Characterization of 2D Rational Local Conformal Nets

and Its Boundary Conditions: the Maximal Case

Marcel Bischoff^{1}, Yasuyuki Kawahigashi^{2}, Roberto Longo^{3}

Received: September 2, 2015

Communicated by Joachim Cuntz

Abstract. Let A be a completely rational local Möbius covariant net on S^{1}, which describes a set of chiral observables. We show that local Möbius covariant nets B_{2} on 2D Minkowski space which contains A as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category DHR(A). The Möbius covariant boundary conditions with symmetry A of such a net B_{2} are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions.

To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric ∗-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren’s construction (generalized Longo-Rehren construction, α-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.

2010 Mathematics Subject Classification: 81T40, 18D10, 81R15, 46L37 Keywords and Phrases: Conformal Nets, Boundary Conditions, Qsystem, Full Center, Subfactors, Modular Tensor Categories.

1Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) by the DFG Research Training Group 1493 “Mathematical Structures in Modern Quantum Physics” until August 2014.

2Supported by the Grants-in-Aid for Scientific Research, JSPS.

3Supported in part by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, PRIN-MIUR and GNAMPA-INdAM.

1138 Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo

Contents

1 Introduction 1138

2 Preliminaries 1141 2.1 Endomorphisms of type III factors and Q-systems . . . . . . . 1141 2.2 UMTCs in End(N) and braided subfactors . . . . . . . . . . . 1146 2.3 Braided subfactors and α-induction . . . . . . . . . . . . . . . . 1148

3 Morita equivalence for braided subfactors 1149 3.1 Module categories, modules and bimodules . . . . . . . . . . . 1149 3.2 The Morita equivalence class of a braided subfactor . . . . . . . 1153

4 α-induction construction and the full center 1154 4.1 The full center and Rehren’s construction coincide . . . . . . . 1154 4.2 The adjoint functor of the full center . . . . . . . . . . . . . . . 1162

5 Modular invariance and Q-systems in _{N}C_{N} ⊠ \overline{_{N}C_{N}} 1164 5.1 Characterization of modular invariant Q-systems . . . . . . . . 1164 5.2 Permutation modular invariants . . . . . . . . . . . . . . . . . . 1164 5.3 Maximal chiral subalgebras and second cohomology for modular

invariant Q-systems . . . . . . . . . . . . . . . . . . . . . . . . 1166

6 Conformal nets 1168 6.1 Extensions and Q-systems . . . . . . . . . . . . . . . . . . . . . 1171 6.2 Representation theory of local extensions . . . . . . . . . . . . 1173 6.3 Maximal 2D nets with chiral observables A . . . . . . . . . . . 1173 6.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 1175 6.5 Reducible boundary conditions . . . . . . . . . . . . . . . . . . 1178 6.6 Adding the boundary . . . . . . . . . . . . . . . . . . . . . . . . 1180

1 Introduction

The subject of algebraic quantum field theory has led to many structural results and recently also to interesting constructions and classifications in quantum field theory. Conformal quantum field theory can be conveniently studied in this approach. In particular there is the notion of a conformal QFT on Minkowski space and boundary conformal QFT on Minkowski half-plane x > 0. One can associate with a boundary conformal QFT (boundary theory) a conformal QFT on Minkowski space (bulk theory), but in general several boundary theories can have the same bulk theory, which correspond to different boundary conditions of the bulk theory. In a different framework Fuchs, Runkel and Schweigert gave a general construction, the so-called TFT construction, of a (euclidean) rational full conformal field theory (CFT). The construction can be divided into two steps: first one chooses a certain vertex operator algebra (VOA), whose representation category C is a modular tensor category and which specifies chiral fields. This can be seen as the analytical part. Then with a choice of a special symmetric Frobenius algebra object A ∈ C one can construct correlators on an arbitrary Riemann surface. The bulk field content depends on the Morita equivalence class of A, while A itself fixes a boundary condition. Carpi, and two of the authors gave a general procedure starting from an algebraic quantum field theory on the Minkowski space, to obtain all locally isomorphic boundary conformal QFT nets, in other words to find all possible boundary conditions (with unique vacuum). The main purpose of this paper is to show that there is a similar classification for the boundary conditions for maximal (full) (conformal) local nets on Minkowski space and its boundary conditions as in the afore mentioned TFT construction. Let us consider more concretely a quantum field theory on Minkowski space. By introducing new coordinates x_{±} = t ∓ x we identify the two-dimensional Minkowski space 𝕄 = {(t, x) ∈ ℝ^{2}} with metric ds^{2} = dt^{2} − dx^{2} with the product L_{+} × L_{−} of two light rays L_{±} = {(t, x) : t ± x = 0} with metric ds^{2} = dx_{+}dx_{−}. The densities of conserved quantities (symmetries) are prescribed by left and right moving chiral fields, i.e. fields just depending on x_{+} or x_{−}, respectively. For example for the stress-energy tensor holds T_{00,01} = T_{+}(x_{+}) ± T_{−}(x_{−}) and for the conserved U(1)-current holds j_{0,1}(t, x) = j_{+}(x_{+}) ± j_{−}(x_{−}). In the algebraic setting such conserved quantities are abstractly given by a net A_{2}(O) = A_{+}(I) ⊗ A_{−}(J). In general, there can be other local observables, so the net of observables is a local extension B(O) ⊃ A_{2}(O) of A_{2}. We ask this extension to be irreducible (B(O)∩A_{2}(O)′ = ℂ·1), which is for example true if we assume that A_{2} contains the stress energy tensor of B. We will also assume that the algebras of left and right moving chiral fields are isomorphic, in other words A_{2}(O) = A(I) ⊗ A(J) where O = I × J ⊂ L_{+} × L_{−} and A is a local Möbius covariant net on ℝ. So in this case symmetries are prescribed by the net A. We further assume A to be completely rational, this is for example true for the net Vir_{c} generated by the stress energy tensor with central charge c < 1, SU(N) loop group models, or conformal nets associated with even lattices (lattice compactifications). The category of Doplicher–Haag–Roberts superselection sectors of a completely rational conformal net is a unitary modular tensor category [KLM01]. Fixing A we are, as a first step, interested in classifying all nets B “containing the symmetries described by A”, i.e. to classify all local extensions B_{2} ⊃ A_{2}. It turns out that the maximal ones are classified by Morita equivalence classes of chiral extensions A ⊂ B. Let us look a moment into nets defined on 𝕄_{+} = {(t, x) ∈ M : x > 0}, i.e. nets with a boundary at x = 0. We are interested to prescribe boundary conditions of B_{2} without flow of “charges” associated with A. The vanishing of the chargeflow across the boundary of the charges associated with A is encoded in the algebraic framework via the trivial boundary net A_{+}(O) = A(I) ∨ A(J) with I × J ∈ 𝕄_{+}. This net is locally isomorphic to A_{2} restricted to 𝕄_{+}. In other words A_{+} prescribes the boundary condition of A_{2} such that there is no charge flow across the boundary. Now given a two-dimensional net B_{2} which contains the given rational symmetries described by A, i.e. a local irreducible extension B_{2} ⊃ A_{2}, we are now interested in all boundary conditions with no charge flow associated with A as above. Such a boundary condition is abstractly given [LR04, CKL13] by a net B_{+} ⊃ A_{+} on 𝕄_{+} which is locally isomorphic to B_{2} such that this isomorphism restricts to an isomorphism of A_{+} ≅ A_{2}. A classification gets feasibile by operator algebraic methods. Finite index subfactors N ⊂ M are in one-to-one correspondence with algebra objects (Qsystems) in the unitary tensor category End(N) of endomorphisms of N. Local irreducible extension B ⊃ A of nets with finite index give rise to nets of subfactors A(O) ⊂ B(O) and the corresponding Q-system (up to isomorphism) is independent of O and is in the category of localized DHR endomorphisms. Conversely, every such Q-system gives a relatively local extension, which is local if and only if the Q-system is commutative. In particular, one has a one-to-one correspondence between Q-systems and relatively local extensions. This situation can be abstracted to the setting of braided subfactors, namely we fix an interval I, set N = A(I) and denote by _{N}C_{N} the category of localized DHR endomorphisms which are localized in I. We can start with a type III factor N and a modular tensor category _{N}C_{N} ⊂ End(I) and look into subfactors N ⊂ M such that the corresponding Q-system is in _{N}C_{N}. We introduce the notion of Morita equivalence of such braided subfactors. As a main technical result we show that a conjecture of Kong and Runkel [KR10] is true. Namely, we show in Prop. 4.18 that the generalized Longo–Rehren construction [Reh00] coincides with the full center construction in the categorical literature (e.g. [FFRS06,KR08]). We give some consequences on the study of braided subfactors and modular invariants. This result opens the possiblity to apply many results from the categorical literature to the braided subfactor and conformal net setting. In particular, we make use of the result that Q-systems are Morita equivalent if and only if they have the same full center [KR08]. Going back to the conformal net setting we get the main result. Namely, maximal 2D extensions B_{2} ⊃ A_{2} are classified by Morita equivalence classes of Q-systems in Rep(A) (see Prop. 6.7 and irreducible boundary conditions of B_{2} are classified by equivalence classes of irreducible Q-systems in the Morita class (see Prop.6.11). We also treat reducible boundary conditions, which were not conisidered before in the literature, and show that we get a classification by reducible Q-systems. The article is structured as follows. In Sec. 2 we give some background on the category of endomorphisms of a type III factor, Q-systems, unitary modular tensor categories (UMTC), braided subfactors and the α-induction construction.

In Sec. 3 we give a notion of Morita equivalence for subfactors and Q-systems in UMTCs. The Morita equivalence class of a subfactor in a UMTC can be described by irreducible sectors in the module category of the subfactor modulo automorphisms of some dual category. In Sec. 4 we show that the α-induction construction in subfactors coincide with the full center construction in the categorical literature. This is the first main technical result. In Sec. 5 we study maximal commutative Q-systems in the category_{N}C_{N} ⊠\overline{_{N}C_{N}} (the Drinfel’d center of_{N}C_{N}) and give a characterization of them. We give some application to the study of modular invariants and examples of inequivalent extensions with same modular invariant, i.e. example of non-vanishing second cohomology. In Sec. 6 we apply our former results to the study of conformal field theory on the Minkowski space in the operator algebraic (Haag–Kastler) framework. We give a proof of a folk theorem about the representation theory of local extensions (Prop. 6.4). Given a completely rational conformal net A, as the main result, we obtain a classification of maximal local CFTs containing the chiral observables described by A and all its boundary conditions. We also discuss reducible boundary conditions, i.e. we drop the assumption that the boundary condition possesses a unique vacuum. Finally, we give a relation to the construction of adding a boundary in [CKL13], which gives an alternative proof for the classification of boundary conditions.

2 Preliminaries

2.1 Endomorphisms of type III factors and Q-systems

Let us look into the following strict 2–C^{∗}-category C. Its 0-cells Ob(C) = {N, M, P, …} are given by a (finite) set of type III factors. The 1-cells are given for M, N ∈ Ob(C) by Mor(M, N), i.e. the set of unital ∗-homomorphisms (morphism) from ρ : M → N with finite (statistical) dimension dρ ≡ d_{ρ} = [N : ρ(M)]^{\frac{1}{2}} , where [N : ρ(M)] denotes the minimal index [Jon83,Kos86]. The 2-cells are intertwiners, i.e. for λ, μ ∈ Mor(M, N) we define Hom(λ, μ) = {t ∈ N : tλ(m) = μ(m)t for all m ∈ M}. Then Hom(λ, μ) is a vector space and we write ⟨λ, μ⟩ = dim Hom(λ, μ) for its dimension. Let ρ ∈ Mor(M, N). We call ρ irreducible if ρ(M)′ ∩ N = ℂ · 1_{N}. A sector is a unitary equivalence class [ρ] = {Ad U ∘ ρ : U ∈ N unitary}. We denote by End(N) = Mor(N, N), which is a 2–C^{∗}-category with only one 0-cell, so a C^{∗}-tensor category. Let ρ_{1}, …, ρ_{n} ∈ Mor(M, N), and let r_{i} ∈ N be generators of the Cuntz algebra O_{n}, i.e. ∑_{i=1}^{n} r_{i}r_{i}^{∗} = 1_{N} and r_{j}^{∗}r_{i} = δ_{ij} · 1_{N}. The morphism

ρ = ∑_{i=1}^{n} Ad r_{i} ∘ ρ_{i} ∈ Mor(M, N),

is called direct sum of ρ_{1}, …, ρ_{n} and we have r_{i} ∈ Hom(ρ_{i}, ρ). The direct sum is unique on sectors and we write it as

[ρ] =: [ρ_{1}] ⊕ ⋯ ⊕ [ρ_{n}] =: ⨁_{i=1}^{n}[ρ_{i}] ,

and for the multiple direct sum we introduce the notation:

n[σ] := ⨁_{i=1}^{n}[σ] ,  n ∈ ℕ, σ ∈ Mor(M, N) .

We say that a full and replete subcategory C of Mor(M, N) has subobjects, if every object is a finite direct sum of irreducible sectors in C. Similarly, we say it has direct sums, if ρ_{1}, …, ρ_{n} ∈ C implies that also their direct sum is in C. Let us assume C has subobjects. If e ∈ Hom(ρ, ρ) is a (not necessarily orthogonal) projection (idempotent), then there exists a ρ′ ∈ C and s ∈ Hom(ρ′, ρ) and t ∈ Hom(ρ, ρ′) such that s · t = e and t · s = 1_{ρ′} ≡ 1_{N}. We note that if we have e ∈ Hom(θ, θ) we have an orthonormal projection p = e(1 + e − e^{∗})^{−1} ∈ Hom(θ, θ) with the same range. If [ρ] = ⨁_{i=1}^{m}[ρ_{i}] and [σ] = ⨁_{j=1}^{n}[σ_{j}] we can decompose t ∈ Hom(ρ, σ) as

t = ⨁_{ij} t_{ij} := s_{i} · t_{ij} · r_{i}^{∗},  t_{ij} ∈ Hom(ρ_{i}, σ_{j}) ,

where r_{i} ∈ Hom(ρ_{i}, ρ) and s_{j} ∈ Hom(σ_{j}, σ) are isometries as above. Similarly, one can decompose t ∈ Hom(ρ, στ) etc. Let us briefly explain the graphical notation (string diagrams) [JS91, BEK99, BEK00, Sel11, BDH14] which we will use. The 0-cells N, M, … are drawn as shaded two-dimensional regions, with different shadings for each factor. A 1cell ρ ∈ Mor(N, M) is a vertical line (one dimensional) between the region M and N and composition of 1-cells correspond to horizontal concatenation. The identity id_{N} ∈ End(N) is not drawn. The 2-cells t ∈ Hom(ρ, σ) are vertices between two lines. Sometimes we draw also boxes and again the identity 1_{ρ} ≡ 1 ∈ Hom(ρ, ρ) is in general not drawn. The composition of intertwiners is vertical concatenation and the monoidal product horizontal concatenation. We use a Frobenius rotation invariant convention for trivalent vertices, namely for an isometry e ∈ Hom(ν, λμ) we introduce the diagram

\begin{matrix}ν \\ μ \end{matrix} ; \begin{matrix}e \\ λ \end{matrix} =: \sqrt[4]{\frac{dλdμ}{dν}}e .

Let C ⊂ End(N) and D ⊂ End(M) be two full subcategories. We define the Deligne product C ⊠ D to be the completion of C ⊗_{ℂ} D under subobjects and direct sums cf. [LR97, Appendix].

A morphism ρ̄: N → M is said to be a conjugate to ρ: M → N if there exist intertwiners R ∈ (id_{M}, ρ̄ρ) and R̄ ∈ (id_{N}, ρρ̄) such that the conjugate equations hold:

(1_{ρ} ⊗ R^{∗}) · (R̄ ⊗ 1_{ρ}) ≡ ρ(R^{∗}) · R̄ = 1_{ρ}  (1)
(1ρ̄ ⊗ R̄^{∗}) · (R ⊗ 1ρ̄) ≡ ρ̄(R̄^{∗}) · R = 1ρ̄ .  (2)

The 2–morphisms R, R̄ will graphically be represented by

R̄ = ρ ; ρ̄ ; id_{N} R = ρ̄ ; ρ ; id_{M}

and the above equations (1), (2) are sometimes called zig-zag identities, because in diagrams they are given by

\begin{matrix}ρ \\ ρ \end{matrix} = \begin{matrix}ρ \\ ρ \end{matrix} ,  ρ̄ ; ρ̄ = ρ̄ ; ρ̄ .

If ρ is irreducible we ask the solution R, R̄ to be normalized, i.e. ∥R∥ = ∥R̄∥. In the case that ρ is not irreducible we further ask that R, R̄ is a standard solution of the conjugate equation, i.e. R (and similar R̄) is of the form

R = ∑_{i}(W̄_{i} ⊗ W_{i}) · R_{i} ≡ ⨁_{i} R_{i} ,

where R_{i} ∈ (id_{M}, ρ̄_{i}ρ_{i}) is a normalized solution for an irreducible object ρ_{i} ≺ ρ and W_{i} ∈ (ρ_{i}, ρ) and W̄_{i} ∈ (ρ̄_{i}, ρ) are isometries expressing ρ and ρ̄ as direct sums of irreducibles. We note that for the dimension d_{ρ} ≡ dρ of ρ we have R^{∗}R = d_{ρ} · 1_{M} and dρ = dρ̄. For N ≠ M we may always choose R̄_{ρ} = Rρ̄. If we have a subcategory _{N}C_{N} ⊂ End(N) we may choose a system _{N}Δ_{N} of representants for every sector in _{N}C_{N} and choose R_{ρ} for every ρ ∈ _{N}Δ_{N} such that for [ρ] ≠ [ρ̄] we have R̄_{ρ} = Rρ̄. For [ρ̄] = [ρ] the intertwiners R_{ρ} and R̄_{ρ} are intrinsically related, namely R̄_{ρ} = ±R_{ρ} holds, where the sign ±1 is called the Frobenius–Schur indicator. In this case the sector [ρ] is called real for +1 and pseudo-real for −1. Although [ρ] and [ρ̄] might be represented by the same ρ ∈ _{N}Δ_{N} we still use ρ̄ in the diagrammatically notation to distinguish between R_{ρ} and R̄_{ρ}. A triple Θ = (θ, w, x) with θ ∈ End(N) and isometries w: id_{N} → θ and x: θ → θ^{2}, which we will graphically display as

\sqrt[4]{dθ} w = \begin{matrix}θ \\ w \end{matrix}  \sqrt[4]{dθ} x = \begin{matrix}θ \\ θ \\ θ \end{matrix} ; x

is called a Q-sytem (cf. [Lon94, LR97]) if it fulfills

xx = θ(x)x  (x ⊗ 1_{θ})x = (1_{θ} ⊗ x)x  (associativity)
w^{∗}x = θ(w^{∗})x = λ1_{θ} (w^{∗} ⊗ 1_{θ})x = (1_{θ} ⊗ w^{∗})x = λ1_{θ}  (unit law)

where λ = \sqrt{dθ}^{−1}. In graphical notation this reads:

θ ; θ ; \begin{matrix}θ \\ θ \end{matrix}  =  θ ; θ ; \begin{matrix}θ \\ θ \end{matrix} ;  \begin{matrix}θ \\ θ \end{matrix} = \begin{matrix}θ \\ θ \end{matrix} = \begin{matrix}θ \\ θ \end{matrix} .

Two Q-systems Θ = (θ, w, x) and Θ̃ = (θ̃, w̃, x̃) in End(N) are called equivalent, if there is a unitary u ∈ Hom(θ, θ̃), such that

x̃u = (u ⊗ u)x ≡ uθ(u)x;  uw̃ = w

hold, or graphically:

θ̃ ; θ̃ ; θ ; x̃ ; u = θ̃ ; θ̃ ; θ ; x ; u ; u ;  ^{θu w̃^{∗}} = ^{θw^{∗}} .

A Q-system in a C^{∗}-tensor category automatically [LR97] fulfills the “Frobenius law”

(x^{∗} ⊗ 1_{θ})(1_{θ} ⊗ x) ≡ x^{∗}θ(x) = xx^{∗} = (1_{θ} ⊗ x^{∗})(x ⊗ 1_{θ}) ≡ θ(x^{∗})x

or graphically:

\begin{matrix}θ \\ θ \end{matrix} ; θ ; θ  =  \begin{matrix}θ \\ θ \end{matrix} ; \begin{matrix}θ \\ θ \end{matrix} =  \begin{matrix}θ \\ θ \end{matrix} ; θ ; θ .

This means a Q-system is a special symmetric ∗-Frobenius algebra object, but we prefer to use the name Q-system which is most common in the subfactor context, (other names would be monoid, algebra object, monoidal algebra). We say a Q-system Θ = (θ, w, x) is irreducible (called haploid in the Frobenius algebra context) if ⟨id_{N}, θ⟩ = 1.

Definition 2.1. Every irreducible a ∈ Mor(M, N) defines an irreducible Qsystem

Θ_{a} = (θ_{a}, w_{a}, x_{a}) := (aā, r̄_{a}, a(r_{a}))

in End(N), where r_{a}: id_{M} → āa and r̄_{a}: id_{N} → aā are isometries such that R̄_{a} = \sqrt{da} · r̄_{a} and R_{a} = \sqrt{da} · r_{a} fulfill the conjugate equations (1,2) for a. In graphical notation:

θ_{a} = \begin{matrix}a \\ a \end{matrix} ; ā ; ā ,  \sqrt{da} w_{a} = a ; ā ,  \sqrt{da} x =  a ; ā ; a ; ā ; a ; ā .

We remark that up to this point everything can abstractly be defined in a 2–C^{∗}-category. Consider now a finite index irreducible subfactor N ⊂ M with inclusion ι: N → M then Θ := Θῑ gives dual canonical Q-system of N ⊂ M (and Γ = Θ_{ι} the canonical Q-system). The endomorphism θ ≡ ῑι ∈ End(N) is called the dual canonical endomorphism of N ⊂ M (γ ≡ ιῑ ∈ End(M) is called the canonical endomorphism). Conversely, starting from an irreducible Q-system Θ in End(N), there is a subfactor N_{1} ⊂ N, where N_{1} is defined to be the image N_{1} := E(N) of the conditional expectation E( · ) = x^{∗}θ( · )x and there is subfactor (extension) N ⊂ M defined by the Jones basic construction N_{1} ⊂ N ⊂ M (cf. [LR95]). One can make the construction of M explicit (cf. [BKLR15]) and obtains this way a dual morphism ῑ: M → N of the inclusion ι: N → M such that Θ = Θῑ. The upshot of this discussion is that there is a one-to-one correspondence (cf. [Lon94]) of

• Q-systems in End(N) up to equivalence.

• Irreducible finite index subfactors N ⊂ M up to conjugation.

Remark 2.2. We note that θ alone does not fix N ⊂ M, which can be seen as a cohomological obstruction. Izumi and Kosaki [IK02] define the second cohomology H^{2}(N ⊂ M) to be all equivalence classes of Q-systems Θ = (θ, w, x) with θ the dual canonical endomorphism of N ⊂ M (their definition uses actually the canonical endomorphism). We say the second cohomology of N ⊂ M vanishes if there up to equivalence is just one Q-system Θ = (θ, x, w), where θ is the dual canonical endomorphism of N ⊂ M. We finally note that Θ is a Q-system in the full C^{∗}-tensor subcategory with subobjects generated by θ. The Q-system becomes “trivial”, i.e. is of the form Θῑ, in the 2–C^{∗}-category formed of 0-cells {N, M} and full and replete subcategories _{L}C_{P} ⊂ Mor(P, L) with subobjects and direct sums, which is generated by {ι, ῑ}. We remark that this is actually a general feature of Frobenius algebra object in rigid tensor categors, in particular the obtained 2–C^{∗}-category together with the 1-morphisms ι: N → M and ῑ: M → N appears in [Müg03a] under the name Morita context. In the general situation having a special symmetric Frobenius algebra A in a rigid tensor category C one can find a bicategory C̃ ⊃ C giving a Morita context in which the Frobenius algebra becomes trivial, cf. [Müg03a] for details.

2.2 UMTCs in End(N) and braided subfactors

Let us fix a type III factor N and write _{N}C_{N} ⊂ End(N) for a full and replete subcategory _{N}C_{N} of End(N), such that each object is a finite direct sum of irreducible objects and _{N}C_{N} is closed under taking finite direct sums. We use this notation to stress that it is a category of N-N morphisms. We may choose an endomorphism for each irreducible sector and denote the set of these endomorphisms by _{N}Δ_{N}. Let us assume the following properties:

1. id_{N} ∈ _{N}Δ_{N}.

2. There are only finitely many irreducible sectors in _{N}C_{N}, i.e. |_{N}Δ_{N}| < ∞.

3. If σ ∈ _{N}Δ_{N} then also a conjugate (dual) σ̄ ∈ _{N}Δ_{N}.

4. If ρ, σ ∈ _{N}Δ_{N}, then ρ ∘ σ ∈ _{N}C_{N}, in other words we have that

[μ ∘ ν] = ⨁ N_{μν}^{ρ}[ρ],  N_{μν}^{ρ} = ⟨ρ, μν⟩,

where N_{μν}^{ρ} are called fusion rule coefficients.

This means that_{N}C_{N} is a finite rigid C^{∗}–tensor category [LR97], i.e. a unitary fusion category. We associated with _{N}C_{N} a finite dimensional vector space K_{0}(_{N}C_{N}) ⊗_{ℤ} ℂ ≅ ℂ^{|_{N} Δ_{N} |}, where |_{N}Δ_{N}| denotes the cardinality of the system NΔ_{N} and K_{0}(_{N}C_{N}) is the Grothendieck group of the monoidal category _{N}C_{N}. We define the global dimension dim_{N}C_{N} of _{N}C_{N} to be

dim_{N}C_{N} = ∑_{ρ∈_{N} Δ_{N}}(dρ)^{2} .

We remark that for convenience we assume_{N}C_{N} to be a subcategory of End(N). But it turns out that this is not a lost of generality, because every countable generated rigid C^{∗}–tensor can be embedded in End(N) by the result of [Yam03]. We will need more structure on _{N}C_{N}, in particular we additionally assume:

5. There is a natural family {ε(μ, ν) ∈ Hom(μν, νμ) : μ, ν ∈ _{N}C_{N}} fulfilling:

ε(λ, μν) = (1_{μ} ⊗ ε(λ, ν)) · (ε(λ, μ) ⊗ 1_{ν}) ≡ μ(ε(λ, ν)) · ε(λ, μ)
ε(λμ, ν) = (ε(λ, ν) ⊗ 1_{μ}) · (1_{λ} ⊗ ε(μ, ν)) ≡ ε(λ, ν) · λ(ε(μ, ν)).

Naturality means, that for s: σ → σ′ and t: τ → τ′

(t ⊗ s) · ε(σ, τ) ≡ t · τ(s) · ε(σ, τ)
= ε(σ′, τ′) · (s ⊗ t) ≡ ε(σ′, τ′) · s · σ(t).

We note that this family is determined by {ε(μ, ν) ∈ Hom(μν, νμ) : μ, ν ∈ NΔ_{N}}.

That means that _{N}C_{N} is a braided unitary fusion category which has automatically the structure of a unitary ribbon fusion category. We then say that _{N}C_{N} ⊂ End(N) is a URFC. The braiding ε^{+}(λ, μ) := ε(λ, μ) always comes along with an opposite braiding ε^{−}(λ, μ) := ε(μ, λ)^{∗} which in general is different from ε^{+}(λ, μ). We will graphically denote the braiding by:

ε^{+}(λ, ν) = \begin{matrix}ν \\ ν \end{matrix} ; \begin{matrix}λ \\ λ \end{matrix} ; \begin{matrix}λ \\ λ \end{matrix}  ε^{−}(λ, ν) = \begin{matrix}λ \\ λ \end{matrix} ; \begin{matrix}λ \\ λ \end{matrix} ; \begin{matrix}ν \\ ν \end{matrix} .

We denote by \overline{_{N}C_{N}} the braided category obtained by interchanging the braiding with the opposite braiding. Finally, most of the time we will also use the following additional assumption:

6. The braiding is non-degenerate, i.e. ε^{+}(λ, μ) = ε^{−}(λ, μ) for all μ ∈ _{N}Δ_{N} implies [λ] = [id_{N}].

We then say _{N}C_{N} is modular. In other words _{N}C_{N} is a unitary modular tensor category (UMTC). We define (see [BEK99]) for λ, μ ∈ _{N}Δ_{N}

Y_{λμ} =  λ̄ ; μ̄  ;  ω_{λ} · 1_{λ} = \begin{matrix}λ \\ λ \end{matrix}

and the following |_{N}Δ_{N}| × |_{N}Δ_{N}|-matrices

S_{λμ} = (dim_{N}C_{N})^{− \frac{1}{2}} Y_{λ,μ} ,  T_{λμ} = e^{−πic/12}δ_{λμ}ω_{λ} ,  (3)

where

z = ∑_{ρ∈_{N} Δ_{N}}(dρ)^{2}ω_{ρ};  c = 4 arg(z)/π .

They obey the relations of the partial Verlinde modular algebra: T ST ST = S, CT C = T, and CSC = S, where C_{μν} = δ_{μ,ν̄} is the charge conjugation matrix. The property (6) is equivalent to:

(6’) Z(_{N}C_{N}) ≅ _{N}C_{N} ⊠ \overline{_{N}C_{N}}, where Z(_{N}C_{N}) is the Drinfeld center of _{N}C_{N} [Müg03b, Corollary 7.11] and

(6”) the matrix S = (S_{λμ}) is unitary.

In particular, in the modular case we have ([BEK99, Prop. 2.5]):

S^{∗}S = T^{∗}T = 1 ,  (ST)^{3} = S^{2} = C ,  CT C = T ,

i.e. S and T define a unitary representation of SL(2, ℤ) ≅ ℤ_{6} ∗_{ℤ_{2}} ℤ_{4} on ℂ^{|_{N} Δ_{N} |} if and only if _{N}C_{N} is modular.

2.3 Braided subfactors and α-induction

Let N be a type III factor, _{N}C_{N} ⊂ End(N) a URFC and let ι(N) ⊂ M be an irreducible subfactor such that θ ≡ ῑι ∈ _{N}C_{N}. We call the data (ι(N) ⊂ M,_{N}C_{N}) a braided subfactor. If _{N}C_{N} ⊂ End(N) happens to be a UMTC we call the braided subfactor a non-degenerately braided. There is an obvious one-to-one correspondence between (the equivalence classes of) braided subfactors in _{N}C_{N} and Q-systems in _{N}C_{N}. For ρ ∈ _{N}C_{N} we define its α-induction by

α_{λ}^{±} = ῑ^{−1} ∘ Ad(ε^{±}(λ, θ)) ∘ λ ∘ ῑ ∈ End(M) .

We define the module category _{N}C_{M} to be the full subcategory with subobjects and direct sums of Mor(M, N), which is generated by _{N}C_{N} ῑ ≡ {ρῑ: ρ ∈ NC_{N}} and choose a set of representatives of irreducible sectors _{N}Δ_{M}. In the same way we define _{M}C_{N} and the dual category _{M}C_{M} generated by ι_{N}C_{N} and ι_{N}C_{N} ῑ, respectively. Finally we define _{M}C_{M}^{±} to be generated by α^{±}(_{N}C_{N}), respectively, and the ambichiral category _{M}C_{M}^{0} = _{M}C_{M}^{+} ∩_{M}C_{M}^{−}. Again we choose a set of representatives of irreducible sectors _{M}Δ_{N},_{M}Δ_{M},_{M}Δ_{M}^{±},_{M}Δ_{M}^{0} in the respective categories. It turns out that _{M}C_{M}^{±} ⊂ _{M}C_{M} and that _{M}C_{M}^{+} ∪_{M}C_{M}^{−} generates _{M}C_{M} [BEK99, Thm. 5.10]. It will be convenient to work in the 2-category generated by NC_{N} ∪_{N}C_{M} ∪_{M}C_{N} ∪_{M}C_{M}. As shown in [BEK99, Prop. 3.1], we have for a ∈ _{N}C_{M}, λ ∈ _{N}C_{N}:

ε^{±}(λ, aι) ∈ Hom(λa, aα_{λ}^{±})  E^{±}(λ, ā) ∈ Hom(α_{λ}^{±}ā, āλ) ,

where E^{±}(λ, ā) := T^{∗}ι(ε^{±}(λ, ν̄))α_{λ}^{±}(T) for a ∈ _{N}C_{M} with ā ≺ ῑν for some ν ∈ _{N}C_{N} and T ∈ (ā, ῑν) an isometry. The definition does not depend on the choice of ν and T. We set E^{±}(ā, λ) := (E^{∓}(λ, ā))^{∗}. We represent this graphically—where we use thin lines for morphisms in _{M}C_{N} and _{N}C_{M}, normal lines for endomorphisms in_{N}C_{N} and thick lines for endomorphisms in_{M}C_{M}—as follows:

ε^{+}(λ, aι) = \begin{matrix}a \\ a \end{matrix} λ α_{λ}^{+} ;  E^{+}(λ, ā) = ā ; ā λ α_{λ}^{+} .

The intertwining braided fusion equations (IBFE’s) [BEK99, Prop. 3.3] hold, namely

ρ(t) ε^{±}(λ, ρ) = ε^{±}(aι, ρ) a(E^{±}(b̄, ρ)) t ,
t ε^{±}(ρ, λ) = a(E^{±}(ρ, b̄)) ε^{±}(ρ, aι) ρ(t) ,
ρ(y) ε^{±}, (aι, ρ) = ε^{±}(λ, ρ) λ(ε^{±}(bι, ρ)) y ,
y ε^{±}(ρ, aι) = λ(ε^{±}(ρ, bι) ε^{±}(ρ, λ)) ρ(y) ,
α^{∓}(Y) E^{±}(ā, ρ) = E^{±}(b̄, ρ) b̄(ε^{±}(λ, ρ)) Y ,
Y E^{±}(ρ, ā) = b̄(ε^{±}(ρ, λ)) E^{±}(ρ, b̄) α_{ρ}^{±}ρ(Y) ,

where λ, ρ ∈ _{N}C_{N}, a, b ∈ _{N}C_{M} with conjugates ā, b̄ ∈ _{M}C_{N}; t ∈ Hom(λ, ab̄), y ∈ Hom(a, λb) and Y ∈ Hom(ā, b̄λ). The IBFE’s have simple graphical interpretation, e.g. the first and sixth equations are represented by:

λ b̄ ; t ; a ρ ρ α− ρ =  ^{λ b̄ ; t ; a ρ ρ}  ;  λ^{ā Y b̄} ρ α− ρ =  λ^{ā Y b̄} ρ α− ρ .

For details we refer to [BEK99, Sect. 3.3]. There is a relative braiding [BEK00, p. 738]

E_{r}(β_{+}, β_{−}) := S^{∗}α_{μ}(T^{∗})ε(λ, μ)α_{λ}^{+}(S)T ∈ Hom(β_{+}β_{−}, β_{+}β_{−}) ,  (4)

where for fixed β_{±} ∈ _{M}C_{M}^{±}, we choose λ, μ ∈ _{N}C_{N}, such that β_{+} ≺ α_{λ}^{+}, β_{−} ≺ α_{ν}^{−}and isometries S, T, such that T ∈ Hom(β_{+}, α^{+}μ) and S ∈ Hom(β_{−}, α_{μ}^{−}). The definition is independent of the particular choice of λ, μ, S, T. The relative braidings give a non-degenerate braiding ε( · , · ) := E_{r}( · , · ) on _{M}C_{M}^{0} by [BEK00, Sec. 4], so in particular _{M}C_{M}^{0} becomes a UMTC. In general for two braided subfactors ι_{a}(N) ⊂ M_{a} and ι_{b}(N) ⊂ M_{b} in _{N}C_{N} we define _{M_{a}}C_{M_{b}} as a full subcategory of Mor(M_{b}, M_{a}) with subobjects and direct sums generated by ι_{a}_{N}C_{N} ῑ_{b}.

3 Morita equivalence for braided subfactors

3.1 Module categories, modules and bimodules

In this section we give the notion of Morita equivalent non-degenerately braided subfactors. We adapt the following definitions from [Ost03].

Definition 3.1. A (strict) module category over a tensor category C is a category M together with an exact bifunctor ⊗: C × M → M such that (X ⊗ Y) ⊗ M = X ⊗ (Y ⊗ M) for all X, Y ∈ C and M ∈ M.

Let M_{1}, M_{2} be two module categories over C. A (strict) module functor from M_{1} to M_{2} is a functor F: M_{1} → M_{2} such that F (X ⊗M) = X ⊗F (M). Two module categories M_{1} and M_{2} over C are called isomorphic if there exist a module functor, which is an isomorphism of categories.

Let _{N}C_{N} ⊂ End(N) be a UFC and let Θ = (θ, w, x) be a Q-system in _{N}C_{N} corresponding to N ⊂ M. A (right) Θ-module (cf. [EP03]) is a pair (ρ, r) with ρ ∈ _{N}C_{N} and r̃ ∈ Hom(ρ ∘ θ, ρ), such that r^{∗} is an isometry and r̃ = \sqrt[4]{dθ} r satisfies

r̃ · (1_{ρ} ⊗ m) = r̃ · (ẽ ⊗ 1_{θ})  ⇔  r̃ · ρ(m) = ρ̃(r̃^{2})
r̃ · (1_{ρ} ⊗ r) = 1_{ρ}  ⇔  r̃ · ρ(e) = 1_{ρ}

where m = \sqrt[4]{dθ}x^{∗} the multiplication and e = \sqrt[4]{dθ}w the unit of the (Frobenius) algebra object corresponding to Θ. Graphically this means:

\begin{matrix}ρ \\ ρ \end{matrix} \begin{matrix}θ \\ θ \end{matrix}rx^{∗}  = \begin{matrix}ρ \\ ρ \end{matrix} ; θ ; θ ; r ; r ;  \begin{matrix}ρ \\ ρ \end{matrix} ; r ; w = \begin{matrix}ρ \\ ρ \end{matrix} .

A left Θ-module can be defined similarly. We note that because we are working in C^{∗}-categories and ask r^{∗} to be an isometry, that a module is also a co-module by the action r^{∗}. The endomorphism ρθ with ρ ∈ _{N}C_{N} has the structure of a right Θ-module, where the action is given by r̃ = 1_{ρ}⊗m ≡ ρ(m) ≡ \sqrt[4]{dθ}·ρ(x^{∗}) ∈ Hom(ρθθ, ρθ) in other words r = ρ(x^{∗}), graphically:

\begin{matrix}ρθ \\ ρθ \end{matrix} ; θ ; r := ^{\begin{matrix}ρ \\ ρ \end{matrix} \begin{matrix}θ \\ θ \end{matrix}θx^{∗}} .

It is called the induced module. Any irreducible right Θ-module is equivalent to a submodule of an induced module cf. [Ost03]. The Θ-modules form a category with Hom_{Θ}(ρ, σ) ≡ Hom_{Θ}((ρ, r), (σ, s)) = {t ∈ Hom(ρ, σ) : tr = st}, so the arrows are arrows of the objects which intertwine the actions. There is a correspondence between projections p ∈ Hom_{Θ}(ρ, ρ) and submodules, namely we can choose ρ_{p} and t ∈ Hom(ρ_{p}, ρ) with t^{∗}t = 1_{ρ_{p}}, tt^{∗} = p and define r_{p} = t^{∗}rt. Let Θ_{a} = (θ_{a}, w_{a}, x_{a}) and Θ_{b} = (θ_{b}, w_{b}, x_{b}) be two Q-systems in _{N}C_{N}. A Θ_{a}- Θ_{b} bimodule is a triple (ρ, r_{a}, r_{b}) with ρ ∈ _{N}C_{N} and ρ_{a} ∈ Hom(θ_{a}ρ, ρ) and ρ_{b} ∈ Hom(ρθ_{b}, ρ), such that (ρ, r_{a}) is a left Θ_{a}-module and (ρ, r_{b}) is a (right) Θ_{b}-module and which commute, i.e.

r_{a} · θ_{a}(r_{b}) = r_{b} · r_{a}.

We can define:

r := r_{a} · (1_{θ_{a}} ⊗ r_{b}) = r_{b} · (r_{a} ⊗ 1_{θ_{a}}) ∈ (θ_{a} ∘ ρ ∘ θ_{b}, ρ).

Let ρ = (ρ, r_{a}, r_{b}) and σ = (σ, s_{a}, s_{b}) be two Θ_{a}–Θ_{b} bimodules. An intertwiner t: ρ → σ is an Θ_{a}–Θ_{b} bimodule intertwiner, if t intertwines the actions r and s, i.e.

tr = s(1_{θ_{a}} ⊗ t ⊗ 1_{θ_{b}}) ≡ sθ_{a}(t) .

Let us denote by Bim(Θ_{a}, Θ_{b}) the category of bimodules with Hom_{Θ_{a}−Θ_{b}}(ρ, σ) Θ_{a}-Θ_{b} bimodule intertwiner. We note that one can give Q-systems, bimodules and intertwiners the structure of a bicategory, by introducing a relative tensor product between bimodules. We set Mod(Θ) = Bim(1, Θ) to be the category of (right) Θ-modules. The category Mod(Θ) has a natural structure of a (strict) left _{N}C_{N} module category, where the functor _{N}C_{N} × Mod(Θ) is given by (μ, ρ) ↦ μρ where μρ is a right-module with r_{μσ} = μ(r_{ρ}) and Hom_{Mod(Θ)}(ρ, σ) ∋ T ↦ μ(T) ∈ Hom_{Mod(Θ)}(μρ, μσ).

Proposition 3.2 ([EP03, Lemma 3.1.]). Let _{N}C_{N} be a UMTC and Θ_{a}, Θ_{b} irreducible Q-systems in _{N}C_{N}. The category of Θ_{a}-Θ_{b} bimodules is equivalent to the category _{M_{a}}C_{M_{b}}. The functor Φ maps β ∈ _{M_{a}}C_{M_{b}} to ῑ_{a} ∘ β ∘ ι_{b} and t ∈ Hom(β, β′) to ῑ_{a}(t) ∈ Hom_{Θ_{a}-Θ_{b}}(Φ(β), Φ(β′)).

Proof. In [EP03, Lemma 3.1.] is shown that the functor Φ is fully faithful. It is also shown that is is essentially surjective, so it gives an equivalence of categories.

The functor Φ is graphically given as follows, where ρ = Φ(β) r̃ ∈ Hom(θ_{a}ρθ_{b}, ρ) the action:

Φ: \begin{matrix}β & β′ \\ t \end{matrix} ↦  β β′ ; ῑ_{a} ; ῑ_{a} ; \begin{matrix}ι_{b} \\ ι_{b} \end{matrix} ; t ,  r̃ =  \begin{matrix}ρ \\ ρ \end{matrix} ; θ_{a} ; θ_{b} :=  \begin{matrix}β \\ β \end{matrix} ; ῑ_{a} ; ῑ_{a} ; \begin{matrix}ι_{b} \\ ι_{b} \end{matrix} ; ι_{b} ; ῑ_{b} ; ῑ_{a} ; ι_{a}  .

Remark 3.3. Let Θ = (θ, w, x) be a Q-system in a UMTC _{N}C_{N} with corresponding subfactor ι(N) ⊂ M. The bimodule Φ(α_{λ}^{±}) ≡ ῑα_{λ}^{±}ι ≡ ῑιλ is the object θλ with left action the induced action x^{∗} and right action by x^{∗}ε^{±}(λ, θ), namely for the +-case:

r̃ =  \begin{matrix}θ \\ θ \end{matrix} ; θ ; θ ; \begin{matrix}λ \\ λ \end{matrix} =  \begin{matrix}α_{λ}^{+} \\ α_{λ}^{+} \end{matrix} ; ῑ ; ῑ ; \begin{matrix}ι \\ ι \end{matrix} ; ι ; ῑ ; ῑ ; ι  ,

where equality can be seen easily using ιλ = α_{λ}^{+}ι, Θ = Θῑ and the IBFEs by pulling the λ-string between ῑ and ι. The −-case works analogous using the opposite braiding. The obtained bimodules coincide with the notion of α-induction in the categorical literature.

The category Bim(Θ, Θ) becomes a tensor category, where ρ ⊗_{Θ} σ is the object associated to the projection in P_{ρ⊗_{Θ}σ} ∈ Hom(ρσ, ρσ) given by:

P_{ρ⊗_{Θ}σ} = \frac{1}{\sqrt{dθ}} \begin{matrix}ρ \\ σ \end{matrix} .

and it is easy to check that Φ is a tensor functor. Thus, Bim(Θ, Θ) and _{M}C_{M} are equivalent as tensor categories. We note that this category is non-strict. We can define the categories Bim^{±}(Θ, Θ) to be the image of _{M}C_{M}^{±} under Φ and Bim^{0}(Θ, Θ) = Bim^{+}(Θ, Θ) ∩ Bim^{−}(Θ, Θ). In the special case M_{a} = N and M_{b} = M and θ_{a} = θ we have an equivalence of the category _{N}C_{M} and the category Mod(Θ) of right Θ-modules given by ā ↦ āι. The category of right Θ-modules Mod(Θ) becomes a module category over _{N}C_{N} using the monoidal structure inherent from End(N). The same is true for _{N}C_{M}. In particular, it follows:

Proposition 3.4. Let _{N}C_{N} ⊂ End(N) be a UMTC and Θ be a Q-system in _{N}C_{N} with corresponding subfactor N ⊂ M. Then Mod(Θ) and _{N}C_{M} are equivalent as module categories.

Proof. It follows directly from the properties of the monoidal structure, that the functor Φ (in the case of M_{a} = N and M_{b} = M and θ_{a} = θ) in the proof of Prop. 3.2 is a module functor, so in particular a module isomorphism, between the two module categories Mod(Θ) and _{N}C_{M} over _{N}C_{N}.

We remark that in general in the definition of module it is not assumed that r is a (multiple) of an isometry, because the existence of a unitary structure is not assumed. But since every module in the general sense is equivalent to a submodule of an induced module and the submodule can chosen to have a multiple of an isometry as action, we can without lost of generality restrict to modules where r is a multiple of an isometry. This can also be shown directly [BKLR15]. Let a ∈ _{N}C_{M} be irreducible and consider the subfactor N ⊂ M_{a} given by the Q-system Θ_{a} (see Def. 2.1). Let M_{a} be the factor which is given by Jones basic construction a(M) ⊂ N ⊂ M_{a} and denote the inclusion map ι_{a}: N ↪ M_{a}. Because the subfactors ῑ_{a}(M_{a}) ⊂ N and a(M) ⊂ N have by definition the same Q-system and thus are conjugated by a unitary in N, we may and do choose ῑ_{a}: M_{a} → N, such that ῑ_{a}(M_{a}) = a(M). This implies that α = ῑ_{a}^{−1} ∘ a: M → M_{a} is an isomorphism with conjugate α^{−1} = a^{−1} ∘ ῑ_{a}: M_{a} → M.

Lemma 3.5 (cf. [LR04, Eva02]). Let _{N}C_{N} ⊂ End(N) be a UMTC and Θ be a Q-system in _{N}C_{N} with corresponding subfactor N ⊂ M. For a ∈ _{N}C_{M} irreducible let Θ_{a} be the canonical Q-system (Θ_{a} = aā, w_{a}, x_{a}) and N ⊂ M_{a} the corresponding subfactor. Then _{N}C_{M} and _{N}C_{M_{a}} are isomorphic as module categories of _{N}C_{N}. The isomorphism is given by Ψ: b ↦ b ∘ a^{−1} ∘ ι_{a} and Hom_{_{N} C_{M}} (b, c) ∋ t ↦ t ∈ Hom_{_{N} C_{Ma}} (Ψ(b), Ψ(c)).

Remark 3.6. Given a ∈ _{N}C_{M} we have the Q-systen Θ_{a} with θ_{a} = aā. Let β = Φ(a) ∈ Mod(Θ), then β̄ is a Θ left module and there is another way to construct a Q-system [KR08] denoted by β̄ ⊗_{Θ} β, and it is easy to check that β̄ ⊗_{Θ} β ≅ āa and that the obtained Q-systems are equivalent.

3.2 The Morita equivalence class of a braided subfactor

In the following we use the definition of Morita equivalence for module categories as in [Ost03, Def. 3.3]. Let _{N}C_{N} ⊂ End(N) be a UMTC. We remember that we call a pair (N ⊂ M,_{N}C_{N}) where N ⊂ M is a subfactor whose Q-system Θ is in _{N}C_{N} a non-degenerately braided subfactor.

Definition 3.7. Let _{N}C_{N} ⊂ End(N) be a UMTC. Two irreducible Q-systems Θ_{a} and Θ_{b} in _{N}C_{N} are called Morita equivalent if one of the following equivalent statements hold:

• Mod(Θ_{a}) and Mod(Θ_{b}) are equivalent as module categories over _{N}C_{N}.

• _{N}C_{M_{a}} and _{N}C_{M_{b}} are equivalent as module categories over _{N}C_{N}, where N ⊂ M_{•} is corresponding to Θ_{•}.

We say that the subfactors N ⊂ M_{a} and N ⊂ M_{b} are Morita equivalent if their Q-systems Θ_{a} and Θ_{b}, respectively, are Morita equivalent.

Let (ι(N) ⊂ M,_{N}C_{N}) be a non-degenerately braided subfactor. It follows directly that for a, b ∈ _{N}C_{M} irreducible Θ_{a} and Θ_{b} are Morita equivalent and in particular are Morita equivalent to Θῑ. But it can also happen that Θ_{a} and Θ_{b} are equivalent for [a] ≠ [b]. If C is a UTFC, we denote by Pic(C) the full and replete subcategory (2-group) with objects {ρ ∈ C : dρ = 1} (not completed under direct sums).

Proposition 3.8 ([GS15]). Given two irreducible objects a, b ∈ _{N}C_{M}. Then the Q-systems Θ_{a} and Θ_{b} are equivalent if and only if there is an automorphism β ∈ Pic(_{M}C_{M}) such that bβ = a.

Now we can give a characterization of the Morita equivalence class of a nondegenerately braided subfactor.

Proposition 3.9. Let _{N}C_{N} ⊂ End(N) be a UMTC and let Θ be a Q-system in _{N}C_{N}. Then there is a one-to-one correspondence between

1. equivalence classes [Θ_{a}] of irreducible Q-systems Morita equivalent to Θ, 2. irreducible sectors [a] with a ∈ _{N}C_{M} up the identification: [a] ∼ [b] if

there is an automorphism β ∈ _{M}X_{M}, such that [a] = [βb],

3. elements in _{N}Δ_{M}/ Pic(_{M}C_{M}).

Proof. Statement (3) is just a reformulation of (2). Let a ∈ _{N}X_{M} then we obtain a canonical Q-system Θ_{a} in _{N}C_{N} which is Morita equivalent to Θ by Lemma 3.5. Conversely given a Q-system Θ_{a} Morita equivalent to Θ then NC_{M} is equivalent to_{N}C_{M_{a}}. The element a ∈ _{N}C_{M} corresponding to ι_{a} ∈ _{N}C_{M_{a}} under this equivalence is the corresponding element in_{N}C_{M}, cf. [Ost03, Remark 3.5]. The rest follows by Prop. 3.8.

4 α-induction construction and the full center

4.1 The full center and Rehren’s construction coincide

Let N be a type III factor and _{N}C_{N} ⊂ End(N) a UMTC. As before let _{N}Δ_{N} = {id_{N}, ρ_{1}, …, ρ_{n}} a set of representatives for each sector. Given ν, λ, μ ∈ _{N}Δ_{N}, we can choose a set of isometries B(ν, λμ) := {e_{i}}_{i=1,...,⟨ν,λμ⟩} with e_{i} ∈ Hom_{_{N} C_{N}} (ν, λμ), such that {e_{i}} form an orthonormal basis with respect to the scalar product (e, f) = Φ_{ν}(e^{∗}f) defined by the left inverse Φ_{ν} of ν [LR97] or equivalently defined by (e, f) · 1_{ν} = e^{∗}f. We define for an isometry e ∈ Hom_{_{N} C_{N}} (ν, λμ) an isometry ē ∈ Hom\overline{_{N} C_{N}} (ν̄, λ̄μ̄) by

ν̄ ; μ̄ ē λ̄ :=  e^{∗} ; λ̄ ; ν̄ ; μ̄  .

Definition 4.1 (Longo–Rehren construction). Let _{N}C_{N} ⊂ End(N) a URFC. There is a Q-system Θ_{LR} = (θ_{LR}, w_{LR}, x_{LR}) in _{N}C_{N} ⊠ \overline{_{N}C_{N}} given by:

[θ_{LR}] = ⨁_{ρ∈_{N} C_{N}}[ρ ⊠ ρ̄],  x_{LR} = \frac{1}{\sqrt{dθ}} ⨁_{λμν} ∑_{e∈B(ν,λμ)} \sqrt{\frac{dλdμ}{dνdθ}} e ⊠ ē,

= ⨁_{λμν} ∑_{e∈B(ν,λμ)}  \begin{matrix}ν \\ μ \end{matrix} ; \begin{matrix}e \\ λ \end{matrix} ⊠ ν̄ ; μ̄ ē λ̄ .

More general, for an equivalence of braided categories ϕ: _{N}C_{N} → _{N}C′_{N}, we define the Q-system Θ_{LR}^{ϕ} = (θ_{LR}^{ϕ}, w_{LR}^{ϕ}, x_{LR}^{ϕ}) in _{N}C_{N} ⊠ \overline{_{N}C′_{N}} by

[θ_{LR}^{ϕ}] = ⨁_{ρ∈_{N} C_{N}}[ρ ⊠ ϕ(ρ̄)],  x_{LR}^{ϕ} = ⨁_{λμν} ∑_{e∈B(ν,λμ)} \sqrt{\frac{dλdμ}{dνdθ}}e ⊠ \overline{ϕ(e)} .

Definition 4.2. Let _{N}C_{N} ⊂ End(N) be a URFC. A Q-system Θ = (θ, w, x) in _{N}C_{N} is called commutative if ε(θ, θ)x = x. Diagrammatically:

\begin{matrix}θ \\ θ \end{matrix} ; θ = \begin{matrix}θ \\ θ \end{matrix} ; θ ; θ ; θ .

Proposition 4.3 ([LR95]). The Q-system obtained by the Longo–Rehren construction is commutative.

Definition 4.4 (Product Q-system). Let Θ_{i} = (θ_{i}, w_{i}, x_{i}) with i = 1, 2 be two Q-systems in a URFC category _{N}C_{N}. Then we define two Q-systems Θ_{1} ∘^{±} Θ_{2} = (θ_{1} ∘ θ_{2}, w_{1}w_{2}, x_{±}) in _{N}C_{N}, where x_{±} = θ_{1}(ε^{±}(θ_{1}, θ_{2}))x_{1}θ_{1}(x_{2}), graphically:

\begin{matrix}θ_{1}θ_{2} \\ θ_{1}θ_{2} \\ θ_{1}θ_{2} \end{matrix} ; x_{+} =  \begin{matrix}θ_{2} \\ θ_{2} \\ θ_{2} \end{matrix} ; x_{2} ; \begin{matrix}θ_{1} \\ θ_{1} \\ θ_{1} \end{matrix} ; \begin{matrix}θ_{1} \\ θ_{1} \\ θ_{1} \end{matrix} ; x_{1}  .

Definition 4.5. For Θ ≡ (θ, w, x) a Q-system in _{N}C_{N} and ρ ∈ _{N}C_{N}, we define

P_{Θ}^{l}(ρ) = \frac{1}{\sqrt{dθ}} ·  \begin{matrix}ρ \\ ρ \end{matrix} ; θ ; θ ≡  \begin{matrix}ρ \\ ρ \end{matrix} ; \begin{matrix}θ \\ θ \end{matrix} ∈ Hom(θρ, θρ)

and P_{Θ}^{l} := P_{Θ}^{l}(id_{N}). Similarly, we define P_{Θ}^{r}(ρ) and P_{Θ}^{r} by interchanging the braiding with the opposite braiding.

Lemma 4.6. P_{Θ}^{l/r}(ρ) is a projection.

Proof. That P_{Θ}^{l}(ρ)^{2} = P_{Θ}^{l}(ρ) is proven as in [FRS02, Lemma 5.2], see also [BKLR15]. We just remark that we have a prefactor due to another normalization and that one can check that P_{Θ}^{l}(ρ) is selfadjoint.

Proposition 4.7 (Sub-Q-system cf. [BKLR15]). Let p ∈ Hom(θ, θ) be an orthogonal projection satisfying pθ(p)xp = θ(p)xp = pxp = pθ(p)x and w^{∗}p = w^{∗}.

Let θ_{p} ≺ θ corresponding to p, i.e. there a isometry s ∈ Hom(θ_{p}, θ), such that s^{∗}s = 1_{θ_{p}} and ss^{∗} = p. Then Θ_{p} = (θ_{p}, w_{p}, x_{p}) with

w_{p} := s^{∗}w,  x_{p} := \sqrt{\frac{dθ}{dθ_{p}}} · s^{∗}θ(s^{∗})xs

is a Q-system.

Graphically, the conditions are given by:

\begin{matrix}θ \\ θ \end{matrix} ; θ ; p ; p ; p =  \begin{matrix}θ \\ θ \end{matrix} ; θ ; p ; p =  \begin{matrix}θ \\ θ \end{matrix} ; θ ; p ; p =  \begin{matrix}θ \\ θ \end{matrix} ; θ ; p ; p ,  \begin{matrix}θ \\ p \end{matrix} = θ .

Remark 4.8. The notion of sub-Q-system Θ_{p} of Θ corresponds to the notion of intermediate subfactor L with N ⊂ L ⊂ M where Θ is the dual canonical Q-system of N ⊂ M. Namely, the properties of the sub-Q-system are just a reformulation of [ILP98, Corollary 3.10]. Namely, they consider subspaces K_{ρ} ⊂ Hom(ι, ιρ) for each ρ ∈ _{N}Δ_{N}, which correspond to a projection p ∈ Hom(θ, θ) if we identify the Hilbert spaces Hom(ρ, θ) and Hom(ι, ιρ) by Frobenius reciprocity. Remark 4.9 (cf. [BKLR15]). If one drops the condition w^{∗}p = w^{∗} in Prop. 4.7 then we obtain a more general “sub” Q-system Θ_{p} = (θ_{p}, w_{p}, x_{p}) with

w_{p} := λ^{−1} · s^{∗}w,  x_{p} := λ · \sqrt{\frac{dθ}{dθ_{p}}} · s^{∗}θ(s^{∗})xs

where λ = \sqrt{w^{∗}pw}.

Definition 4.10. We denote by C_{l}(Θ) = (C_{l}(θ), C_{l}(w), C_{l}(x)) the left center of Θ, which is defined to be the sub-Q-sytem associated with the projection P_{Θ}^{l} ∈ Hom(θ, θ). Analogously, the right center C_{r}(Θ) is defined using P_{Θ}^{r}.

Remark 4.11 ([FFRS06, Lemma 2.30]). The Q-system C_{l/r}(Θ) is a maximal commutative sub-Q-system of Θ. Remark 4.12. The intermediate factor N ⊂ M_{+} ⊂ M defined in [BE00] is given by the Q-system C_{l}(Θ). Namely, the characterization of P_{Θ}^{l} in [FFRS06, Lemma 2.30] is the characterization in [BE00, Lemma 4.1] in terms of subspaces H_{ρ} ⊂ Hom(ι, ιρ) of “charged intertwiners”. Similarly, N ⊂ M_{−} ⊂ M is given by C_{r}(Θ).

Definition 4.13 (cf. [FFRS08]). Let _{N}C_{N} be a UMTC. The full center of a Q-system Θ is defined to be the Q-system Z(Θ) ≡ (Z(θ), Z(w), Z(x)) = C_{l}((Θ ⊠ id_{N}) ∘^{+} Θ_{LR}) in _{N}C_{N} ⊠ \overline{_{N}C_{N}}.

In particular we have Z(id_{N}) = Θ_{LR}.

Definition 4.14. Let _{N}C_{N} be a URFC and Θ = (θ, w, x) a Q-system in _{N}C_{N}. We define

Hom_{loc}(θρ, σ) = {t ∈ Hom(θρ, σ) : t · P_{θ}^{l}(ρ) = t} ,
Hom_{loc}(σ, θρ) = {t^{∗} ∈ Hom(σ, θρ) : P_{θ}^{l}(ρ) · t^{∗} = t^{∗}} .

In particular, the spaces Hom_{loc}(θρ, σ) and Hom_{loc}(σ, θρ) are anti-isomorphic, due to the self-adjointness of P_{θ}^{l}(ρ).

Lemma 4.15. The isometry ψ ∈ Hom (Z(θ), (θ ⊠ id_{N})θ_{LR}) with ψψ^{∗} = P_{(Θ⊠id_{N})∘^{+}Θ_{LR}}^{l} and ψ^{∗}ψ = 1 is of the form:

ψ = ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}} ⨁_{m∈B(θλ_{2},λ_{1})_{loc}} m^{∗} ⊠ id_{λ_{2}} ∈ Hom (Z(θ), (θ ⊠ id_{N})θ_{LR}) ,

where the sum over m goes over an ONB of Hom_{loc}(θλ_{2}, λ_{1}). In particular:

[Z(θ)] = ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}}⟨θλ_{2}, λ_{1}⟩_{loc} [λ_{1} ⊠ \overline{λ}_{2}] ,

where ⟨ · , · ⟩_{loc} = dim Hom_{loc}( · , · ).

Proof. We first note that u ∈ Hom (R(θ), (θ ⊠ 1)θ_{LR}) given by

u := ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}} ⨁_{m∈B(θλ_{2},λ_{1})} m^{∗} ⊠ id_{λ_{2}} ∈ Hom (R(θ), (θ ⊠ id_{N})θ_{LR}) ,
R(θ) := ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}}⟨θλ_{2}, λ_{1}⟩λ_{1} ⊠ \overline{λ}_{2}

is a unitary interwiner. It can be shown that

P_{(Θ⊠id_{N})∘^{+}Θ_{LR}}^{l} · u = P_{Θ⊠id_{N}}^{l} (θ_{LR}) · u ≡ ( ⨁_{λ∈_{N} Δ_{N}} P_{Θ}^{l}(λ) ⊠ 1\overline{λ}) · u .

The equality is the statement [FFRS06, Prop. 3.14(i)], namely it is proven that C_{l}((Θ ⊠ id_{N}) ∘^{+} Θ_{LR}) which is associated with P_{(Θ⊠id_{N})∘^{+}Θ}^{l} is associated with the projection P_{Θ⊠id_{N}}^{l} (C_{l}(θ_{LR})) ≡ P_{Θ⊠id_{N}}^{l} (θ_{LR}). We can conclude by eventually choosing another basis that a maximal isometry invariant w.r.t. P_{(Θ⊠id_{N})∘^{+}Θ_{LR}}^{l} is given by summing just over ONB’s of Hom_{loc}(θλ_{2}, λ_{1}).

Given a Q-system Θ in _{N}C_{N} and ι(N) ⊂ M its associated subfactor with the inclusion map ι: N → M, we will constantly use that the Q-system Θ is of the form Θῑ as in Def. 2.1, in other words the Q-system Θ becomes trivial in the 2–C^{∗}-category generated by _{N}C_{N}, ι, ῑ. This simplifies many graphical proofs.

Lemma 4.16. Let _{N}C_{N} ⊂ End(N) be a UMTC, Θ a Q-system in _{N}C_{N} and N ⊂ M the corresponding subfactor. Let ρ, σ ∈ _{N}C_{N} be irreducible. The spaces Hom_{loc}(θρ, σ) and Hom(α_{ρ}^{−}, α_{σ}^{+}) are isomorphic by the map:

Hom_{loc}(θρ, σ) ⟶ Hom(α_{ρ}^{−}, α_{σ}^{+})

θ ; \begin{matrix}σ \\ ρ \end{matrix} ⟼ \frac{1}{\sqrt[4]{dθ}}  \begin{matrix}α_{ρ}^{−} \\ α_{σ}^{+} \end{matrix}

\frac{1}{\sqrt[4]{dθ}}  \begin{matrix}θ \\ σ \\ ρ \end{matrix}  ⟵ \begin{matrix}α_{σ}^{+} \\ α_{ρ}^{−} \end{matrix}.

In the same way Hom_{loc}(ρ, θσ) is isomorphic to Hom(α_{ρ}^{+}, α_{σ}^{−}). This gives a unitary equivalence between the Hilbert spaces Hom_{loc}(ρ, θσ) with scalar product (e, f) = Φ_{σ}(e^{∗}f) and Hom(α_{ρ}^{+}, α_{σ}^{−}) with scalar product (e′, f′) = Φ_{α_{σ}^{+}}(e^{′∗}f′), where Φ_{σ} and Φ_{α_{σ}^{+}} denote the unique left inverse and unique standard left inverse, respectively.

Proof. We first check that the map is well defined, namely the image is an element in Hom_{loc}(θρ, σ) and we have (“=” denotes the trivial intertwiner identifying θ = ῑι)

\frac{1}{\sqrt{dθ}}  \begin{matrix}ρ \\ θ \\ σ \\ = \end{matrix} ≡ \frac{1}{\sqrt{dθ}}  \begin{matrix}θ \\ θ \\ ρ \\ σ \end{matrix}  =  \begin{matrix}θ \\ σ \\ ρ \end{matrix} ,

where we used in the first equation that Θ is of the form Θῑ and in the second equation that the closed string can be contracted which cancels the prefactor. So we conclude that the image is actually in Hom_{loc}(θρ, σ).

We have to show that both maps are inverse to each other:

θ ; \begin{matrix}σ \\ ρ \end{matrix} ⟼  \begin{matrix}α_{ρ}^{−} \\ α_{σ}^{+} \end{matrix} ⟼ \frac{1}{\sqrt{dθ}}  \begin{matrix}θ \\ σ \\ ρ \end{matrix}  =  θ ; \begin{matrix}σ \\ ρ \end{matrix}

\begin{matrix}α_{σ}^{+} \\ α_{ρ}^{−} \end{matrix} ⟼ \frac{1}{\sqrt[4]{dθ}}  \begin{matrix}θ \\ σ \\ ρ \end{matrix}  ⟼ \frac{1}{\sqrt{dθ}}  \begin{matrix}α_{ρ}^{−} \\ α_{σ}^{+} \end{matrix} = \begin{matrix}α_{σ}^{+} \\ α_{ρ}^{−} \end{matrix},

where the last equation in the first line is exactly the fact that the intertwiner is in Hom_{loc}(θρ, σ), namely the diagram can be deformed to obtain P_{Θ}^{l}(ρ) which can be omitted; in the last equation of the second line the closed string can again be contracted to a dimension cancelling the prefactor.

Finally, unitarity can be seen as follows:

∥ \frac{1}{\sqrt[4]{dθ}}  \begin{matrix}θ \\ σ \\ ρ \end{matrix}  ∥^{2} = \frac{1}{\sqrt{dθ} dσ}  \begin{matrix}θ \\ σ \\ ρ \end{matrix} = ∥ \begin{matrix}α_{σ}^{+} \\ α_{ρ}^{−} \end{matrix} ∥^{2} ,

where in the last equation we use that the string diagram can be deformed to give the standard left inverse for α_{σ}^{+} (cf. [Reh00, Lemma 2.2]).

Definition 4.17 (α-induction construction [Reh00]). For a braided subfactor ι(N) ⊂ M in _{N}C_{N} there is a Q-system Θ_{M} = (θ_{M}, w_{M}, x_{M}) in _{N}C_{N} ⊠ \overline{_{N}C_{N}} given by:

[θ_{M}] = ⨁_{ρ,σ∈_{N} Δ_{N}} Z_{μν}[μ ⊠ ν̄],
Z_{μν} = ⟨α_{μ}^{+}, α_{ν}^{−}⟩
x_{M} = ⨁_{lmn} ∑_{e_{1},e_{2}} \sqrt{\frac{dλ_{2}dμ_{2}}{dθ_{M}dν_{2}}} Φ_{ν_{1}}^{1}[ι(e_{1}^{∗})(ϕ_{l}^{∗} ⊗ ϕ_{m}^{∗})ι(e_{2})ϕ_{n}] · e_{1} ⊠ ē_{2},

= ⨁_{lmn} ∑_{e_{1},e_{2}} \frac{1}{\sqrt{dθ_{M}}} \sqrt[4]{\frac{dλ_{2}dμ_{2}dν_{1}}{dλ_{1}dμ_{1}dν_{2}}} Φ_{ν_{1}}^{1}[⋯] \begin{matrix}ν_{1} \\ μ_{1} \end{matrix} ; \begin{matrix}e_{1} \\ λ_{1} \end{matrix} ⊠ ν̄_{2} ; μ̄_{2} ē_{2} λ̄_{2}

where l is considered as a multi-index (λ_{1} ∈ _{N}Δ_{N}, λ_{2} ∈ _{N}Δ_{N}, l = 1, ⋯, Z_{λ_{1},λ_{2}}) and e_{i} stands for an ONB in Hom(ν_{i}, λ_{i}μ_{i}) and ϕ_{l} an ONB in Hom(α_{λ_{1}}^{+}, α_{λ_{2}}^{−}) with respect to the induced left inverse Φ_{λ_{1}}^{1}.

The following result was conjectured in [KR10]. It can be seen as the main technical result. It allows to apply a lot of results obtained in the categorical literature to the braided subfactor and conformal net setting.

Proposition 4.18. Let _{N}C_{N} be a UMTC. The α-induction construction for (ι(N) ⊂ M,_{N}C_{N}) coincides with the full center Z(Θ) of the corresponding Q-system Θ.

Proof. It is already clear that the two constructions give equivalent objects, namely

[Z(θ)] = ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}}⟨θλ_{2}, λ_{1}⟩_{loc}[λ_{1} ⊠ λ̄_{2}] = ⨁_{λ_{1},λ_{2}∈_{N} Δ_{N}}⟨α_{λ_{1}}^{+}, α_{λ_{2}}^{−}⟩[λ_{1} ⊠ λ̄_{2}] = [θ_{M}]

follows from Lemma 4.15 and Lemma 4.16. We have to show that the two intertwiners Z(x) and x_{M} of the two respective constructions are equivalent. We decompose Z(x) w.r.t. an ONB to show that we obtain the same coefficients as in the α-induction construction for x_{M}. Using Lemma 4.15 we have:

\sqrt{dθ_{LR}}\sqrt{dθ}Z(x) = ⨁_{lmn} ∑_{e_{2}} \sqrt[4]{\frac{dλ_{1}}{dλ_{2}} \frac{dμ_{1}}{dμ_{2}} \frac{dν_{1}}{dν_{2}}}  e_{2} μ_{1} m^{∗} λ_{1} l^{∗} ν_{1} n  ⊠  ν̄_{2} ; μ̄_{2} ē_{2} λ̄_{2} , (5)

where l, m, n run over an ONB of Hom_{loc}(λ_{1}, θλ_{2}), Hom_{loc}(μ_{1}, θμ_{2}) and Hom_{loc}(ν_{1}, θν_{2}), respectively. We use the following expansion of an arbitrary intertwiner t ∈ Hom(ν, λμ) with respect to an ONB {e} of

\begin{matrix}ν \\ λ \\ μ \\ t \end{matrix} = ∑_{e} Φ_{ν}(e^{∗}t)e = \frac{1}{\sqrt{dλdμdν}} ∑_{e}  e^{∗} ; t ; \begin{matrix}ν \\ μ \end{matrix} ; \begin{matrix}e \\ λ \end{matrix}

with respect to an orthonormal basis {e} of Hom(ν, λμ). The rhs of Eq. (5) becomes

= ⨁_{lmn} ∑_{e_{1},e_{2}} \sqrt{dλ_{1}dμ_{1}dν_{1}}^{\sqrt[4]{\frac{dλ_{1}}{dλ_{2}} \frac{dμ_{1}}{dμ_{2}} \frac{dν_{1}}{dν_{2}}}} _{e_{2} μ_{1} m^{∗} λ_{1} l^{∗} ν_{1} n e_{1}^{∗}} · \begin{matrix}ν_{1} \\ μ_{1} \end{matrix} ; \begin{matrix}e_{1} \\ λ_{1} \end{matrix} ⊠ ν̄_{2} ; μ̄_{2} ē_{2} λ̄_{2} .

We calculate:

\sqrt[4]{\frac{dλ_{1}}{dλ_{2}} \frac{dμ_{1}}{dμ_{2}} \frac{dν_{1}}{dν_{2}}}  _{e_{2} μ_{1} m^{∗} λ_{1} l^{∗} ν_{1} n e_{1}^{∗}} = \sqrt[4]{\frac{dλ_{1}}{dλ_{2}} \frac{dμ_{1}}{dμ_{2}} \frac{dν_{1}}{dν_{2}}} (dθ)^{− \frac{3}{2}}  _{e_{2} μ_{1} m^{∗} λ_{1} l^{∗} ν_{1} n e_{1}^{∗}}  =

= _{e_{2} m^{∗} l^{∗} n e_{1}^{∗}} = dν_{1}\sqrt{dθ}\sqrt[4]{\frac{dλ_{1}dλ_{2}dμ_{1}dμ_{2}}{dν_{1}dν_{2}}} Φ_{ν_{1}}^{1}[⋯] ,

where we first use that the intertwiners l, m, n are in Hom_{loc}( · , · ) and then replace by Lemma 4.16 with an orthonormal basis in Hom(α_{λ_{1}}^{+}, α_{λ_{2}}^{−}) and in the second step deform the ι string to obtain the left inverse of α_{nu_{1}}^{+} and Φ_{ν_{1}}^{1}[⋯] is the expression of Def. 4.17. This shows that Z(x) has the same coefficients as x_{M} from the α-induction construction.

We need the following general result as a main tool in the following sections.

Proposition 4.19 (cf. [KR08]). Let Θ_{a} and Θ_{b} be irreducible in a UMTC NC_{N}. Then Θ_{a} and Θ_{b} are Morita equivalent if and only if Z(Θ_{a}) and Z(Θ_{b}) are equivalent.

4.2 The adjoint functor of the full center

We have a tensor functor T as follows: the map

T (⨁_{i} λ_{i} ⊠ μ̄_{i}) = ⨁_{i} λ_{i} ∘ μ̄_{i}  (6)

is an extension of the monoidal product (which by definition is a bifunctor). We have T (id_{N} ⊠ id_{N}) = id_{N} and the family of morphisms

μ_{(ρ_{1}⊠σ̄_{1}),(ρ_{2}⊠σ̄_{2})} : T (ρ_{1} ⊠ σ̄_{1}) ∘ T (ρ_{2} ⊠ σ̄_{2}) ⟶ T (ρ_{1}ρ_{2} ⊠ σ̄_{1}σ̄_{2})
μ_{(ρ_{1}⊠σ̄_{1}),(ρ_{2}⊠σ̄_{2})} := (1_{ρ_{1}} ⊗ ε(ρ_{2}, σ̄_{1})^{∗} ⊗ 1σ̄_{2}) ≡ ρ_{1}(ε(ρ_{2}, σ̄_{1})^{∗})  (7)

extends to a family

μ_{(β_{1}),(β_{2})} : T (β_{1}) ∘ T (β_{2}) ⟶ T (β_{1} ∘ β_{2}),  β_{1}, β_{2} ∈ _{N}C_{N} ⊠ \overline{_{N}C_{N}}

and makes the following diagram commute:

\begin{matrix}T (β_{1}) ∘ T (β_{2}) ∘ T (β_{3}) & ⟶ & T (β_{1}) ∘ T (β_{2} ∘ β_{3}) \\ ↓ & & ↓ \\ T (β_{1} ∘ β_{2}) & ⟶ & T (β_{1} ∘ β_{2} ∘ β_{3}) \end{matrix}.

This means T is a (strict with respect to the unity but in general non-strict for associativity, i.e. μ_{•,•} ≠ 1) strong monoidal functor (tensor functor). It is well known that strong monoidal functors map monoids into monoids, by this we can conclude that for Θ_{2} = (θ_{2}, w_{2}, x_{2}) a Q-system in _{N}C_{N} ⊠ \overline{_{N}C_{N}} we obtain a (reducible) Q-system T (Θ_{2}) = (T (θ_{2}), w_{T (Θ_{2})}, x_{T (Θ_{2})}) by

w_{T (Θ_{2})} = T (w_{2}),  x_{T (Θ_{2})} = μ_{θ_{2},θ_{2}}^{∗} · T (x_{2})

or explicitely by (t_{i}^{jk} ∈ Hom(ρ_{i} ⊠ σ̄_{i}, ρ_{j}ρ_{k} ⊠ σ̄_{j}σ̄_{k}))

θ = ⨁_{i} ρ_{i} ⊠ σ̄_{i}  x = ⨁_{ijk} t_{i}^{jk}
T (θ_{2}) = ⨁_{i} ρ_{i}σ̄_{i}  x_{T (Θ)} = ⨁_{ijk} \underbrace{ρ_{j}(ε(ρ_{k}, σ̄_{j})) · T (t_{i}^{jk})}_{∈Hom(ρ_{i}σ̄_{i},ρ_{j} σ̄_{j}ρ_{k}σ̄_{k})} .

We note that even if Θ is commutative T (Θ) is in general not commutative, because the functor is not braided.

We introduce the notion of a direct sum for Q-systems (cf. [EP03, p. 321]). Let _{N}C_{N} ⊂ End(N) be a URFC and {Θ_{i} = (θ_{i}, w_{i}, x_{i})}_{i=1,...,n} be Q-systems in NC_{N}. The direct sum Q-system Θ = (θ, w, x) with θ = ⨁_{i=1}^{n} θ_{i} is defined by

θ = ∑_{i=1}^{n} Ad T_{i} ∘ θ_{i} , w = \frac{1}{\sqrt{d(θ)}} ∑_{i=1}^{n} d_{i} · T_{i} · w_{i} , x = ∑_{i=1}^{n} θ(T_{i})T_{i}x_{i}T_{i}^{∗} ,

where d_{i} = \sqrt{d(θ_{i})} = d(ι_{i}) and T_{i} are generators of the Cuntz algebra with n elements, i.e. T_{i}^{∗}T_{j} = δ_{ij} · 1 and ∑_{i} T_{i}T_{i}^{∗} = 1. If (θ_{i}, w_{i}, x_{i}) corresponds to the subfactor N ⊂ M_{i} with inclusion map ι_{i}, then (θ, w, x) corresponds to the inclusion N ⊂ ⨁_{i=1}^{n} M_{i}. The p_{i} = T_{i}T_{i}^{∗} give a decomposition in the sense of Remark 4.9. The following identity has been proven on the level of objects in [Eva02, Prop. 3.3.]. We remark that a priori it is not clear that this “curious identity” holds also on the level of Q-systems. It is directly related to the adding the boundary construction in [CKL13] as we discuss in Sect. 6.6.

Proposition 4.20 (cf. [KR08, Prop. 4.3]). Let _{N}C_{N} ⊂ End(N) be a UMTC and Θ a Q-system in _{N}C_{N} with corresponding subfactor N ⊂ M. Then we have an equivalence of Q-systems:

T (Z(Θ)) ≅ ⨁_{a∈_{N} Δ_{M}} Θ_{a}.

Our first aim was to prove this identity directly for the α-induction construction. We had a graphical proof for the trivial Q-system. Because the αinduction construction coincides with the full center it follows now easily from the general results of [KR08].

Proof. We note (see Rem. 3.6) that the Q-system Θ_{a} for some a ∈ _{N}C_{M} or equivalently ā ∈ _{M}C_{N} corresponds on the nose with the Q-system Φ(ā)^{∨} ⊗_{Θ} Φ(ā) = Φ(a) ⊗_{Θ} Φ(ā) constructed in [KR08], where Φ: _{M}C_{N} → Bim(Θ, id) is the functor in Prop. 3.2. Then one can directly apply [KR08, Prop. 4.3].

As a corollary this implies the “curious identity” which was proven in [Eva02, Prop. 3.3.] and shows that behind this identity indeed sits more structure.

Corollary 4.21 (cf. [Eva02, Prop. 3.3.], see also [BEK99, Cor 6.13.]). Let N ⊂ M be a non-degenerately braided type III subfactor and Z_{λμ} = ⟨α_{λ}^{+}, α_{μ}^{−}⟩ for λ, μ ∈ _{N}Δ_{N}. Then we have

⨁_{a∈_{N} Δ_{M}}[aā] = ⨁_{ρ,σ∈_{N} Δ_{N}} Z_{ρσ}[ρσ̄]  (8)

and in particular the number of elements in _{N}Δ_{M} or _{M}Δ_{N} is given by

|_{N}Δ_{M}| = |_{M}Δ_{N}| = ∑_{ρ∈_{N} Δ_{N}} Z_{ρρ} .

Remark 4.22. The functor T ( · ) gives a (left) adjoint to the full center Z( · ), namely Θ is a sub-Q-system of T (Z(Θ)).

5 Modular invariance and Q-systems in _{N}C_{N} ⊠ \overline{_{N}C_{N}}

5.1 Characterization of modular invariant Q-systems

Let _{N}C_{N} ⊂ End(N) be a UMTC. Given a Q-system Θ and the corresponding extension ι(N) ⊂ M let Z_{μν} = ⟨α_{μ}^{+}, α_{ν}^{−}⟩ for μ, ν ∈ _{N}Δ_{N}. The matrix Z = (Z_{μν})_{μ,ν∈_{N} Δ_{N}} is a modular invariant [BEK99], i.e. it commutes with S and T from (3). It is called normalized because Z_{00} = 1 and sufferable because it comes from an inclusion ι(N) ⊂ M. The α-induction construction or equivalently the full center gives a Q-system Θ_{2} in _{N}C_{N} ⊠ \overline{_{N}C_{N}} with [θ_{2}] = ⨁_{μ,ν∈_{N} Δ_{N}} Z_{μν}[μ ⊠ ν̄]. It is sometimes convenient to write the matrix (Z_{μν}) formally in character form as Z = ∑_{μ,ν∈_{N} Δ_{N}} Z_{μ,ν}χ_{μ}χ̄_{ν}.

Lemma 5.1 ([BEK00], see also [KO02, Thm 4.5]). Let _{N}C_{N} be a UMTC. If Θ is an irreducible commutative Q-system in _{N}C_{N}, then dim_{M}C_{M}^{0} = dim_{N}C_{N}/(dΘ)^{2}. In particular, dΘ ≤ dim(_{N}C_{N})^{\frac{1}{2}} .

Proof. The first statement is a combination of Thm. 4.2 and Prop. 3.1 in [BEK00]. The second statement follows from the first, using dim_{M}C_{M}^{0} ≥ 1. Using Remark 3.3 and 5.6, this also follows from [KO02, Thm 4.5].

Proposition 5.2 ([KR09, Thm. 3.4, Prop. 3.22]). Let Θ_{2} be an irreducible commutative Q-system in _{N}C_{N} ⊠ \overline{_{N}C_{N}}, then the following are equivalent:

1. dΘ_{2} = dim(_{N}C_{N})

2. Z = (Z_{μν}) is a modular invariant

3. Θ_{2} ≡ Z(Θ) for some irreducible Q-system Θ in _{N}C_{N}.

Proof. (3) are equivalent (1) by [KR09, Thm. 3.4, Prop. 3.22] (see also [Müg10, Thm 3.4], [DMNO13]). The notion of modular invariance in [KR09, Thm. 3.4] is a bit different. But by [LR04, Appendix C] we obtain that (2) implies (1), namely the argument shows that if dθ < dim(_{N}C_{N}) then Z cannot be modular invariant. Together with Lemma 5.1 this gives the statement.

(3) implies (2) is clear by the fact that Z_{μν} = ⟨α_{μ}^{+}, α_{ν}^{−}⟩ defines a modular invariant and that Z(Θ) coincides with the α-induction construction Prop. 4.18.

5.2 Permutation modular invariants

Let _{N}C_{N} ⊂ End(N) be a UMTC. A non-negative integer valued matrix Z = (Z_{μν})_{μ,ν∈_{N} Δ_{N}} with Z_{id_{N} ,id_{N}} = 1 is called a modular invariant if it commutes with the matrices S and T constructed in Subsect. 2.2. It is called realizable (sufferable) if there exists a braided subfactor (ι(N) ⊂ M,_{N}C_{N}) such that Z_{μν} = ⟨α_{μ}^{+}, α_{ν}^{−}⟩.

Proposition 5.3. Let _{N}C_{N} ⊂ End(N) be a UMTC and ϕ ∈ Aut(_{N}Δ_{N}) which only fixes the sector [id_{N}] and which extends to a braided automorphism of NC_{N}. Then there is a braided subfactor N ⊂ M_{ϕ} in _{N}C_{N} with

[θ_{ϕ}] = ⨁_{ν} n_{ν}[ν], n_{ν} = ∑_{μ}⟨μϕ(μ̄), ν⟩

which realizes the permutation modular invariant Z_{μν} = δ_{ν,ϕ(μ)}.

Proof. By the Longo–Rehren construction Def. 4.1 there is a Q-system Θ_{LR}^{ϕ} with:

[θ_{LR}^{ϕ}] = ⨁_{μ}[μ ⊠ ϕ(μ̄)] .

We define the Q-system Θ_{ϕ} := T (Θ_{LR}^{ϕ}) in _{N}C_{N} with

[θ_{ϕ}] := ⨁_{μ}[μϕ(μ̄)] = ⨁_{ν} n_{ν}[ν], n_{ν} = ∑_{μ}⟨μϕ(μ̄), ν⟩

as above which is irreducible because 0 = ⟨μϕ(μ̄), id_{N}⟩ for [μ] ≠ [id_{N}] by the assumption about ϕ not having non-trivial fixed points. Because T ( · ) is leftadjoint to Z( · ) the subfactor N ⊂ M_{ϕ} given by the Q-system Θ_{ϕ} has the modular invariant Z_{μν} = δ_{ν,ϕ(μ)}.

A particular case is, if _{N}C_{N} has no non-trivial self-conjugate sectors besides the trivial sector, in this case the charge conjugation C might fulfill the assumptions and the obtained subfactor realizes the charge conjugation modular invariant Z = C. We therefore can answer a particular case of the question how Z = C is realized, namely the case that there are no non-trivial self-conjugate charges.

Example 5.4. The UMTC E_{6,1} for example obtained by positive energy representation of loop groups, has 3 sectors {ρ_{0}, ρ_{1}, ρ_{2}} with ℤ_{3} fusion rules, i.e. [ρ_{i}ρ_{j}] = [ρ_{i+j mod 3}] for 0 ≤ i, j ≤ 2, and the charge conjugation transposes the two non-trivial charges. Then Prop. 5.3 yields a Q-system with [θ] = [ρ_{0}] ⊕ [ρ_{1}] ⊕ [ρ_{2}] which realizes Z = C, i.e. Z = |χ_{0}|^{2} + χ_{1}χ̄_{2} + χ_{2}χ̄_{1}.

If there is fixed point in the permutation the same construction as in the proof of Prop. 5.3 is possible but we do not know how a dual canonical endomorphism of an irreducible Q-system giving the modular invariant would look, because the “adjoint functor” gives a reducible Q-system. Nevertheless, we can conclude that for a permutation matrix Z of _{N}Δ_{N} which gives rise to a braided automorphism, there exists a braided subfactor ι(N) ⊂ M in _{N}C_{N} which has Z as a modular invariant, i.e. such permutation modular invariants are realizable. The category _{N}C_{N} is called pointed if all irreducible objects are invertible, i.e. have dimension 1 or in other words _{N}C_{N} = Pic(_{N}C_{N}).

Lemma 5.5. Let _{N}C_{N} ∈ End(N) be a pointed UMTC and let Θ_{1} and Θ_{2} be Q-systems. If Θ_{1} and Θ_{2} are Morita equivalent, then they are equivalent.

Proof. Let Θ_{1} and Θ_{2} be irreducible Q-systems in_{N}C_{N} which are Morita equivalent. Without lost of generatlity, we may assume that Θ_{1} = Θῑ comes from a subfactor ι(N) ⊂ M and Θ_{2} = Θ_{a} with a ∈ _{N}C_{M} irreducible. Because _{N}C_{N} is pointed the sectors form an abelian (due to the braiding) group denoted G. The multiplication in G is given by the fusion rules, i.e. NΔ_{N} = {λ_{g} : g ∈ G} with [λ_{g}λ_{h}] = [λ_{gh}] for all g, h ∈ G and [λ_{g^{−1}}] = [λ̄_{g}]. We note that ιλ_{g} is irreducible, namely by Frobenius reciprocity ⟨ιλ_{g}, ιλ_{g}⟩ = ⟨θ, λ_{g}λ̄_{g}⟩ = ⟨θ, id_{N}⟩ = 1. Therefore _{N}Δ_{M} ⊂ {λ_{g}ῑ : g ∈ G} (because there can be [λ_{g}ῑ] = [λ_{h}ῑ]). So we may assume that a = λ_{g}ῑ and can conclude that [θ_{a}] = [λ_{g}ῑιλ̄_{g}] = [θλ̄_{g}λ_{g}] = [θ]. It is easy to check that using ε(λ_{g}, θ) we can construct a unitary intertwiner θ_{a} → θλ_{g}λ̄_{g} → θ, which gives an equivalence of the two Q-systems. Alternatively, we can use that ᾱ_{λ_{g^{−1}}}^{±} is an automorphism satisfying aᾱ_{λ_{g^{−1}}}^{±} =

λ_{g}ῑα_{λ_{g^{−1}}}^{±} = λ_{g}λ_{g^{−1}} ῑ = ῑ. Then Prop. 3.8 gives an alternative proof of the statement.

Let _{N}C_{N} ⊂ End(N) be a pointed UMTC and Θ be a Q-system and Z_{μν} = ⟨α_{μ}^{+}, α_{ν}^{−}⟩. Then Lemma 5.5 shows that T (Z(Θ)) is equivalent to ⨁_{i=1}^{tr Z} Θ. Therefore in this case we obtain an easy formula for θ in terms of its modular invariant matrix Z = (Z_{μν}):

[θ] = \frac{1}{tr Z} ⨁_{ρ∈_{N} Δ_{N}} ∑_{μ,ν∈_{N} Δ_{N}} Z_{μν}N_{μν}^{ρ}[ρ] ,

see also [Pin07].

5.3 Maximal chiral subalgebras and second cohomology for modular invariant Q-systems

Let us assume that Θ is a commutative Q-system in _{N}C_{N} and N ⊂ M the associated subfactor. The category Mod(Θ) forms a (non-strict) tensor category as follows. Let ρ, σ be two right Θ-modules. Because Θ is commutative, we obtain a left action on ρ and σ using the braiding, which makes them bimodules. Then the tensor product ρ ⊗ σ is defined to be the object ρ ⊗_{Θ} σ as in Remark 3.3, which we see as right module by forgetting the left action. Let Mod_{0}(Θ) the subcategory of dyslectic modules (see [Par95, KO02]), i.e. modules (ρ, r), such that rε(θ, ρ)ε(ρ, θ) = r, graphically:

r ; \begin{matrix}ρ \\ ρ \end{matrix} ; θ = r ; \begin{matrix}ρ \\ ρ \end{matrix} ; θ .

It can easily be seen that if we give the induced right Θ-module ρθ the structure of a bimodule using the braiding that it becomes equivalent to the α-induction Φ(α_{ρ}^{±}) in Remark 3.3, where the sign is depending on the choice of the braiding. We obtain that Bim^{±}(Θ, Θ) ≅ Mod(Θ) as tensor categories, but we will just need the following fact.

Remark 5.6. The map obtained by restricting bimodules to right modules

Bim^{0}(Θ, Θ) → Mod_{0}(Θ)

is an equivalence of categories. Namely, an object in Bim^{0}(Θ, Θ) gives a dyslectic module, because using the fact that it is contained both, in the image of α^{+} and α^{−}, we can “unwind” the double braid. Conversely, if a module is dyslectic, the left action obtained by the both braidings coincide, so it must come from Bim^{0}(Θ, Θ).

For β ∈ _{M}C_{M} we define the σ-restriction σ_{β} = ῑβι ∈ _{N}C_{N}. Given Θ_{±} commutative Q-systems corresponding to N ⊂ M_{±} it follows that _{M_{±}}C_{M_{±}}^{0} are again UMTCs. Let us assume there is a braided equivalence ϕ: _{M_{+}}C_{M_{+}}^{0} → _{M_{−}}C_{M_{−}}^{0}. Now we consider the Q-system Θ_{LR}^{ϕ} in M_{+}C_{M_{+}}^{0} ⊠ \overline{_{M_{−}}C_{M_{−}}^{0}}. By composing ι_{LR} with ι_{1} ⊠ ι_{2} we obtain a Q-system

Θ_{(Θ_{+},Θ_{−},ϕ)} = Θ_{(ῑ_{1}⊠ῑ_{2})∘ῑ_{LR}^{ϕ}}

with

[θ_{LR}^{ϕ}] = ⨁_{α∈_{M_{+}} Δ_{M_{+}}^{0} ,β∈_{M_{−}} Δ_{M_{−}}^{0}} Z̃_{αβ}[α ⊠ β̄], Z̃_{αβ} = δ_{α,ϕ(β)}

[θ_{(Θ_{+},Θ_{−},ϕ)}] = ⨁_{μ,ν∈_{N} Δ_{N}} Z_{μν}[μ ⊠ ν̄],  Z_{μν} = ∑_{αβ} Z_{αβ}⟨σ_{α}^{+}, μ⟩⟨σ− ; β̄, μ̄⟩

= ∑_{τ} b_{τ,μ}^{+}b_{ϕ(τ),ν}^{−}

where b_{τ,μ}^{±} = ⟨σ_{τ}^{±}, μ⟩ for τ ∈ _{M_{±}}C_{M_{±}}^{0}. All maximal commutative Q-systems in NC_{N} ⊠ \overline{_{N}C_{N}} are of this form:

Proposition 5.7 ([DNO13, Prop. 3.7, Cor. 3.8]). There is a one-to-one correspondence between

1. Equivalence classes of commutative irreducible Q-systems Θ_{2} in _{N}C_{N} ⊠ \overline{_{N}C_{N}} with dθ_{2} = dim(_{N}C_{N}).

2. Isomorphism classes of triples (Θ_{+}, Θ_{−}, ϕ) where Θ_{±} are commutative irreducible Q-systems in _{N}C_{N} and ϕ: _{M_{+}}C_{M_{+}}^{0} → _{M_{−}}C_{M_{−}}^{0} is an equivalence of braided categories.

3. Indecomposable module categories over _{N}C_{N}.

Proof. This statement is proven in a more general setting in [DNO13, Prop. 3.7, Cor. 3.8]. They call the objects in point 1) Lagrangian algebras. We use that by Remark 3.3 and 5.6 (see also [Müg10, Thm 3.1]) the category _{M_{+}}C_{M_{+}}^{0} is equivalent to the category of dyslectic modules.

We note that there can exist inequivalent ϕ_{1}, ϕ_{2} giving the same modular invariant Z = (Z_{μν}). Namely if ⟨σ_{ϕ_{1}(τ)}, μ⟩ = ⟨σ_{ϕ_{2}(τ)}, μ⟩ holds for all τ ∈ _{M_{+}}C_{M_{+}}^{0} and μ ∈ _{N}C_{N} for which b_{τ,μ}^{+} ≠ 0. Because ϕ_{1} and ϕ_{2} are inequivalent the Q-systems Θ_{LR}^{ϕ_{1}} and Θ_{LR}^{ϕ_{2}} are inequivalent. This (or using Prop. 5.7) implies that also Θ_{(Θ_{+},Θ_{−},ϕ_{1})} and Θ_{(Θ_{+},Θ_{−},ϕ_{2})} are inequivalent. This means that the second cohomology (see Rem. 2.2) of Θ_{(Θ_{+},Θ_{−},ϕ_{1,2})} does not vanish in this case.

Example 5.8. Let us consider for _{N}C_{N} the UMTC obtained by SU(3)_{9} and Θ_{+} coming from the conformal inclusion SU(3)_{9} ⊂ E_{6,1}. As in Ex. 5.4 the UMTC category E_{6,1} has three sectors _{M_{+}}Δ_{M_{+}}^{0} = {β_{0}, β_{1}, β_{2}} and we obtain an extension M_{+} ⊂ M̃ with [θ̃] = [β_{0}] ⊕ [β_{1}] ⊕ [β_{2}], which gives the permutation modular invariant interchanging β_{1} ↔ β_{2}. Now σ_{β_{1}}^{+} = σ_{β_{2}}^{+}, so both inclusions N ⊂ M_{+} and N ⊂ M̃ give by the above discussion the same modular invariant with respect to SU(3)_{9}, which is Z = |χ_{0,0} + χ_{9,0} + χ_{0,9} + χ_{4,1} + χ_{1,4} + χ_{4,4}|^{2} + 2|χ_{2,2} + χ_{5,2} + χ_{2,5}|^{2}. This example appeared in [BE01], cf. [EP09, EP11]. So we can conclude that Θ_{(Θ_{+},Θ_{+},id)} and Θ_{(Θ_{+},Θ_{+},ϕ)} in _{N}C_{N} ⊠ \overline{_{N}C_{N}} have isomorphic endomorphisms [θ_{(Θ_{+},Θ_{+},id)}] = [θ_{(Θ_{+},Θ_{+},ϕ)}] but the Q-systems are not equivalent. So we have an example where the second cohomology does not vanish. The same happens for the inclusion^{4} G_{2,3} ⊂ E_{6,1} where Z = |χ_{00} + χ_{11}|^{2} + 2|χ_{02}|^{2}.

6 Conformal nets

We now apply the results to conformal nets. Let \overline{ℝ} = ℝ∪\{∞\} be the one-point compactification of the real line ℝ, which we can by the Cayley map \overline{ℝ} ∋ x ↦ z = \frac{i−x}{i+x} ∈ 𝕊^{1} identify with the circle 𝕊^{1} ⊂ ℂ. We denote by Möb the Möbius group which is isomorphic to both:

• PSL(2, ℝ), which acts naturally on the real line \overline{ℝ}, and

• PSU(1, 1), which acts naturally on the circle 𝕊^{1} ⊂ ℂ.

The universal covering group of Möb is denoted by M̃öb. We denote by Möb_{±} = Möb ⋊ ℤ_{2} where the action of ℤ_{2} is given by the reflection r: z ↦ z̄ on 𝕊^{1}. The rotations R(ϑ)z = e^{iϑz} on 𝕊^{1}, the dilations δ(s)x = e^{s}x on ℝ, and the translations τ(t)x = x + t on ℝ give three distinguished one-parameter subgroups of Möb which generate Möb.

4This was told to us by V. Ostrik via mathoverflow

Documenta Mathematica 20 (2015) 1137–1184

We denote by I ∈ I the set of all proper intervals on 𝕊^{1}, i.e. all open, connected, non-dense, non-empty intervals I ⊂ 𝕊^{1}.

Definition 6.1. A local Möbius covariant net (conformal net) A on 𝕊^{1} is a family {A(I)}_{I∈I} of von Neumann algebras on a Hilbert space H_{A}, with the following properties:

A. Isotony. I_{1} ⊂ I_{2} implies A(I_{1}) ⊂ A(I_{2}).

B. Locality. I_{1} ∩ I_{2} = ∅ implies [A(I_{1}), A(I_{2})] = {0}.

C. Möbius covariance. There is a unitary representation U of Möb on H such that U(g)A(I)U(g)^{∗} = A(gI).

D. Positivity of energy. U is a positive energy representation, i.e. the generator L_{0} (conformal Hamiltonian) of the rotation subgroup U(R(θ)) = e^{iθL_{0}} has positive spectrum.

E. Vacuum. There is a (up to phase) unique rotation invariant unit vector Ω ∈ H which is cyclic for the von Neumann algebra ⋁_{I∈I} A(I).

The Reeh–Schlieder property automatically holds [FJ96], i.e. Ω is cyclic and separating for any A(I) with I ∈ I. Furthermore, we have the Bisognano–Wichmann property [GF93, BGL93] saying that the modular operators with respect to Ω have geometric meaning; e.g. the modular operators for the upper circle I_{0} are given by the dilation Δ^{it} = U(δ(−2πt)) and reflection J = U(r), where here U is extended to Möb_{±}. For a general interval I ∈ I the modular operators are given by a special conformal transformation δ_{I} and a reflection r_{I} both fixing the endpoints of I. The Bisognano–Wichmann property implies Haag duality

A(I)′ = A(I′)  I ∈ I

and it can be shown (see e.g. [GF93]) that each A(I) is a type III_{1} factor in Connes’ classification [Con73]. A conformal net is additive [FJ96], i.e. for intervals I ∈ I and I_{1}, …, I_{n} ∈ I we have

I ⊂ ⋃_{i} I_{i} ⟹ A(I) ⊂ ⋁_{i} A(I_{i}) .

A local Möbius covariant net on A on 𝕊^{1} is called completely rational if it

F. fulfills the split property, i.e. for I_{0}, I ∈ I with \overline{I_{0}} ⊂ I the inclusion A(I_{0}) ⊂ A(I) is a split inclusion, namely there exists an intermediate type I factor M such that A(I_{0}) ⊂ M ⊂ A(I).

G. is strongly additive, i.e. for I_{1}, I_{2} ∈ I two adjacent intervals obtained by removing a single point from an interval I ∈ I the equality A(I_{1})∨A(I_{2}) = A(I) holds.

H. for I_{1}, I_{3} ∈ I two intervals with disjoint closure and I_{2}, I_{4} ∈ I the two components of (I_{1} ∪ I_{3})′, the μ-index of A

μ(A) := [(A(I_{2}) ∨ A(I_{4}))′ : A(I_{1}) ∨ A(I_{3})]  (9)

(which does not depend on the intervals I_{i}) is finite.

Example 6.2. Examples of completely rational local Möbius covariant nets are:

• Diffeomorphism covariant nets with central charge c < 1 [KL04a].

• The nets A_{L} where L is a positive even lattice [DX06] which contain as a special case [Bis12] loop group nets A_{G,1} at level 1 for G a compact connected, simply connected simply-laced Lie group.

• The loop group nets A_{SU(n),ℓ} for SU(n) at level ℓ. [Xu00].

Further examples of rational conformal nets can be obtained from these as follows:

• Finite index extensions and subnets of completely rational conformal nets. Namely, let A ⊂ B be a finite subnet i.e. [B(I) : A(I)] < ∞ for some (then all) I ∈ I, then A is completely rational iff B is completely rational [Lon03], in particular orbifolds A^{G} of completely rational nets A with G a finite group are completely rational.

• Let A ⊂ B be a co-finite subnet , i.e. [B(I), A(I) ∨ A^{c}(I)] < ∞ for some (then all) I ∈ I, where the coset net A^{c} is defined by A^{c}(I) = A′∩B(I) with A′ = (∨_{I∈I}A(I))′. Then B is completely rational iff A and A^{c} are completely rational [Lon03]. This gives many example of completely rational nets coming from the coset construction.

A separable (non-degenerated) representation of a strongly additive local Möbius covariant net is a family π = {π_{I}: A(I) → B(H_{π})}_{I∈I} of unital representations (∗-homomorphisms) π_{I} of A(I) on a common separable Hilbert space H_{π}, which are compatible, i.e.

π_{I_{2}} ↾ A(I_{1}) = π_{I_{1}},  I_{1} ⊂ I_{2} .

Such a representation is automatically normal, i.e. all π_{I} are strongly continuous. We denote by DHR(A) the category of separable representations, where morphisms in Hom(π^{1}, π^{2}) are given by intertwiners V ∈ B(H_{π^{1}}, H_{π^{2}}), such that V π_{I}^{1}(a) = π_{I}^{2}(a)V for all I ∈ I and a ∈ A(I). Let us denote by DHR^{0}(A) the representations π with finite statistical dimension dπ, which is defined to be

dπ := [π_{I′}(A(I′))′ : π_{I}(A(I))]^{\frac{1}{2}}

for some I ∈ I, where [M : N] is the minimal index. The definition of dπ does not depend on the choice of I.

Let us from now on fix a completely rational local Möbius covariant net A on 𝕊^{1}. The category DHR^{0}(A) is a (unitary) modular tensor category [KLM01]. Every π ∈ DHR^{0}(A) is equivalent to a representation localized in a given I_{0} ∈ I, i.e. it exists a ρ ≅ π such that H_{ρ} = H_{A} and ρ_{I′_{0}} = id_{A(I′_{0})}. Namely, π_{I′_{0}}(A(I′_{0})) on H_{π} is spatially isomorphic to A(I′_{0}) on H_{A}, by the type III property. Let U: H_{π} → H_{A} be a unitary implementing this isomorphism, then ρ = {ρ_{I} := Ad U ∘ π_{I}}_{I∈I} does the job. This implies that the category DHR^{I_{0}}(A) of representations with finite statistical dimensions which are localized in I_{0} has the same irreducible sectors as DHR^{0}(A). By Haag duality ρ ∈ DHR^{I_{0}}(A) implies ρ_{I}(A(I)) ⊂ A(I) for every I ⊃ I_{0}, that means such a representation is an endomorphism and dρ = [A(I_{0}) : ρ_{I_{0}}(A(I_{0}))]^{\frac{1}{2}} equals the dimension of the endomorphism. Together with strong additivity it follows that all intertwiners are in A(I_{0}). In particular, this means that DHR^{I_{0}}(A) can naturally be seen as a full subcategory of End(A(I_{0})) and that DHR^{I_{0}}(A) is equivalent to DHR^{0}(A). We note that the family {ρ_{I}} is determined by ρ_{I_{0}} by using strong additivity and it is really enough to consider DHR^{I_{0}}(A) as a full and replete subcategory of End(A(I_{0})) and we will drop the index I_{0}. Repleteness is just the fact that for U ∈ A(I_{0}) also Ad_{U} ∘ρ is localized in I_{0}. The braiding (also called statistics operator) is given by:

ε(ρ_{1}, ρ_{2}) = ρ_{2}(U_{1}^{∗})U_{2}^{∗}U_{1}ρ_{1}(U_{2}) ,

where U_{i} ∈ Hom(ρ_{i}, ρ̃_{i}) and ρ̃_{i} ∈ [ρ_{i}] is localized in I_{i}. Here I_{1}, I_{2} ⊂ I_{0} are two disjoint intervals such that I_{1} > I_{2} (I_{2} sits clockwise after I_{1} inside I_{0}). We also write ε^{+} for ε and define the opposite braiding by ε^{−}(ρ_{1}, ρ_{2}) = ε^{+}(ρ_{2}, ρ_{1})^{∗}. We will interpret A as the chiral observables or as chiral symmetries. For example A = Vir_{c} with c < 1 is the net generated by the chiral stress energy tensor T (x). We want to look into CFTs on Minkowski space containing the chiral observables A and boundary conditions on 𝕄_{+} which “preserve” these observables.

6.1 Extensions and Q-systems

Let M be a spacetime, e.g. Minkowski space and K a set of open spacetime regions in M, e.g. the set of double cones. Let G be a group acting locally on M and let G(O) be the set of all g ∈ G, such that there is a continuous path γ in G from the identity to g such that γ(t)O ∈ K.

Definition 6.3. A local G-covariant net A on M is a family {A(O)}_{O∈K} of von Neumann algebras on a Hilbert space H, with the following properties:

A. Isotony. O_{1} ⊂ O_{2} implies A(O_{1}) ⊂ A(O_{2}).

B. Locality. [A(O_{1}), A(O_{2})] = {0} for all pairwise spacelike separated O_{1}, O_{2} ∈ K.

C. G-covariance. There is a unitary positive energy representation U of G on H, such that U(g)A(O)U(g)^{∗} = A(gO) for all g ∈ G(O)

D. Vacuum. There is a (up to phase) unique G- invariant unit vector Ω ∈ H which is cyclic and separating for A(O) for all O ∈ K.

A G-covariant DHR representation of A is a compatible family π = {π_{O}: A(O) → B(H_{π})}_{O∈K} of representations on a Hilbert space H_{π}, such that for all O ∈ K there exists a unitary V : H_{π} → H, such that the representation ρ := Ad V ∘π is localized in O, i.e. ρ_{O_{0}} = id_{A(O_{0})} for O_{0} spacelike to O, and that there is a unitary projective representation U_{π} of G, such that Ad U_{π}(g)∘π_{O} = π_{gO} ∘ Ad U(g) for all g ∈ G(O). Given two local G-covariant nets A and B on Hilbert spaces H_{A} and H_{B}, respectively, an arrow A → B is an isometry V : H_{A} → H_{B} and a compatible family of embeddings (representation) {π_{O}: A(O) ↪ B(O)} such that for all O ∈ K we have V a = π_{O}(a)V, V U_{A}(g) = U_{B}(g)V for all g ∈ G and V Ω_{A} = Ω_{B}. A and B are called unitary equivalent if V is a unitary and π_{O} are isomorphisms. Let us assume that we have a subnet A_{0} of B, i.e. A_{0}(O) ⊂ B(O) for all O and U(g)A_{0}(O)U(g)^{∗} = A_{0}(gO). Then A = A_{0}e with e the Jones projection on \overline{∨A_{0}(O)Ω} is a G-local net on H_{A} := eH, in other words we have an arrow A → B in the above sense. We say that A is a subnet of B and B is a local extension of A. By abuse of notation we will not distinguish between the net A and its representation on the bigger Hilbert space H and write A ⊂ B or B ⊃ A for an inclusion/extension of nets. For every connected region we have a subfactor A(O) ⊂ B(O). If the subfactor is irreducible, we call the extension irreducible and if the index is finite we call the extension finite. If we have a finite irreducible extension B of A then the corresponding Q-system of A(O) ⊂ B(O) is a commutative irreducible Q-system in DHR^{O}(A) and conversely if we have a commutative irreducible Qsystem Θ in DHR^{O}(A) we obtain a finite local extension B of A. In particular we have a one-to-one correspondence between [LR95]:

• local finite irreducible extensions B ⊃ A up to unitary equivalence and

• commutative irreducible Q-systems Θ in DHR^{O}(A) up to equivalvence.

If we assume Θ to be only irreducible, we still have a relatively local extension, i.e. [A(O_{1}), B(O_{2})] = {0} for O_{1} and O_{2} spacelike separated. We call such an extension B ⊃ A also non-local extension to stress the fact that we do not assume locality of B. There is a one-to-one correspondence between [LR95]:

• finite irreducible extensions B ⊃ A up to unitary equivalence and

• irreducible Q-systems Θ in DHR^{O}(A) up to equivalence.

6.2 Representation theory of local extensions

The following is well-known to experts [Müg10].

Proposition 6.4. Let A ⊂ B a finite index inclusion of local Möbius covariant nets on 𝕊^{1} and let either net be completely rational. Then A and B are both completely rational and the inclusion is irreducible. Further, let I ∈ I be an interval N := A(I) ⊂ B(I) =: M and _{N}C_{N} = DHR^{I}(A), and Θ be the Q-system in _{N}C_{N} associated with N ⊂ M. Then DHR^{I}(B) = _{M}C_{M}^{0} as UMTCs and in particular DHR(B) is equivalent to Mod^{0}(Θ) and Bim^{0}(Θ, Θ).

Proof. Both _{M}C_{M}^{0} and DHR^{I}(B) being full and replete subcategories of End(M), the only thing which needs to be checked is that both have the same irreducible sectors. The braiding on _{M}C_{M}^{0} can be checked to give the braiding on Rep(B) since the braiding is fixed by the universal property ε(ρ_{1}, ρ_{2}) = 1 if I_{2} sits clockwise after I_{1} inside I. A sector [β] ∈ _{M}Δ_{M} is a DHR sector if and only it is in _{M}Δ_{M}^{0} (see [LR95, BE98]), which implies MC_{M}^{0} ⊂ DHR^{I}(B). To see equality, we realize that global dimensions coincide, namely dim DHR^{I}(B) ≡ μ(B) = [M : N]^{−2}μ(A) ≡ dim_{N}C_{N}/(dθ)^{2} by [KLM01] and dim_{M}C_{M}^{0} = dim_{N}C_{N}/(dθ)^{2} by Lemma 5.1.

Remark 6.5. Commutative Q-systems Θ in a UMTC_{N}C_{N} are also called quantum subgroups, so finding quantum subgroups in a given UMTCs _{N}C_{N} and finding finite index local extensions of a local Möbius covariant A net with DHR^{0}(A) ≅ _{N}C_{N} is equivalent. The representation theory of the extensions can be completely understood on a categorical level. An analogous statement for inclusions of rational VOAs appeared recently in [HKL15].

6.3 Maximal 2D nets with chiral observables A

Let A be a local Möbius covariant net on 𝕊^{1} ≅ \overline{ℝ}. By restriction we can and will see A as a net on ℝ. Then Haag duality of A on ℝ is equivalent to strong additivity of A. We will assume that A is completely rational, therefore this holds automatically. We denote by 𝕄 the two-dimensional Minkowski space and by K the set of double cones O ⊂ 𝕄. Each double cone is of the form

O = I × J := {(t, x) : t − x ∈ I, t + x ∈ J},

where I, J ∈ I_{0} are two intervals on the light-rays L_{±} = {(t, x) : t ± x = 0}. The action of Möb ≅ PSL(2, ℝ) on \overline{ℝ} gives a local action of M̃öb on ℝ as in [KL04a]. We define G_{2} = M̃öb × M̃öb which acts locally on Minkowski space 𝕄. For O ∈ K we denote by G_{2}(O) all g ∈ G_{2} such that there is a path γ: [0, 1] → G_{2} from the identity element e to g with γ(t)O ⊂ M for all t ∈ [0, 1].

We denote by A_{2} the net on H_{A} ⊗ H_{A} given by

A_{2}(I × J) := A(I) ⊗ A(J).

It is a local Möbius covariant net on M as in [KL04a]. Every DHR representation of A_{2} with finite index is a direct sum of representations of the form ρ ⊗ σ where ρ ∈ DHR(A) and σ ∈ DHR(A). The braiding is given by ε(ρ_{1} ⊗ σ_{1}, ρ_{2} ⊗ σ_{2}) = ε^{+}(ρ_{1}, ρ_{2}) ⊗ ε^{−}(σ_{1}, σ_{2}). Therefore the category of DHR representations of A_{2} with finite statistical dimensions is equivalent to DHR^{I}(A) ⊠ \overline{DHR^{J}(A)}. Let us write B_{2} ⊃ A_{2} for a local, Möbius covariant, irreducible extension of A_{2}, i.e. a local Möbius covariant net B_{2} on Minkowski space 𝕄 on the Hilbert space H_{B_{2}} with irreducible vacuum vector Ω which is extending A_{2} ≅ A ⊗ A, more precisely there is a representation π of A_{2} on H_{B_{2}}, such that π(A_{2}(O)) ⊂ B_{2}(O) is an irreducible inclusion of factors and U(g)π(A(O))U(g)^{∗} = π(A(gO)) for all double cones O ∈ K and all g ∈ G(O). By abuse of notation we will omit the π. We remember that there is a one-to-one correspondence between local irreducible extensions B_{2} ⊃ A_{2} (up to unitary equivalence) and irreducible commutative Q-systems Θ_{2} in DHR^{I}(A) ⊠ \overline{DHR^{J}(A)} (up to equivalence).

Proposition 6.6. Let B_{2} ⊃ A_{2} be a local extension. Then the following statements are equivalent:

1. The net B_{2} is a maximal local irreducible extension, i.e. if B̃_{2} ⊃ B_{2} is a local irreducible extension, then B_{2} = B̃_{2}.

2. The index [B_{2} : A_{2}] = μ_{2}(A) ≡ dim(DHR(A)).

3. The matrix (Z_{λμ}) is a modular invariant.

4. The μ-index of B_{2} is 1.

5. The net B_{2} has no non-trivial superselection sectors.

Proof. To show (2) ⇒ (1) let Θ_{2} be a Q-system in DHR^{I}(A)⊠\overline{DHR^{J}(A)} giving the extension A(I) ⊗ A(J) ⊂ B_{2}(I × J) and let us assume that [B_{2}(I × J) : A(I) ⊗ A(J)] = μ_{2}(A). By Lemma 5.1 we have the following inequality:

dΘ_{2} ≡ [B_{2} : A_{2}] ≤ dim(DHR(A ⊗ A))^{\frac{1}{2}} ≡ dim (DHR(A) ⊠ \overline{DHR(A)})^{\frac{1}{2}}
= dim(DHR(A)) ≡ μ_{2}(A) .

This implies maximality. For showing (1) ⇒ (2), let us assume that [B_{2} : A_{2}] < μ_{2}(A). We need to show that there is an extension B̃_{2} ⊋ B_{2}. This we obtain by adding the boundary [CKL13], i.e. from B_{2} we obtain a possible reducible boundary net (see Subsec. 6.6) of which we choose an irreducible subnet B_{+}. We claim B_{+} cannot be Haag dual, but this follows because [B_{+} : A_{+}] = [B_{2} : A_{2}] < μ_{2}(A) and then [LR04, Prop. 2.13] implies [B_{+}^{d} : B_{+}] > 1. So we have an inclusion A_{+} ⊂ B_{+} ⊊ B_{+}^{d} and a corresponding locally isomorphic inclusion A_{2} ⊂ B_{2} ⊊ B̃_{2} as in [LR04], in particular B_{2} was not maximal. The statements (2) and (3) are equivalent by Prop. 5.2 and the implication (5) ⇒ (1) is clear.

(2) ⇒ (4) follows by calculating the μ index [KLM01] and likewise the implication (4) ⇒ (5) is [KLM01, Corollary 32].

Proposition 6.7. There is a one-to-one correspondence between:

1. maximal local irreducible extensions B_{2} ⊃ A_{2} up to unitary equivalence.

2. Θ_{2} commutative irreducible Q-systems in DHR^{I}(A) ⊠ \overline{DHR^{I}(A)} with dθ_{2} = μ_{2}(A) up to equivalence.

3. (Non-local) irreducible extensions B ⊃ A up to Morita equivalence.

4. Irreducible Q-systems Θ in DHR^{I}(A) up to Morita equivalence.

5. Indecomposable _{N}C_{N} module categories, where N = A(I) and _{N}C_{N} = DHR^{I}(A).

6. Local chiral extensions A_{L} ⊃ A, A_{R} ⊃ A together with a braided equivalence ϕ: DHR(A_{L}) → DHR(A_{R}).

Proof. The correspondence between (1) and (2) is Prop. 6.6, the one between (3) and (4) [LR95]. Starting with (4) we obtain (2) by applying the full center and it is well defined on Morita equivalence classes and injective by Prop. 4.19. It is surjective by Prop. 5.2, so (2) and (4) are equivalent. Equivalently, one can start with B_{2} and add the boundary to obtain a Haag dual boundary net (as in the proof before) which correspond to a non-local extension. The α-induction construction gives back the original net. The correspondence between (4), (5) and (6) is just Prop. 5.7, where (6) is (2) of Prop. 5.7 reformulated in the language of nets, cf. [Müg10].

Remark 6.8. We know how the Morita equivalence looks like, see Subsec. 3.2.

6.4 Boundary conditions

Let A be a completely rational local Möbius covariant net on 𝕊^{1}, which we will see as a net on ℝ by restriction. Let 𝕄_{+} = {(t, x) ∈ 𝕄 : x > 0} be Minkowski half-plane and let K_{+} be the set of double cones O ⋐ 𝕄_{+}. Double cones O ∈ K_{+} are in one-to-one correspondence with pairs of proper intervals I, J ⊂ ℝ such that I < J. We write O = I × J. Let A_{+} be the net on 𝕄_{+} given by

A_{+}(O) = A(I) ∨ A(J)  O = I × J

which is locally covariant w.r.t. G_{+} the universal covering of Möb, namely

U(g)A_{+}(O)U(g)^{∗} = A_{+}(gO)  g ∈ G_{+}(O)

where G_{+} acts locally on O = I × J ∈ K_{+} by gO = gI × gJ and G_{+}(O) is the set of all g ∈ G_{+} such that there is a continuous path γ from the identity to g such that γ(t)O ∈ K_{+}. By the split property it follows that A_{+}(O) is spatially isomorphic to A_{2}(O) ≡ A(I) ⊗ A(J). This implies that the net A_{+} is locally isomorphic to the net A_{2} restricted to 𝕄_{+}. A boundary net B_{+} associated with A is a local, (locally) G_{+}-covariant net B_{+}, which is an irreducible extension B_{+} ⊃ A_{+}. Starting with B_{+} ⊃ A_{+}, we define the generated net B_{+}^{gen} ⊃ A on ℝ by

B_{+}^{gen}(I) = ⋁_{\begin{matrix}O∈K_{+} \\ O⊂W_{I} \end{matrix}} B_{+}(O) ⊃ A(I) ,

where W_{I} = {(t, x) : t ± x ∈ I} is the left wedge, such that its intersection on the t-axis is I. Conversely, given B ⊃ A a (non-local) extension on ℝ, we define

B_{+}^{ind}(O) = B(L) ∩ B(K)′ ,

where O = I × J and L ⋐ K, such that L ∩ K′ = I ∪ J or equivalently O = W_{L} ∩ W′_{K}. The dual net is defined by B_{+}^{d}(O) = B_{+}(O′)′ and B_{+}^{d} = B_{+} if and only if B_{+} is Haag dual. Then (B_{+}^{ind})^{gen} = B and (B_{+}^{gen})_{+}^{ind} = B_{+}^{d} = B_{+} provided B_{+} was already Haag dual. Together we have:

Proposition 6.9 ([LR04, LR95]). There is a one-to-one correspondence between the equivalence classes of:

1. boundary nets B_{+} associated with A, such that B_{+} is Haag dual.

2. boundary nets B_{+} associated with A, such that A_{+} ⊂ B_{+} is maximal.

3. (Non-local) extensions B ⊃ A on ℝ.

4. Q-systems in _{N}C_{N}, where N = A(I) and _{N}C_{N} = DHR^{I}(A).

Definition 6.10. Let B_{2} ⊃ A_{2} be local extension, i.e. a CFT on Minkowski space. A (Möbius covariant) boundary condition of B_{2} ⊃ A_{2} with chiral symmetry A is a unitary equivalence class of boundary nets B_{+} ⊃ A_{+}, where B_{2} ↾ 𝕄_{+} is locally covariantly isomorphic to B_{+}, more precisely there is a compatible family of isomorphisms Φ_{O}: B_{+}(O) → B_{2}(O) such that it restricts to an isomorphism A_{+}(O) → A_{2}(O) for all O ∈ K_{+} and that Φ is covariant respect to the covariance U_{B_{+}} of \overline{Möb} and U_{B_{2}} of \overline{Möb} × \overline{Möb} (where \overline{Möb} is the diagonal subgroup of \overline{Möb} × \overline{Möb}).

Proposition 6.11. Let B_{2} ⊃ A_{2} maximal and let A ⊂ B given by Prop. 6.7. Then there is a one-to-one correspondence between:

1. Boundary conditions of B_{2} ⊃ A_{2} with chiral symmetry A.

2. Unitary equivalence classes of B_{a} ⊃ A Morita equivalent to B ⊃ A.

3. Sectors in

NC_{M}/Pic(_{M}C_{M}) ,

where N = A(I), M = B(I) and _{N}C_{N} = DHR^{I}(A).

In particular the number of boudary conditions of B_{2} ⊃ A_{2} with chiral symmetry A is less or equal than

|_{N}Δ_{M}| ≡ ∑_{λ∈_{N} Δ_{N}} Z_{λλ} .

Proof. The following diagram commutes [LR09, Cor. 2]

{B_{+} ⊃ A_{+} maximal} {B ⊃ A} {B_{2} ⊃ A_{2} ≡ A ⊗ A} removing the boundary ∼ α-induction .

Given a boundary condition, i.e. a boundary net B_{a,+} ⊃ A_{+} let B_{a} ⊃ A be the corresponding chiral extension. We note that B_{a,+} is Haag dual (cf. [LR09, App. C]), because B_{2} is modular invariant. If we remove the boundary we obtain B_{2} ⊃ A_{2}, because the extensions are locally isomorphic and therefore isomorphic, see [LR09]. We conclude by commutativity of the above diagram that B ⊃ A and B_{a} ⊃ A are Morita equivalent, namely the α-induction construction gives equivalent two-dimensional extensions, which means the full centers are equivalent, which is equivalent to the Morita equivalence of B ⊃ A and B_{a} ⊃ A. Conversely, if we have given a chiral extension B_{b} ⊃ A Morita equivalent to B ⊃ A, then B_{b,+} ⊃ A_{+} is locally equivalent to B_{b,2} ⊃ A_{2} ↾ 𝕄_{+} obtained by α-induction. But B_{2,b} ⊃ A_{2} is isomorphic to B_{2} ⊃ A_{2} by Morita equivalence, so we get a boundary condition (this follows also from [LR04], realizing that the DHR orbit exhausts the Morita equivalence class). Choosing N = A(I), M = B(I) and _{N}C_{N} = DHR^{I}(A) the Q-systems Θ_{a} corresponding to B_{a} ⊃ A which is Morita equivalent to B ⊃ A are in one-to-one correspondence with _{N}C_{M}/Pic(_{M}C_{M}) by Prop. 3.9.

Example 6.12. We can give several cases as an example.

• If A is holomorphic, i.e. DHR(A) just contains the vacuum sector or equivalently μ(A) = 1, then B_{2} = A_{2} is maximal and the only 2D net and A_{+} is the only boundary condition. The family of holomorphic nets contains for example the conformal nets A_{L} associated with even selfdual lattices [DX06] like the E_{8} lattice, Leech lattice etc., the Moonshine net A_{♮} [KL06] and certain framed nets [KS14].

• For A from the family of conformal nets, for which DHR(A) is pointed, it follows from Lemma 5.5 that there is always just one boundary condition for each B_{2} ⊃ A_{2}. This family for example contains all conformal nets A_{L} coming from an even lattice L [DX06], which include all loop group conformal nets A_{G,1} of compact, connected, simply connected, simply laced Lie groups G (the simple one being in one-to-one correspondence with A-D-E Dynkin diagrams) at level 1 [Bis12].

• If A is any completely rational net and B_{2} = A_{LR} ⊃ A_{2} given by the trivial Longo-Rehren extension, then _{N}C_{M} ≅ _{N}C_{N} ≅ DHR(A) and the boundary conditions are given by DHR sectors of A modulo DHR automorphisms of A. This case is sometimes also called the Cardy case.

• For A = A_{SU(2),k} the two-dimensional extensions are in one-to-one correspondence with Dynkin diagrams of A-D-E type with Coxeter number k +2. The boundary conditions are given by orbits [ν] of a marked vertex ν under the automorphism group of the Dynkin diagram cf. [KLPR07].

• For A = Vir_{c} with c < 1, the only possible values for c are c = 1−6/m(m+ 1) with m = 2, 3, 4, . . .. The maximal two-dimensional extensions are in one-to-one correspondence with pairs (G_{1}, G_{2}) of Dynkin diagrams of A-D-E type with Coxeter number m and m + 1, respectively, cf. [KL04b]. The boundary conditions are given by pairs ([ν_{1}], [ν_{2}]) with [ν_{i}] the orbit of a marked vertex on G_{i} under the automorphism group of G_{i} (i = 1, 2). This result now follows also from [KLPR07].

The invertible objects (automorphisms) in _{M}C_{M} have to do with invertible defects (see for an interpretation of invertible defects in a different framework [DKR11]). The difference between two inequivalent a, b ∈ _{N}C_{M} related by an invertible β ∈ _{M}C_{M} gets important if we also consider also reducible boundary conditions in the next section.

6.5 Reducible boundary conditions

With the notation as before, let us assume B_{2} ⊃ A_{2} is a maximal extension of A_{2}. Using Prop. 6.7 we can choose a (non-local) extension B ⊃ A such that B_{2} is given by the α-induction construction of B ⊃ A. Let I be an interval, N = A(I), _{N}C_{N} = DHR^{I}(A), M = B(I) and Θ the Q-system in _{N}C_{N} giving N ⊂ M. Then every a ∈ _{N}C_{M} gives a in general reducible Q-system Θ_{a} and an extension B_{a} ⊃ A.

We can define as before

B_{a+}(O) = B_{a}(L) ∩ B_{a}(K)′ .

This net fulfills all the properties of a boundary CFT in [LR04], but the uniqueness of the vacuum and the joint irreducibility.

Proposition 6.13. Let a ∈ _{N}C_{M} possibly reducible. Then the (reducible) boundary net B_{a,}+ ⊃ A_{+} is a (reducible) boundary condition for B_{2} ⊃ A_{2}, which is given by the Q-system Z(Θ_{a}).

Proof. If a is irreducible this is already proven. Let a be reducible and let Θ_{a} = ῑι be the Q-system with inclusion ι(A(I)) ⊂ B_{a}(I). Let {p_{i}}_{i=1}^{n} be a set of minimal projections in ι(A(I))′ ∩ B_{a}(I) = Hom(ι, ι) with ∑_{i=1}^{n} p_{i} = 1 with corresponding morphisms ι_{i} ≺ ι. By the usually Reeh–Schlieder argument, the projection do not depend on the choice of I. The inclusion ι(A(I)) ⊂ B_{a}(I) is conjugated to

\{(\begin{matrix}ι_{1}(a) \\ & . & . & . \\ & & & & ι_{n}(a) \end{matrix}) : a ∈ A(I)\} ⊂ B_{a}(I) ⊗ M_{n}(ℂ) ≅ B_{a}(I) .

With the same notation A_{+}(O) ⊂ B_{a,+}(O) is conjugated to:

\{(\begin{matrix}ι_{1}(a) \\ & . & . & . \\ & & & & ι_{n}(a) \end{matrix}) : a ∈ A_{+}(O)\} ⊂ \{(\begin{matrix}b \\ & . & . & . \\ & & & & b \end{matrix}) : b ∈ B_{a,+}(O)\} . (10)

Because Θ_{2} := Z(Θ_{a}) and Z(Θῑ_{i}) are equivalent (by Prop. 4.19) every B_{i,+} ⊃ A_{+} is a boundary condition for B_{2} ⊃ A_{2}. But then also the inclusion B_{2} ⊃ A_{2} is locally isomorphic to B_{a,+} ⊃ A_{+} by (10) and the isomorphism restricted to A_{2} gives a local isomorphism of A_{2} restricted to 𝕄_{+} and A_{+}.

Note that in the reducible case the vacuum Ω of B_{+} is neither cyclic nor unique and that Ω = ∑_{i=1}^{n} Ω_{i} with Ω_{i} = p_{i}Ω. The restriction of B_{+} to the subspace \overline{B_{+}(O)Ω_{i}} is unitarily equivalent to the boundary condition coming from ι_{i}. In other words, _{N}C_{M} ∋ a ↦ B_{a,+} maps direct sums of sectors to direct sums of boundary conditions.

Example 6.14. Consider a, b ∈ _{N}C_{M} irreducible and mutually inequivalent but related by an automorphism β ∈ _{M}C_{M}, or equivalently Θ_{a} ≅ Θ_{b}. This means the boundary conditions coming from a and b are the same, but for example the boundary conditions coming from c := a ⊕ a and d := a ⊕ b are different. This can be seen for example by regarding the relative commutants of the subfactors associated with Θ_{c} and Θ_{d}, namely c̄(N)′ ∩ N ≅ ℂ ⊕ ℂ, while d̄(N)′ ∩ N ≅ M_{2}(ℂ).

6.6 Adding the boundary

In [CKL13] a purely operator algebraic construction of all boundary conditions is given. As a result a boundary net is obtained which is the direct sum of all boundary conditions. Let us consider the inclusion

A(I) ⊗ A(J) ⊂ B_{2}(O)

for some fixed O = I × J ⋐ W and let Θ_{2} be the associated Q-system in DHR^{I}(A) ⊠ \overline{DHR^{J}(A)}. Let Ω be the vacuum in H_{A} and let us define the state φ_{0}(x ⊗ y) = (Ω, xyΩ) for x ∈ A(I), y ∈ A(J) and let ε_{O}: B_{2}(O) → A_{2}(O) ≅ A_{+}(O) be the conditional expectation. This gives a state φ = φ_{0} ∘ ε_{0} on B_{2}(O) (which can be extended to a state on 𝔄_{2}(W)). Using the GNS representation one get an inclusion A_{+}(O) ⊂ B_{+}(O) on a bigger Hilbert space and which is by construction isomorphic to A_{2}(O) ⊂ B_{2}(O). This construction extends to 𝔄_{2}(W) and gives a (reducible) boundary net {B_{+}(O)}_{O∈K_{+}}. Let us define B(I) = ⋁_{K_{+}∋O⊂W (I)} B_{+}(O) where W (I) is the left wedge such that its intersection with the time axis x = 0 is equals I. This gives a non-local extension B ⊃ A. Let us fix L ⊃ I ∪ J, then the Q-system of B(L) ⊃ A(L) can be chosen to be localized in I ∪ J and it can be in particular trivially extended from the inclusion A_{+}(O) ⊂ B_{+}(O) using strong additivity. Let’s denote its Q-system by Θ̃.

Proposition 6.15. Let B_{2} ⊃ A_{2} be a local irreducible extension with Q-system Θ_{2}. The Q-system of the inclusion A(I) ⊂ B(I), where B = B_{+}^{gen} and B_{+} is obtained by adding the boundary is equivalent to the Q-system T (Θ_{2}).

Proof. We have to show that Θ̃ is equivalent to T (Θ_{2}), where we see Θ_{2} as a Q-system by the equivalence _{N}C_{N} ⊠ \overline{_{N}C_{N}} ≅ DHR^{O}(A_{2}). An endomorphism ρ^{I} ⊠ σ̄^{J} gives an endomorphism ρ^{I}σ̄^{J} ∈ End(A(I) ∨ A(J)) and this gives actually an isomorphism of tensor categories

End(A(I) ⊗ A(J)) ≅ End(A(I) ∨ A(J)) .

Starting from an object in DHR^{O}(A_{2}) the image is a localized endomorphism of A(I) ∨ A(J) which can by strong additivity be extended to a localized endomorphism of End(A(L)), so we get a tensor functor

T̃: DHR^{I}(A_{2}) → DHR^{L}(A) ≡ _{N}C_{N}

where we choose N := A(L) and _{N}C_{N} = DHR^{L}(A). We note that the μ from (7) is trivial as is ε(ρ_{2}, σ̄_{1}) because of the order of localization. So the functor

NC_{N} ⊠ \overline{_{N}C_{N}} ≅ DHR^{O}(A_{2}) → DHR^{L}(A) ≡ _{N}C_{N}

is by construction equivalent to the tensor T from Subsec. 4.2 and, in particular Θ̃ is equivalent to T (Θ_{2}).

2D Conformal Nets and Its Boundary conditions 1181

This gives as an alternative proof of Prop. 6.11. Let us assume B_{2} was modular invariant/maximal. All boundary conditions are obtained by the adding the boundary construction, and by Prop. 4.20 we can conclude:

Corollary 6.16. All boundary conditions of B_{2} come from an a ∈ _{N}Δ_{M}, where N = A(I), M = B(I), _{N}C_{N} = DHR^{I}(A) and B ⊂ A is any (non-local) extension giving B_{2} by the α-induction construction.

Acknowledgements

The authors would like to thank Karl-Henning Rehren for discussions and remarks on the manuscript. We thank Ingo Runkel and Christoph Schweigert for E-mail correspondence. M.B. would like to thank David E. Evans, Noah Snyder and Chenchang Zhu for discussions.

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Marcel Bischoff Vanderbilt University Department of Mathematics 1326 Stevenson Center, Nashville, TN 37240, USA marcel.bischoff@vanderbilt.edu Yasuyuki Kawahigashi Graduate School of Mathematical Sciences The University of Tokyo Komaba, Tokyo 153-8914, Japan and Kavli IPMU (WPI) The University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan yasuyuki@ms.u-tokyo.ac.jp

Roberto Longo

Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 I-00133 Roma, Italy longo@mat.uniroma2.it