14) is valid for all L ∈ I:

Adz_{ρ}(g)^{∗} · ρ_{L}(a) = α_{g} · ρ_{g^{−1}L} · α_{g^{−1}}(a), a ∈ A(L).  (15)

Indeed, if L ⊃ I then the above equation holds by additivity [18]. Thus it holds for sub-intervals L_{0} ⊂ L. Again by additivity, the equation is then satisfied for all L ∈ I.

If ρ is a covariant representation of A localized in I_{0} then for any given interval I ⊃ I_{0} there exists a local cocycle w.r.t. ρ localized in I. Indeed if U is a symmetric neighborhood of the identity of PSL(2, ℝ)^{}˜ such that I_{0} ∪ gI_{0} ⊂ I, ∀g ∈ U, then by Haag duality the unitaries

z_{ρ}(g) ≡ U_{ρ}(g)U(g)^{∗}

belong to A(I) for all g ∈ U and clearly verify the local cocycle property (13,14).

Notice now that, taking I_{0}, I ∈ I_{0}, a local cocycle is expressed in terms of the DHR endomorphism of A_{0} associated with ρ. The converse construction is made in the following.

Proposition 12. Let ρ be a DHR endomorphism of A_{0} localized in the interval I_{0} ∈ I_{0} and ρ̃ the representation of A extending ρ given by Proposition 11. Then ρ̃ is covariant iff there exists a local cocycle z_{ρ} w.r.t. ρ (i.e. properties (12,13,14) hold with I replaced by I_{0}).

Proof We need only to show that ρ̃ is covariant if there exists a local cocycle z_{ρ}. By the above arguments eq. (15) holds. Now set Uρ̃(g) = z_{ρ}(g)U(g) for g in a suitable neighborhood of the identity of PSL(2, ℝ)^{}˜ . Then Uρ̃ is a local representation of PSL(2, ℝ)^{}˜ , hence it extends to a unitary representation of PSL(2, ℝ)^{}˜ because PSL(2, ℝ)^{}˜ is simply connected. The local covariance then gives

U_{ρ}(g)ρ_{L}(a)U_{ρ}^{∗}(g) = ρ_{gL}(α_{g}(a)), a ∈ A(L),

for any L ∈ I. The covariance then follows by the group property of U_{ρ}, see also [21]. □

Before concluding this section we recall that, if ρ is a localizable representation of A, the (statistical) dimension of ρ is d(ρ) ≡ [ρ_{I′}(A(I′))′ : ρ_{I}(A(I))]^{\frac{1}{2}} , independently on I ∈ I, and this clearly coincides with [A(I) : ρ_{I}(A(I))]^{\frac{1}{2}} if ρ is localized in I [31].

If ρ is Möbius covariant and d(ρ) < ∞, then ρ has positive energy [2].

3.3 Subnets

Let A be a local irreducible conformal net of von Neumann algebras on S^{1} as above and U the associated unitary positive energy representation of PSL(2, ℝ) on the vacuum Hilbert space H.

By a conformal subnet we shall mean a map

I ∈ I → B(I) ⊂ A(I)

that associates to each interval I ∈ I a von Neumann subalgebra B(I) of A(I), which is isotone

B(I_{1}) ⊂ B(I_{2}), I_{1} ⊂ I_{2} ,

and Möbius covariant w.r.t. the representation U, namely

U(g)B(I)U(g)^{−1} = B(gI)

for all g ∈ PSL(2, ℝ) and I ∈ I.

Let H_{B} be the closure of (∨_{I∈I}B(I))Ω and E the orthogonal projection of H onto H_{B}. By the Reeh-Schlieder theorem \overline{B(I)Ω} = H_{B} for each fixed I ∈ I.

Clearly H_{B} is U-invariant and Ω is unique U↾_{H_{B}}-invariant, thus by Lemma 10 the restriction of B to H_{B} is an irreducible local conformal net on H_{B} where U↾_{H_{B}} is the associated unitary representation of PSL(2, ℝ).

As Ω is separating for A(I), Ω is also separating for B(I), I ∈ I. Thus the restriction map b ∈ B(I) → b↾_{H_{B}} is is one-to-one, so we will often identify B with its restriction to H_{B}; should we need to specify, we shall talk on the net B on H or on H_{B}. Note that each B(I) is a factor.

Lemma 13. For each I ∈ I there is a vacuum preserving conditional expectation ε_{I} : A(I) → B(I) such that εĨ↾_{A(I)} = ε_{I} if I ⊂ Ĩ. Thus B is a standard net of subfactors in the sense of [34].

Proof By the Bisognano-Wichmann property B(I) is globally invariant under the modular group of (A(I), Ω), hence by Takesaki’s theorem there exists a conditional expectation ε_{I} : A(I) → B(I) given by

ε_{I}(a)E = EaE, a ∈ A(I) .

As E is independent of I, we have that εĨ↾_{A(I)} = ε_{I} if I ⊂ Ĩ. □

By Möbius covariance the index [A(I) : B(I)] is independent of the interval I ∈ I and will be denoted by [A : B]. The following lemma is contained in [5] with the strong additivity assumption and in [11] in the conformal case.

Lemma 14. If [A : B] < ∞ then B(I)′ ∩ A(I) = ℂ, I ∈ I.

Proof By the Bisognano-Wichmann property and the uniqueness of the vacuum the modular group of A(I) w.r.t. Ω acts ergodically on B(I)′∩A(I), hence B(I)′∩A(I) = ℂ because B(I)′ ∩ A(I) is finite-dimensional. □

We shall make a variation of the analysis made in [34], which is needed because our nets are not directed.

Lemma 15. Let [A : B] < ∞ and I_{0} ⊂ I. There exists a canonical endomorphism γ_{I} : A(I) → B(I) with associated Q-system (γ_{I}, T, S) such that T ∈ A(I_{0}) and S ∈ B(I_{0}) and γ_{I}↾_{A(I_{0})} is a canonical endomorphism of A(I_{0}) into B(I_{0}).

Proof Let C(I_{0}) = ⟨A(I_{0}), E⟩ and C(I) = ⟨A(I), E⟩ be the Jones extensions and ε′_{I_{0}} : C(I_{0}) → A(I_{0}), ε′_{I} : C(I) → A(I) the dual expectations. Since every X ∈ C(I_{0}) can be written as X = ∑_{i} x_{i}Ey_{i} with x_{i}, y_{i} ∈ A(I_{0}) [27, 35], we then have

ε′_{I}(X) = ∑_{i} x_{i}ε′_{I}(E)y_{i} = λ ∑_{i} x_{i}y_{i} = ∑_{i} x_{i}ε′_{I_{0}}(E)y_{i} = ε′_{I_{0}}(X), X ∈ A(I),

where λ ≡ [A : B]^{−1}, namely ε′_{I}↾_{A(I_{0})} = ε′_{I_{0}}.

Let V ∈ C(I_{0}) be an isometry V V^{∗} = E. Then a canonical endomorphism γ_{I} : A(I) → B(I) is given by

γ_{I}(a)E = V aV^{∗} .  (16)

Now

T = λ^{−1}ε′_{I}(V), S = λ^{−1}ε_{I}(T),

are the isometries in the Q-system for γ_{I} and T ∈ A(I_{0}) and S ∈ B(I_{0}) by the compatibility of the expectations. □

Notice the formula

γ_{I}(a) = λ^{−1}ε′_{I}(V aV^{∗}), a ∈ A(I)  (17)

which is obtained applying ε′_{I} to both members of eq. (16).

Proposition 16. Let A be a local irreducible conformal net on S_{1} and B ⊂ A a conformal subnet. Given an interval I_{0} ∈ I the dual canonical endomorphism θ_{I_{0}} ≡ γ_{I_{0}}↾_{B(I_{0})} extends to a representation θ of B localized in I_{0}.

Proof First we assume [A : B] < ∞. By Proposition 11 it is sufficient to show that θ_{I_{0}} extends to a DHR endomorphism of B localized in I_{0} (properties (a), (b), (c) with I replaced by I_{0}). This is soon verified by applying Lemma 15. Indeed property (a) is an immediate consequence of this lemma. Concerning property (b) notice the formula [32]

θ_{I}(b) = λ^{−1}ε_{I}(T bT^{∗}), b ∈ B(I),

with T as in Lemma 15, that follows similarly to the formula (17). Then (b) follows because T ∈ A(I_{0}).

Finally property (c) is immediate by the uniqueness up to inners of the canonical endomorphism [30].

The general case can be obtained along the same lines making use of [34, Theorem 3.2] instead of Lemma 15. □

Proposition 17. The representation θ of B on H_{B} in Prop. 16 is unitarily equivalent to the identity representation of B on H. In particular θ is covariant with positive energy.

Proof Indeed the isometry V ∈ C(I) with V V^{∗} = E (as in Lemma 15) satisfies the equation

θĨ(b)E = V bV^{∗}, b ∈ B(Ĩ) ,  (18)

if Ĩ is an interval containing I.

To show that V implements the desired unitary equivalence we need to further show that the above equation (18) holds true with Ĩ replaced with an interval I_{1} not containing I. This is certainly true if I_{1} ∩ I = ∅, because in this case θ acts trivially on B(I_{1}) and V ∈ C(I) commutes with B(I_{1}) because C(I) = ⟨A(I), E⟩ and B(I_{1}) commute.

So we may assume that I_{1} ⊃ I′, extending I_{1} if necessary. Choose then an interval I_{0} ⊂ I with I_{0} ∩ I_{1} = ∅. By Lemma 15 we can find a canonical endomorphism A(I_{0}) → B(I_{0}) with dual canonical endomorphism extending to a representation θ′ of A localized in I_{0} with a unitary u ∈ B(I) such that θ′ = Adu·θ. Then the isometry V′ associated with θ′ belongs to C(I_{0}) and is given V′ = uV. Therefore

θ(b)E = uθ′(b)u^{∗} = uV′bV′^{∗}u^{∗} = V bV^{∗}, b ∈ B(I_{1})

as desired. □

The following corollary is a consequence of the equivalence between local and global intertwiners for a finite-index covariant representation [23].

Corollary 18. Assume [A : B] < ∞ and let θ be the representation of B on H_{B} in Proposition 16. Then θ has a finite direct sum decomposition

θ = ⨁_{i=0}^{N} N_{i}ρ_{i} ,  (19)

where the ρ_{i}’s are positive-energy covariant irreducible representation of B on H_{B} localized in I_{0}. Thus θ_{I} = ⊕_{i=0}^{N}N_{i}ρ_{iI} is a decomposition of the canonical endomorphism for any interval I ⊃ I_{0}.

Proof As d(θ) = [A : B] < ∞ we may decompose θ into irreducible representations ρ_{i} as above localized in I_{0}; moreover the ρ_{i} are covariant because θ is covariant, see [23]. Then by [23, Theorem 2.3] each ρ_{iI} is an irreducible endomorphism of B(I) if I ⊃ I_{0}. □

Thus, if [A : B] < ∞ then the identity representation of B on H has finite statistical dimension. The converse is also true: as we have the inclusions B(I) ⊂ A(I) ⊂ B(I′)′, if the identity representation of B on H has finite statistical dimension, namely [B(I′)′ : B(I)] < ∞, then [A(I) : B(I)] < ∞.

Corollary 19. Assume [A : B] < ∞. With the above notations, let K_{i} ⊂ A(I_{0}) be the Hilbert spaces of isometries corresponding to ρ_{iI_{0}} as in eq. (2) for the inclusion B(I_{0}) ⊂ A(I_{0}). If I is an interval and I ⊃ I_{0}, then A(I) is generated by B(I) and K_{i} as in (4).

If R_{i} ∈ K_{i} then α_{g}(R_{i}) = z_{ρ_{i}}(g)^{∗}R_{i}, for all g ∈ PSL(2, ℝ)˜ such that gI_{0} ⊂ I, where z_{ρ_{i}}(g) ∈ B(I) are unitaries in the local cocycle associated with ρ_{i} (12).

Proof By Corollary 18 we have (ρ_{iI}, θ_{I}) = (ρ_{iI_{0}}, θ_{I_{0}}) ⊂ B(I_{0}), therefore by formula (3) and Lemma 15 K_{i} ⊂ A(I_{0}) is also the Hilbert space associated with B(I) ⊂ A(I) as in (4). By Lemma 1 A(I) is then generated by B(I) and the K_{i}’s.

Let g ∈ PSL(2, ℝ)^{}˜ such that gI_{0} ⊂ I. Then α_{g} · ρ_{i} · α_{g^{−1}} is localized in I and α_{g} · ρ_{i} · α_{g^{−1}} = Adz_{ρ_{i}}(g)^{∗} · ρ_{i} by formula (14) and Cor. 18, where z_{ρ_{i}}(g) ∈ B(I). Therefore α_{g}(K_{i}) = z_{ρ_{i}}(g)^{∗}K_{i}, namely α_{g}(R_{i}) = z_{ρ_{i}}(g)^{∗}D(g)^{∗}R_{i} for all R_{i} ∈ K_{i}, where D(g) ∈ B(K_{i}). It is immediate to check that D locally satisfies the cocycle property with respect to α_{g}^{ρ_{i}} ≡ Adz_{ρ_{i}}(g) · α_{g}, namely D(gh) = D(g)α_{g}^{ρ_{i}}(D(h)) for g, h in a suitable neighborhood of PSL(2, ℝ)^{}˜ . But α^{ρ_{i}}↾_{B(K_{i})} is a finite-dimensional representation of PSL(2, ℝ)^{}˜ , thus it must be trivial because PSL(2, ℝ)^{}˜ has no nontrivial unitary finite-dimensional representations. Thus D is a local finite-dimensional unitary representation of PSL(2, ℝ)^{}˜ on K_{i}, so D is again trivial.

Therefore α_{g}(R_{i}) = z_{ρ_{i}}(g)^{∗}R_{i}, for all g ∈ PSL(2, ℝ)˜ such that gI_{0} ⊂ I as desired. □

If A is a conformal net on a Hilbert space H and B is a conformal subnet, we shall set B′ ≡ (∨_{L∈I}B(L))′, where the commutants are taken on H. For completeness we mention the following partial extension of Cor. 18 to the infinite index case, although it is not used in this paper.

Proposition 20. Let A be a local irreducible conformal net on S^{1} and B ⊂ A a strongly additive conformal subnet. The following are equivalent:

(i) The identity representation of B on H contains the vacuum representation of B with multiplicity one.

(ii) The identity representation of B on H contains the vacuum representation of B with finite multiplicity.

(iii) B^{c} = ℂ, where B^{c}(I) ≡ B′ ∩ A(I).

Proof (iii) ⟹ (i). By Proposition 17 we have to show that the intertwiner space between the representation θ on H_{B} and the identity representation of B on H_{B} is onedimensional. If θ is localized in I then, by Haag duality, any such intertwiner belongs (θ_{I}, ι) namely it belongs to B(I) and intertwines θ_{I} and the identity automorphism of B(I). But (θ_{I}, ι) is one-dimensional because B(I) ⊂ A(I) is an irreducible inclusion of infinite factors with a normal conditional expectation [31, 16].

(i) ⟹ (ii) is obvious, we show (ii) ⟹ (iii). Denote by π the subrepresentation of the identity representation of B on H corresponding to the vacuum representation and K ⊂ H the corresponding subspace. Then we have a decomposition K = H_{0} ⊗ H′_{0} and π = π_{0} ⊗ id where π_{0} is irreducible and dimH′_{0} < ∞ The representation U of PSL(2, ℝ) decomposes as U = U_{0} ⊗ U′_{0} and, since PSL(2, ℝ) has no non-trivial unitary representation, U′_{0} is the identity on H′_{0}. As Ω is unique U-invariant, we then have dimH′_{0} = 1, namely (i) holds.

Thus K = H_{B} and the projection E onto H_{B} belongs to the center of B′. In particular if b ∈ B^{c}(I) then b commutes with E. But E implements the expectation ε_{I} and b ∈ A(I), so b belongs to the center of B(I), thus b is a scalar. □

3.4 Intermediate subnets

Let A be a local irreducible conformal net and B a conformal subnet with finite index. We now show that there exists a one-to-one correspondence between ℑ(B(I_{0}), A(I_{0})) for a fixed interval I_{0} and the set of intermediate conformal nets between B and A.

Theorem 21. Let I_{0} be a fixed interval of S^{1} and R be an intermediate subfactor between B(I_{0}) and A(I_{0}). There exists a unique conformal subnet L on A with B(I) ⊂ L(I) ⊂ A(I) and L(I_{0}) = R.

Proof Let Λ_{I} denote as before the one-parameter group of special conformal transformations preserving I. As is easily seen Λ_{I}(ℝ) is exactly the subgroup of PSL(2, ℝ) of those g with gI = I. Then t → β_{t} = AdU(Λ_{I_{0}}(t))↾_{A(I_{0})} is a one-parameter automorphism group of A(I_{0}) leaving B(I_{0}) globally invariant, hence β_{t}(R) = R by Corollary 4.

Now, for I ∈ I we set

L(I) ≡ α_{g}(R) ,

where g ∈ PSL(2, ℝ) is a Möbius transformation such that I_{0} = gI and α_{g} ≡ AdU(g). L(I) is indeed well-defined because if h ∈ PSL(2, ℝ) is any other element with I_{0} = hI, then h^{−1}gI_{0} = I_{0}, thus h^{−1}g = Λ_{I_{0}}(t) for some t ∈ ℝ and α_{h^{−1}g}(R) = β_{t}(R) = R, thus α_{g}(R) = α_{h}(R).

As L(I) ⊂ A(I) and A is local, L(I_{1}) and L(I_{2}) clearly commute if I_{1} ∩ I_{2} = ∅. To show that I ∈ I → L(I) is a conformal net we need to check the isotony property, namely that L(I_{1}) ⊂ L(I) if I_{1} ⊂ I are intervals. By conformal invariance we may assume that I_{1} = I_{0} and that I = g_{0}I_{0} for some g_{0} ∈ PSL(2, ℝ) and then we need to show that α_{g_{0}}(R) ⊃ R.

Now by Corollary 19 A(I) is generated by B(I) and Hilbert spaces of isometries K_{i} ∈ A(I_{0}) corresponding to the expansion (4). Moreover

α_{g_{0}}(R_{i}) = z_{ρ_{i}}(g_{0})^{∗}R_{i}, ∀R_{i} ∈ K_{i} ,

where z_{ρ_{i}}(g_{0}) ∈ B(I) by Corollary 19. With K′_{i} ≡ K_{i} ∩ R, by Lemma 19 R is generated by B(I_{0}) and the K′_{i}’s, thus

α_{g_{0}}(R) = {α_{g_{0}}(B(I_{0})), α_{g_{0}}(K′_{i})}′′ = {B(I), z_{ρ_{i}}(g_{0})^{∗}K′_{i}}′′ = {B(I), K′_{i}}′′ ⊃ {B(I_{0}), K′_{i}}′′ = R

as desired. □

3.5 Complete rationality is hereditary

To simplify notations, given two different points a, b of S^{1}, we shall write [a, b] for the closure of the set of all z ∈ S^{1} that follow a and precede b in the counterclockwise order, and by (a, b) the interior of [a, b]. Two intervals I_{1}, I_{2} will be called adjacent if there are three different points a, b, c ∈ S^{1} such that Ī_{1} = [a, b], Ī_{2} = [b, c] and Ī_{1} ∪ Ī_{2} belongs to I.

Let A be a conformal net on S^{1}. Recall that A is strongly additive if

A(I_{1}) ∨ A(I_{2}) = A(I) ,

if I_{1}, I_{2} are adjacent intervals and I = Ī_{1} ∪ Ī_{2} ∈ I.

A is split if A(I_{0}) ⊂ A(I) is a split inclusion of von Neumann algebras, namely A(I_{0}) ∨ A(I)′ is naturally isomorphic to A(I_{0}) ⊗ A(I)′, if I_{0}, I ∈ I and Ī_{0} contained in the interior of I.

If A is split and I_{1}, I_{2} are intervals with disjoint closures, then A(I_{1}) ∨ A(I_{2}) ≃ A(I_{1})⊗A(I_{2}) is a factor and we shall denote by μ_{A} the index of the 2-interval inclusion A(I_{1}) ∨ A(I_{2}) ⊂ (A(I_{3}) ∨ A(I_{4}))′ where I_{3}, I_{4} are the two connected components of S^{1} ∖ (I_{1} ∪ I_{2}).

We shall say that A is completely rational if A is split, strongly additive and the index μ_{A} < ∞ where the I_{i}’s are intervals as above.

Lemma 22. Let A be an irreducible local conformal net on S^{1} and B ⊂ A a finiteindex conformal subnet. Then A is split and μ_{A} < ∞ iff B is split and μ_{B} < ∞.

In this case the relation μ_{B} = [A : B]^{2}μ_{A} holds.

Proof The implication A split ⟹ B split is rather immediate. The rest is proved in [14, Prop. 24]. □

Lemma 23. Let A be an irreducible local conformal net on S^{1} and B ⊂ A a finiteindex conformal subnet. Then A is strongly additive if B is strongly additive.

Proof Let I_{1}, I_{2} be adjacent intervals with I ≡ Ī_{1} ∪ Ī_{2} and let T ∈ A(I_{1}) be the isometry in the Q-system for γ_{I_{1}} as in Lemma 15. Then, by applying Lemma 15, T is also the isometry in the Q-system associated with γ_{I}. In particular A(I_{1}) = B(I_{1})T and A(I) = B(I)T, thus

A(I_{1}) ∨ A(I_{2}) = B(I_{1})T ∨ A(I_{2}) ⊃ {B(I_{1}) ∨ B(I_{2}), T }′′ = {B(I), T }′′ = A(I) .

This concludes the proof. □

Theorem 24. Let A be an irreducible local conformal net on S^{1} and B ⊂ A a conformal subnet with [A : B] < ∞. Then A is completely rational iff B is completely rational.

We postpone the proof of this theorem to Subsection 3.5.2.

3.5.1 Complete rationality of coset models

We begin with the following simple lemma.

Lemma 25. Let A_{1} and A_{2} be irreducible local conformal nets on S^{1}. Then A_{1} ⊗A_{2} is split (resp. strongly additive; completely rational) iff both A_{1} and A_{2} are split (resp. strongly additive; completely rational).

Proof All this can be checked directly, see [13, 14]. □

Let A be an irreducible local conformal net on S^{1} and B ⊂ A a conformal subnet. Then

B^{c} : I ∈ I → B^{c}(I) ≡ B′ ∩ A(I)

is clearly a conformal subnet of A and is called the coset net associated with B ⊂ A, cf. [42]. Also B ∨ B^{c} : I ∈ I → B(I) ∨ B^{c}(I) is then a conformal subnet of A.

Following F. Xu [42], we call B cofinite in A if [A : B ∨ B^{c}] < ∞. Indeed in [42] B is strongly additive so B^{c}(I) = B(I)′ ∩ A(I) is the relative commutant of B(I) in A(I).

By Takesaki theorem B(I) ∨ B^{c}(I) is naturally isomorphic to the von Neumann tensor product B(I) ⊗ B^{c}(I).

Corollary 26. Let A be an irreducible local conformal net on S^{1} and B ⊂ A a cofinite conformal subnet. With the above notations, A is completely rational iff both B and B^{c} are completely rational.

If A is ‘split & strongly additive’, so are B and B^{c}.

Proof It is enough to apply Theorem 24 and Lemma 25 and Proposition 34 below. □

To give a first application, suppose now that the net A is diffeomorphism invariant. Then one can consider the conformal subnet A_{Vir(c)} ⊂ A, which is associated with the vacuum representation of the Virasoro algebra with central charge c > 0 see e.g. [8].

Corollary 27. Let A be split, strongly additive and diffeomorphism invariant. If A_{Vir(c)} is cofinite in A, then c ≤ 1.

Proof If A_{Vir(c)} is cofinite in A then by Cor. 26 also A_{Vir(c)} is strongly additive, which is not the case if c > 1 [8]. □

We now turn our attention to coset models. Let G be simply connected semisimple compact Lie group of type A, i.e. G = SU(N_{1})×SU(N_{2})×⋯×SU(N_{n}). If H ⊂ G is a closed subgroup, there is a corresponding inclusion of loop groups LH ⊂ LG. Then the vacuum representation of LG at level k (see [37]) gives an inclusion of conformal nets denoted by H ⊂ G_{k} (where H may also have a suffix denoting the appearing level). Thanks to results of Xu [42, Corollary 3.1], see also the correction in [43], the inclusions of conformal nets associated with

(i) G_{k_{1}+k_{2}+···+k_{m}} ⊂ G_{k_{1}} × G_{k_{2}} × ⋯ G_{k_{m}} where the inclusion is diagonal, k_{i} ∈ ℕ, i = 1, …, m and G = SU(n),

(ii) H_{ℓk} ⊂ G_{ℓ}, if H_{k} ⊂ G_{1} is a conformal inclusion, where k is the Dynkin index, ℓ ∈ ℕ, H is simple and of type A and G is simple,

(iii) H ⊂ G_{m}, where H is the Cartan subgroup of G,

are all cofinite. So we have the following corollary.

Corollary 28. The coset subnets corresponding to the inclusions of nets in (i), (ii), (iii) of the above Xu’s list are completely rational.

Proof As the conformal net SU(N)_{k} is completely rational [44] (the proof of the strong additivity is contained in [40]), and the subnets in Xu’s list are cofinite [42, 43], it is then enough to apply Theorem 24. □

It then follows from [14] that for the above coset nets the tensor category of all represenations is rational and modular, as shown in [43], and the results in [14] apply.

3.5.2 Proof of Theorem 24

The remaining and more difficult part to prove in Theorem 24 is that A split and strongly additive implies that B is strongly additive. In the following we thus assume that A is split and strongly additive and prove that B is strongly additive in a series of Lemmata. The starting argument is similar to the one in [45].

Lemma 29. Let A be an irreducible, split and strongly additive, local conformal net on S^{1}. If B ⊂ A a conformal subnet with [A : B] < ∞ and I_{1} and I_{2} are adjacent intervals, I = Ī_{1} ∪Ī_{2}, then A(I_{1})∨B(I_{2}) ⊂ A(I) is a finite-index irreducible inclusion of factors.

Proof First notice that

(A(I_{1}) ∨ B(I_{2}))′ ∩ A(I) = A(I_{1})′ ∩ A(I) ∩ B(I_{2})′ = A(I_{2}) ∩ B(I_{2})′ = ℂ ,

where A(I_{1})′ ∩ A(I) = A(I_{2}) because A is strongly additive. Thus A(I_{1}) ∨ B(I_{2}) ⊂ A(I) is an irreducible inclusion of factors.

To show that [A(I) : A(I_{1}) ∨ B(I_{2})] < ∞ we set I_{1} = [a, b], I_{2} = [b, c] and take intervals I_{2}^{n} = [b_{n}, c], where lenghtI_{2}^{n} ↗ lenghtI_{2}. Then, by the split property, N_{n} ≡ A(I_{1})∨B(I_{2}^{n}) ⊂ M_{n} ≡ A(I_{1})∨A(I_{2}^{n}) is isomorphic to A(I_{1})⊗B(I_{2}^{n}) ⊂ A(I_{1})⊗A(I_{2}^{n}) and thus [M_{n} : N_{n}] = λ^{−1}. As

N_{n} ↗ A(I_{1}) ∨ B(I_{2}), M_{n} ↗ A(I_{1}) ∨ A(I_{2}) = A(I),

we have [A(I) : A(I_{1}) ∨ B(I_{2})] ≤ λ^{−1} by [14, Proposition 3]. □

As in Xu’s proof for the group case, we consider two adjacent intervals I_{1}, I_{2} and set I = Ī_{1} ∪ Ī_{2}. Then [A(I) : A(I_{1}) ∨ B(I_{2})] < ∞, and we consider an expectation μ : A(I) → A(I_{1}) ∨ B(I_{2}).

Then R(I_{2}) ≡ μ(A(I_{2})) is contained in A(I_{1})′ ∩A(I) and the latter coincides with A(I_{2}) because A is strongly additive [24]. Hence R(I_{2}) is a von Neumann algebra and B(I_{2}) ⊂ R(I_{2}) ⊂ A(I_{2}). The following lemma is contained in [45].

Lemma 30. If R(I_{2}) = A(I_{2}) then B is strongly additive.

Proof If R(I_{2}) = A(I_{2}) then

μ(A(I)) ⊃ A(I_{1}) ∨ R(I_{2}) = A(I_{1}) ∨ A(I_{2}) = A(I) ,

thus μ is the identity and A(I_{1}) ∨ B(I_{2}) = A(I). We then have

B(I) = ε_{I}(A(I)) = ε_{I}(A(I_{1}) ∨ B(I_{2})) = ε_{I}(A(I_{1})) ∨ B(I_{2}) = B(I_{1}) ∨ B(I_{2}) .
□

Lemma 31. Assume R(I_{2}) = B(I_{2}). Given intervals L_{0} ⊂ L, L̄_{0} ≠ L̄, and ϵ > 0 there exists a projection e ∈ A(L) such that

ε_{L_{0}}(a)e = eae, ∀a ∈ A(L_{0}) ,  (20)
(eΩ, Ω) > 1 − ϵ .  (21)

Proof As B(I_{2})′ ∩A(I_{2}) = ℂ, there exists a unique expectation of A(I_{2}) onto B(I_{2}), thus

μ↾_{A(I_{2})} = ε_{I_{2}}

is the vacuum preserving conditional expectation.

In order to show the Lemma we can clearly assume that L_{0} and L have one common boundary point. As the Möbius group acts transitively on the family of three different points of S^{1}, property (20,21) does not depend on the choice of the pair L_{0} ⊂ L.

Let e ≠ 0 be a projection in B(H) implementing μ namely

μ(a)e = eae, a ∈ A(I).

As μ acts identically on A(I_{1}), we have e ∈ A(I_{1})′ = A(I′_{1}).

Setting L = I′_{1}, L_{0} = I_{2} we then have: L ⊃ L_{0} are intervals with one common boundary point and there exists a non-zero projection e ∈ A(L), such that the property in formula (20) holds, i.e.

e ≠ 0 & ε_{L_{0}}(a)e = eae, ∀a ∈ A(L_{0}) .  (22)

Clearly the above property (22) is a fortiori true if we replace L with a larger interval and L_{0} with a smaller interval.

Set L_{0} = [a, b], L = [a, c] and choose sequences of points b_{n} ∈ (a, b) and c_{n} ∈ (c, a) in S^{1}, such that lenght[a, b_{n}] ↘ 0 and lenght[a, c_{n}] ↗ 2π.

As [b_{n}, c_{n}] is an increasing sequence of intervals and ∪_{n}[b_{n}, c_{n}] is dense in S^{1}, it follows that ∪A([b_{n}, c_{n}]) is strongly dense in B(H) (this is a consequence of Haag duality and the factoriality of the local von Neumann algebras). Therefore the unitaries of ∪_{n}A([b_{n}, c_{n}]) are strongly dense in the unitaries of B(H). Given ϵ > 0, then there exists an integer n and a unitary u ∈ A([b_{n}, c_{n}]) such that

(euΩ, uΩ) > 1 − ϵ .

Replacing L with [a, c_{n}], L_{0} with [b_{n}, c_{n}] and e with e′ ≡ u^{∗}eu, equation (21) clearly holds. But also equation (20) is satisfied because

ε_{L_{0}}(a)e′ = u^{∗}ε_{L_{0}}(a)eu = u^{∗}eaeu = u^{∗}euau^{∗}eu = e′ae′, a ∈ A(L_{0}) ,

as u commutes with A(L_{0}). □

Lemma 32. R(I_{2}) ≠ B(I_{2}) unless B = A.

Proof Let’s assume R(I_{2}) = B(I_{2}). Note that by Möbius covariance the equality R(I_{2}) = B(I_{2}) is independent of the choice of I_{1}, I_{2}.

Let I be a fixed interval and I_{n} a decreasing sequence of intervals with a common boundary point with I such that ∩_{n}I_{n} = I and choose a projection e_{n} ∈ A(I_{n}) such that

ε_{I}(a)e_{n} = e_{n}ae_{n}, ∀a ∈ A(I) & (e_{n}Ω, Ω) > 1 − \frac{1}{n} .  (23)

Let e be a weak limit point of {e_{n}}. Then e ∈ ∩A(I_{n}) = A(I) and e ∈ B(I)′, thus e is a scalar, 0 ≤ e ≤ 1. As (e_{n}Ω, Ω) > 1 − \frac{1}{n}, we have e = 1. Thus e_{n} → 1 weakly. As the weak and strong topologies coincide on the set of selfadjoint projections, e_{n} → 1 strongly. Going to the limit in eq. (23) we then have ε_{I}(a) = a, a ∈ A(I), namely B(I) = A(I). □

Lemma 33. Let A be a local irreducible conformal net and B a finite-index subnet. Suppose A is split and strongly additive and B is not strongly additive. Then there exists an intermediate conformal net B ⊂ L ⊂ A such that L is split and strongly additive and L ≠ A.

Proof We use the above notations. By Lemma 32 R(I_{2}) ≠ B(I_{2}) and Lemma 30 R(I_{2}) ≠ A(I_{2}). By Theorem 21 there exists a conformal subnet R intermediate between B and A such that the associated local von Neumann algebra R(I_{2}) is such a factor. Set R_{1} ≡ R. Replacing B by R_{1} and repeating the construction we find a factor R_{2}(I_{2}) between R_{1}(I_{2}) and A(I_{2}). Iterating the procedure we get a sequence of factors R_{n}(I_{2}), coming from a conformal subnets R_{n}, such that

B(I_{2}) ⊂ R_{1}(I_{2}) ⊂ R_{2}(I_{2}) ⊂ ⋯ ⊂ A(I_{2}) .

As [A(I_{2}) : B(I_{2})] < ∞, after finitely many steps the iteration stabilizes, so let n be the smallest integer such that R_{n}(I_{2}) = R_{n+1}(I_{2}). Then n ≥ 2 by the above discussion.

By Lemma 32 we then have R_{n}(I_{2}) = A(I_{2}). Thus L ≡ R_{n−1} is strongly additive by Lemma 30. L is clearly split and, by construction, properly between B and A.

□

Proposition 34. Let A be a local irreducible conformal net and B a finite-index subnet. If A is split and strongly additive then B is split and strongly additive.

Proof Clearly B is split. Suppose that B is not strongly additive. Then by Lemma 33 there exists a strongly additive conformal subnet L_{1} intermediate between B and A such that L_{1} ≠ A. Again by Lemma 33 there exists a strongly additive conformal subnet L_{2} intermediate between B and L_{2} such that L_{2} ≠ L_{1}. By iteration we find a sequence of subnets

A ⊃ L_{1} ⊃ L_{2} ⊃ ⋯ ⊃ B ,

where all inclusions are proper, thus [L_{n+1} : L_{n}] ≥ 2 by Jones’ theorem [27]. So [A : B] = ∞ by the multiplicativity of the index [29], contrary to our assumptions.

□

Proof of Theorem 24 It is now sufficient to apply Lemma 22, Lemma 23 and Proposition 34. □

3.5.3 Case of a Fermi net

Most of our analysis extends to the case of non-local Fermi conformal nets. As there are several examples of local conformal finite-index subnets of Fermi nets, we sketch here how to modify our arguments and reduce to the local situation.

In this subsection A is a twisted local irreducible net of von Neumann algebras on S^{1}, namely A an irreducible net satisfying all properties A to E in Section 3.1, except B which is now replaced by

B′. Twisted locality. There exists a unitary Z commuting with the unitary representation U such that ZΩ = Ω and

ZA(I′)Z^{∗} ⊂ A(I)′

for all intervals I.

The basic results for local nets (modular structure, duality, etc.) have a version for twisted local nets, see [11].

We shall say that a conformal net A is a Fermi net if there exists a self-adjoint unitary V on H such that V Ω = Ω and β(A(I)) = A(I), I ∈ I, where β ≡ AdV, with canonical commutation relations: if I_{1}, I_{2} are disjoint intervals then the commutator or anti-commutator

[a_{1}, a_{2}]_{±} = 0, a_{i} ∈ A(I_{i}) ,

if β(a_{i}) = ±a_{i}; the commutator vanishes if one of the a_{i}’s is a Bose operator (β(a_{i}) = a_{i}) and the anti-commutator vanishes if both the a_{i} are Fermi operators (β(a_{i}) = −a_{i}).

A conformal Fermi net satisfies twisted locality, hence twisted duality, where the unitary Z is given by

Z = \frac{1 + iV}{1 + i} ,

in particular ZbZ^{∗} = b, for all b ∈ ∨_{I}B_{b}(I), see [22], where we denote by B_{b} the Bose subnet A^{β} of A. Note that [A : B_{b}] = 2 and B_{b} is a local conformal net.

Lemma 35. Let B be a local subnet of A. Then B ⊂ B_{b}.

Proof This is obvious since otherwise each B(I) would contain non-zero Fermi operators and these do not commute if they are localized in disjoint intervals. □

Due to the above lemma, the results in the previous sections extend to the case of a local finite-index subnet B of A once we show them in the particular case B = B_{b}. We give here explicitly the extension of Theorem 24

Proposition 36. Let A be an irreducible Fermi conformal net on S^{1} and B ⊂ A a local conformal subnet with [A : B] < ∞. Then A is split and strongly additive iff B is split and strongly additive.

Proof By the above discussion we may assume that B is the Bose subnet. We assume that A is split and strong additive and show that B strong additive, the other implications are obtained essentially as in the local case.

It is enough to show that A(I_{1}) ∨ B(I_{2}) = A(I) if I_{1}, I_{2} are adjacent intervals and I = Ī_{1} ∪ Ī_{2}. The inclusion A(I_{1}) ∨ B(I_{2}) ⊂ A(I) has finite index as in the local case and we assume A(I_{1}) ∨ B(I_{2}) ≠ A(I). We consider an expectation μ : A(I) → A(I_{1}) ∨ B(I_{2}). Let u_{i} ∈ A(I_{i}) be Fermi unitaries. Then Adu_{2} implements an automorphism of B(I_{2}), acts trivially on B(I_{1}) and Adu_{2}(u_{1}) = −u_{1}, therefore Adu_{2} implements an automorphism of A(I_{1})∨B(I_{2}). As A(I) is generated by A(I_{1})∨B(I_{2}) and u_{2}, it follows that A(I) is the crossed product of A(I_{1}) ∨ B(I_{2}) by Adu_{2}, thus μ acts trivially on A(I_{1}) and μ(u_{2}) = 0, so μ(A(I_{2})) = B(I_{2}).

Let e ≠ 0 be a projection implementing μ. Then e ∈ ZA(I′_{1})Z^{∗}. Arguing as in the local case e ∈ A(I_{2})′ ∩ ZA(I_{2})Z^{∗}, but

A(I_{2})′ ∩ ZA(I_{2})Z^{∗} = Z(Z^{∗}A(I_{2})′Z ∩ A(I_{2}))Z^{∗} ⊂ Z(Z^{∗}B(I_{2})′Z ∩ A(I_{2}))Z^{∗} = Z(B(I_{2})′ ∩ A(I_{2}))Z^{∗} = ℂ .

so e = 1 and μ is trivial, which contradicts the assumption A(I_{1}) ∨ B(I_{2}) ≠ A(I). □

Remark. In this paper the positivity of the energy has been used only indirectly, essentially to entail the Reeh-Schlieder property and the factoriality of the local algebras. Thus our results extend to the case of conformal nets on S^{1} with the above properties, without assuming the positivity of the conformal Hamiltonian. We encounter this situation if we consider a local conformal net A on the 1+1-dimensional Minkowski spacetime and look at the corresponding time zero net A_{t=0}. In particular, if A is a finite-index local extension of a 1 + 1-dimensional chiral net A_{1} ⊗ A_{2}, as discussed in [38], then A_{t=0} is split and strongly additive iff both A_{1} and A_{2} are split and strongly additive.

Acknowledgements. This work began while the author was visiting the Mathematical Sciences Research Institute, Berkeley, in November-December 2000, during the Operator Algebra program and he wishes to thank the organizers for the kind invitation and the warm hospitality. He also thanks M. Müger, V. Toledano and F. Xu for stimulating conversations and S. Carpi and R. Conti for critical reading of the manuscript.

References

[1] H. Araki, “Mathematical Theory of Quantum Fields”, International series of monographs on Physics 101, Oxford University Press 1999.

[2] P. Bertozzini, R. Conti & R. Longo, Covariant sectors and positivity of the energy, Commun. Math. Phys. 141 471-492 (1998).

[3] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994) 201–216.

[4] D. Bisch & V.F.R. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997) 89–157.

[5] J. Böckenhauer & D. E. Evans, Modular invariants, graphs and α-induction for nets of subfactors I-III, Commun. Math. Phys. 197 (1998) 361–386, 200 (1999) 57–103 & 205 (1999), 183–228. J. Böckenhauer, D. E. Evans & Y. Kawahigashi, Longo-Rehren subfactors arising from αinduction, math.OA/0002154 .

[6] R. Brunetti, D. Guido & R. Longo, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys. 156 (1993) 201–219.

[7] D. Buchholz & C. D’Antoni, Phase space properties of charged fields in theories of local observables, Rev. Math. Phys. 7, (1995) 527–557.

[8] D. Buchholz & H. Schulz-Mirbach, Haag duality in conformal quantum field theoery, Rev. Math. Phys. 2 (1990) 105.

[9] R. Conti, S. Doplicher & J.E. Roberts, Superselection theory for subsystems, Commun. Math. Phys. (to appear). S. Carpi & R. Conti, Classification of subsystems for local nets with trivial superselection structure, Commun. Math. Phys. 217, (2001) 89–106.

[10] E. Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979) 17–29.

[11] C. D’Antoni, F. Radulescu & R. Longo, Conformal nets, maximal temperature and models from free probability, J. Operator Theory 45, 195-208 (2001).

[12] S. Doplicher, R. Haag & J.E. Roberts, Local observables and particle statistics. I & II, Commun. Math. Phys. 23, 199-230 (1971) & 35, 49-85 (1974).

[13] S. Doplicher & R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 73 (1984) 493-536.

[14] Y. Kawahigashi, R. Longo & M. Müger, Multi-interval subfactors and modularity of representations in conformal field theory, Commun. Math. Phys. (to appear).

[15] D. E. Evans & Y. Kawahigashi, “Quantum Symmetries on Operator Algebras”, Oxford University Press, 1998.

[16] F. Fidaleo & T. Isola, The canonical endomorphism for infinite index inclusions, Z. Anal. und ihre Anwendungen 18 (1999), 47-66.

[17] K. Fredenhagen, Generalization of the theory of superselection sectors, in “The algebraic theory of superselection sectors” D. Kastlers ed. World Scientific 1990.

[18] K. Fredenhagen & M. Jörß, Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion, Commun. Math. Phys. 176 (1996), 541.

[19] J. Frölich & F. Gabbiani, Operator algebras and conformal field theory, Commun. Math. Phys. 155 (1993), 569–640.

[20] P. Goddard, A. Kent & D. Olive, Unitary represenations of the Virasoro and super-Virasoro algebras, Commun. Math. Phys. 103 (1986), 105-119.

[21] D. Guido & R. Longo, Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148 (1992), 521-551.

[22] D. Guido & R. Longo, An algebraic spin and statistics theorem, Commun. Math. Phys. 172, 517-533 (1995).

[23] D. Guido & R. Longo, The conformal spin and statistics theorem, Commun. Math. Phys. 181, 11 (1996).

[24] D. Guido, R. Longo & H.W. Wiesbrock, Extensions of conformal nets and superselection structure, Commun. Math. Phys. 192 217-244 (1998).

[25] R. Haag “Local Quantum Physics”, Springer-Verlag (1996).

[26] M. Izumi, R. Longo & S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 10, (1998) 25–63.

[27] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25.

[28] H. Kosaki, “Type III Factors and Index Theory”, Res. Inst. of Math., Lect. Notes 43, Seoul Nat. Univ. 1998.

[29] H. Kosaki & R. Longo, A remark on the minimal index of subfactors, J. Funct. Anal. 107 (1992), 458-470. R. Longo, Minimal index and braided subfactors, J. Funct. Anal. 109 (1992), 98-112.

[30] R. Longo, Simple injective subfactors, Adv. in Math. 63 (1986), 152-171.

[31] R. Longo, Index of subfactors and statistics of quantum fields I–II, Commun. Math. Phys. 126 (1989) 217–247 & 130 (1990) 285–309.

[32] R. Longo, A duality for Hopf algebras and for subfactors. I, Commun. Math. Phys. 159 (1994) 133–150.

[33] R. Longo, An analogue of the Kac-Wakimoto formula and black hole conditional entropy, Commun. Math. Phys. 186 (1997), 451-479.

[34] R. Longo & K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995) 567–597.

[35] M. Pimsner & S. Popa, Entropy and index for subfactors, Ann. Scient. Éco. Norm. Sup. 19 (1986) 57–106.

[36] S. Popa, Correspondences, INCREST manuscript, 1986.

[37] A. Pressley, G. Segal, “Loop Groups”, Oxford Science publ., Oxford 1986

[38] K.-H. Rehren, Canonical tensor product subfactors, Commun. Math. Phys. 211 (2000) 395-406. K.-H. Rehren, Chiral observables and modular invariants, Commun. Math. Phys. 208 (2000) 689-712.

[39] M. Takesaki, “Theory of Operator Algebras”, I Springer-Verlag 1979; II in preparation.

[40] V. Toledano, “Fusion of positive energy representations of LSpin_{2n}”, Ph.D. dissertation, Univ. of Cambridge, 1997.

[41] Y. Watatani, Lattices of intermediate subfactors, J. Funct. Anal. 140 (1996) 312–334.

[42] F. Xu, Algebraic coset conformal field theory, Commun. Math. Phys. 211 (2000) 1. F. Xu, Algebraic coset conformal field theory II, Publ. RIMS 35 (1999) 795-824.

[43] F. Xu, On a conjecture of Kac-Wakimoto, math.RT /9904098, appeared in Publ. RIMS.

[44] F. Xu, Jones-Wassermann subfactors for disconnected intervals, Commun. Contemp. Math. 2 (2000), 307–347.

[45] F. Xu, Algebraic orbifold conformal field theory, Preprint.