1 Introduction↩︎

Conformal nets and vertex operator algebras (VOAs) are two major mathematical axiomatizations of two dimensional chiral conformal field theories (CFTs). In the conformal net approach, chiral CFTs are formulated and studied under the framework of algebraic quantum field theory [5], [6] which relies heavily on the methods of operator algebras and especially Jones subfactor theory [7]–[9]. VOAs originate from the study of Moonshine conjecture and infinite dimensional Lie algebras [10], [11], and are deeply related to modular forms [12] and the moduli spaces of algebraic curves [13]–[16]. The reason for such differences in the two approaches can be explained partially by the fact that the full-boundary CFTs extending conformal nets live on Minkowskian spacetimes, whereas those extending VOAs live on Euclidean spacetimes (e.g. compact Riemann surfaces with boundaries); see [17] for a detailed discussion. Despite such differences, one has many similar and parallel results in the two approaches; see [1], [2] for an overview.

Tensor categories (cf. for example [18], [19]), which relate rational CFTs with 3d topological quantum field theories (first observed in physics by Witten [20]) in a mathematically rigorous way [21], play an important role in the representation theories of conformal nets [22]–[24] and VOAs [25]–[27]. Moreover, many similar results in the two approaches can be formulated in terms of tensor categories. Therefore, it is desirable to establish a framework under which one can systematically relate conformal nets and VOAs and their representation tensor categories.

The first important progress on this problem was made by Carpi-Kawahigashi-Longo-Weiner in [28]. Using smeared vertex operators, the authors of [28] gave a natural construction of a conformal net \(\mathcal{A}_V\) from a sufficiently nice (i.e. strongly local) unitary simple VOA \(V\), and proved that many familiar unitary VOAs satisfy these nice conditions. Their ideas were later generalized by the author to the intertwining operators of VOAs to prove the unitarity of the braided tensor category \(\mathrm{Rep}^{\mathrm u}(V)\) of unitary \(V\)-modules [29]–[31], and to prove the unitary equivalence of \(\mathrm{Rep}^{\mathrm u}(V)\) with the braided \(C^*\)-tensor category \(\mathrm{Rep}^{\mathrm f}(\mathcal{A}_V)\) of dualizable (i.e. finite-dimensional) \(\mathcal{A}_V\)-modules [3]. As an alternative to smeared vertex operators, in [32]–[34], Tener used Segal CFT [35], [36] to relate \(V\) and \(\mathcal{A}_V\), and proved many similar results as well as new ones. Most remarkably, he gave in [34] the first systematic and complete proof of the complete rationality of the conformal nets associated to unitary affine VOAs and type \(ADE\) discrete series \(W\)-algebras.

1.1 Equivalence of braided \(C^*\)-tensor categories↩︎

In this article, we continue our study of the equivalence of braided \(C^*\)-tensor categories \(\mathrm{Rep}^{\mathrm u}(V)\simeq\mathrm{Rep}^{\mathrm f}(\mathcal{A}_V)\) initiated in [3]. The techniques in this article allow us to finish proving the long-standing conjecture that unitary affine VOAs and their conformal nets have equivalent modular tensor categories. Moreover, our approach is general enough to be applied to many other examples including lattice VOAs, parafermion VOAs, and type \(ADE\) discrete series \(W\)-algebras.

One of the main unsolved problems in [3] is proving the strong braiding of smeared intertwining operators. To be more precise, we have shown in [3] that if \(V\) has sufficiently many intertwining operators satisfying

(a) polynomial energy bounds,

(b) strong intertwining property,

(c) strong braiding,
then \(\mathrm{Rep}^{\mathrm u}(V)\) is equivalent to a braided \(C^*\)-tensor subcategory of \(\mathrm{Rep}(\mathcal{A}_V)\). Although these three conditions are expected to hold for all rational unitary chiral CFTs, the condition of strong braiding is much more difficult to prove than the other two. In [3], the only method for proving strong braiding is to show that sufficiently many intertwining operators satisfy linear energy bounds (see section 3.1 for the precise definition), which is NOT expected to hold even for all unitary affine VOAs. As a consequence, in [3] we were not able to prove the equivalence of tensor categories for type \(BDEF\) affine VOAs. This difficulty is completely resolved in this article: we show that strong braiding follows from polynomial energy bounds and the strong intertwining property, i.e. that (a) and (b) imply (c). See theorem 2 for more details. Consequently, we are able to prove \(\mathrm{Rep}^{\mathrm u}(V)\simeq\mathrm{Rep}^{\mathrm f}(\mathcal{A}_V)\) for a huge number of examples including all WZW models and their regular cosets. The more precise statement is the following:

1.2 Strong braiding of smeared intertwining operators↩︎

The strong braiding of intertwining operators is a generalization of the strong locality of vertex operators [28] and the strong intertwining property of intertwining operators [30]. We now explain the meaning of these notions.

Let us assume that \(V\) is a unitary simple VOA. Let \(Y(v,z)\) be the vertex operator associated to the vector \(v\in V\). For any \(f\in C^\infty(\mathbb{S}^1)\) one can define the smeared vertex operator \[\begin{align} Y(v,f)=\oint_{\mathbb{S}^1} Y(v,z)f(z)\frac{dz}{2\mathbf{i}\pi}=\int_0^{2\pi}Y(v,e^{\mathbf{i}\theta})f(e^{\mathbf{i}\theta})\cdot\frac{e^{\mathbf{i}\theta}}{2\pi}d\theta \end{align}\] which is indeed a closable/closed operator on \(\mathcal{H}_0\), the Hilbert space completion of \(V\). (Note that since \(V\) is unitary, as a vector space \(V\) has an inner product.) Moreover, \(V\) is inside the domain of any (finite) product of smeared vertex operators. ¹

Smeared vertex operators always satisfy weak locality, which means that if \(I,J\) are disjoint (non-dense and non-empty) open intervals of the unit circle \(\mathbb{S}^1\), and if \(u,v\in V,f\in C_c^\infty(I),g\in C_c^\infty(g)\), then \([Y(u,f),Y(v,g)]=0\) when acting on any vector inside \(V\). However, to construct a conformal net \(\mathcal{A}_V\) from \(V\), one needs the strong locality (of vertex operators), which means that for the above \(u,v,f,g\), the von Neumann algebras generated by \(Y(u,f)\) and by \(Y(v,g)\) commute. (In this case, we say that the closable/closed operators \(Y(u,f)\) and \(Y(v,g)\) commute strongly. Weak commutativity does not always imply strong commutativity by the celebrated example of Nelson [37].)

One can generalize the notion of strong locality to smeared intertwining operators. Assume \(V\) is regular (equivalently, rational and \(C_2\)-cofinite [38]), which corresponds to rational chiral CFTs in physics. We also assume that \(V\) is strongly unitary, i.e. any \(V\)-module admits a unitary structure. An intertwining operator of \(V\) is a “charged field" of \(V\). More precisely, if \(W_i,W_j,W_k\) are unitary \(V\)-modules, then a type \(W_k\choose W_iW_j\) (written as \(k\choose i~j\) for short) intertwining operator \(\mathcal{Y}\) associates linearly to each vector \(w^{(i)}\in W_i\) a multivalued function \(\mathcal{Y}(w^{(i)},z)\) on \(z\in\mathbb{C}^\times=\mathbb{C}-\{0\}\) whose values are linear maps from \(W_j\) to the”algebraic completion" of \(W_k\).(See [39] for the rigorous definition.) We say that \(W_i\) is the charge space of \(\mathcal{Y}\). Let \(\mathcal{H}_i,\mathcal{H}_j,\mathcal{H}_k,\dots\) be the Hilbert space completions of \(W_i,W_j,W_k,\dots\). Then one can define the smeared intertwining operator in a similar way as smeared vertex operator, except that the smeared intertwining operator depends not only on the smooth function \(f\in C_c^\infty(I)\) supported in an interval \(I\), but also on a choice of (continuous) argument function \(\arg_I\) of \(I\) (equivalently, a branch of \(I\) in the universal cover of \(\mathbb{S}^1\)). We call \(\widetilde{f}=(f,\arg_I)\) an arg-valued function. Then for any \(w^{(i)}\in W_i\) and for any \(\widetilde{f}\), we have a closable/closed smeared intertwining operator \(\mathcal{Y}(w^{(i)},\widetilde{f})\) mapping from (a dense subspace of) \(\mathcal{H}_j\) to \(\mathcal{H}_k\). Note that a vertex operator is a special kind of intertwining operator. Then the smeared intertwining operator \(\mathcal{Y}(w^{(i)},\widetilde{f})\) and the smeared vertex operators \(Y_j(v,g),Y_k(v,g)\) associated to the vertex operators \(Y_j,Y_k\) of the unitary \(V\)-modules \(W_j,W_k\) satisfy the weak locality: if the supports of \(\widetilde{f}\) and \(g\) are disjoint, then the weak intertwining property \[\begin{align} \mathcal{Y}(w^{(i)},f)Y_j(v,g)=Y_k(v,g)\mathcal{Y}(w^{(i)},f) \end{align}\] holds when acting on \(W_j\). If for any \(\widetilde{f},g,v\) this commutative relation holds “strongly" in the sense of the commutativity of the associated von Neumann algebras, then we say that the charged field \(\mathcal{Y}(w^{(i)},z)\) satisfies the strong intertwining property. (See section 3.3 for details.) More generally, one can study the weak and strong locality of two smeared intertwining operators, which are called the weak and strong braiding of intertwining operators in our paper. (See section 3.5). Moreover, weak braiding is always true and not hard to show,² while strong braiding requires more difficult techniques to be proved.

By saying that \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded, we mean roughly that there exists \(n\in\mathbb{N}\) such that the smeared intertwining operator \(\mathcal{Y}(w^{(i)},\widetilde{f})\) is bounded by \(L_0^n\) for each \(\widetilde{f}\). (See section 3.1 for details.) If \(\mathcal{Y}(w^{{(i)}},z)\) is energy-bounded for each vector \(w^{(i)}\), we say that \(\mathcal{Y}\) is an energy-bounded intertwining operator. We say that \(V\) is strongly energy-bounded if the vertex operators associated to all unitary \(V\)-modules are energy-bounded. We say that \(V\) is strongly unitary if each \(V\)-module admits a unitary structure. The following main theorem of this article implies theorem 1.

Roughly speaking, the assumption on intertwining operators is that \(V\) has sufficiently many intertwining operators that are energy-bounded and satisfy the strong intertwining property. Part (1) of this theorem is not new and follows essentially from [29], [30]. Part (2) claims the equivalence of braided \(C^*\)-tensor structures. (Note that the unitary ribbon structure is uniquely determined by the braided \(C^*\)-tensor structure.) Part (3) claims the strong braiding, which follows immediately from the method used to prove (2).

1.3 Main idea of the proof↩︎

Let us explain the main idea of proving parts (2) and (3) of theorem 2. We first compare this theorem with the main result of [3], which says roughly that the assumptions of theorem 2, together with (3), imply (2). Thus, the main improvement of theorem 2 is that part (3) (strong braiding) is now a consequence but no longer an assumption.

The main idea in [3] is the following. For the conformal net \(\mathcal{A}_V\) and any representations \(\mathcal{H}_i,\mathcal{H}_j\), for suitable \(\xi\in\mathcal{H}_i\) and \(\eta\in\mathcal{H}_j\) “localized" in disjoint arg-valued intervals \(\widetilde{I},\widetilde{J}\), one has fusion product \(\xi\boxtimes\eta\), written as \(L(\xi,\widetilde{I})\eta\) and also as \(R(\eta,\widetilde{J})\xi\). \(L(\xi,\widetilde{I})\) and \(R(\eta,\widetilde{J})\) describe the left and right actions of \(\xi,\eta\) on \(\mathcal{H}_j\) and \(\mathcal{H}_i\) respectively. Moreover, for the vectors \(\xi,\eta\) considered in [3], the left and the right (linear) operators \(L(\xi,\widetilde{I}),R(\eta,\widetilde{J})\) are bounded operators and satisfy braid relations which capture the braided \(C^*\)-tensor structure of \(\mathrm{Rep}(\mathcal{A}_V)\). On the VOA side, the (smeared) intertwining operators of \(V\) also record the braided \(C^*\)-tensor structure of \(\mathrm{Rep}^{\mathrm u}(V)\). To show that \(\mathrm{Rep}^{\mathrm u}(V)\) and \(\mathrm{Rep}(\mathcal{A}_V)\) are compatible, we relate the smeared intertwining operators of \(V\) with the left and right operators. Unfortunately, the smeared intertwining operators are often not bounded. [3] addresses this issue by taking polar decompositions of smeared intertwining operators and relating the partial isometry parts (the phases) with the bounded \(L\) and \(R\) operators. To establish such a relation, one has to first prove that the phases of smeared intertwining operators also satisfy nice braid relations, which relies on the strong braiding of intertwining operators.

Different from the above approach which first takes polar decompositions and then relates bounded linear operators, in this article we relate (possibly) unbounded closed operators directly. To be more precise, we loosen the conditions on \(\xi,\eta\) which yield unbounded closed left and right operators \(\mathscr L(\xi,\widetilde{I}),\mathscr R(\eta,\widetilde{J})\). These \(\mathscr L\) and \(\mathscr R\) operators satisfy the strong intertwining property and (more generally) the strong braiding, and the phases of these operators give (bounded) \(L\) and \(R\) operators. Using the polynomial energy bounds and the strong intertwining properties of \(V\)-intertwining operators, one can relate the smeared intertwining operators of \(V\) with the \(\mathscr L\) and \(\mathscr R\) operators of \(\mathcal{A}_V\) since weak braiding is satisfied on both sides. Therefore, as \(\mathscr L\) and \(\mathscr R\) satisfy strong braiding, so do the smeared intertwining operators of \(V\).

Note that part (3) of theorem 2 also claims that if sufficiently many intertwining operators of \(V\) are energy-bounded and satisfy the strong intertwining operators, then the strong braiding holds not only for these intertwining operators, but also for any other ones satisfying only polynomial energy bounds. In particular, the strong intertwining property will be a consequence of polynomial energy bounds. This result generalizes [28] theorem 8.1, which says that all the vertex operators of \(V\) satisfy strong locality if sufficiently many quasi-primary fields satisfy polynomial energy bounds and strong locality. Similar to that theorem, the proof of our result relies heavily on the Bisognano-Wichmann theorem. See section 2.6. This result can be applied to the following situation: Let \(V\) and \(V'\) satisfy the assumptions of theorem 2. Assume that all the intertwining operators of \(V\) and \(V'\) are energy-bounded and satisfy the strong intertwining property. Then any intertwining operator of \(V\otimes V'\), which is clearly energy-bounded since it is a sum of tensor products of intertwining operators of \(V\) and \(V'\), also satisfies strong braiding and (in particular) the strong intertwining property. (See theorem 84 for the precise statement.)

1.4 Outline of the paper↩︎

This article is organized as follows. In chapter 2, we prove theorem 2 under the general framework of operator algebras and algebraic quantum field theories. (In particular, theorem 35 and corollary 39 prove theorem 2-(2) and (3) respectively.) We are able to do so since the main techniques of proving this theorem are operator algebraic. We hope that people with little background in VOA can still understand the main ideas of the proofs.

In section 2.1 we review the basic facts about the (bounded) \(L\) and \(R\) operators. Then, in section 2.2, we define the unbounded closed \(\mathscr L\) and \(\mathscr R\) operators, and prove that they satisfy many important properties including the strong braiding. The symbol \(\mathcal{H}_i^\mathrm{pr}(I)\) denotes the subspace of all vectors in \(\mathcal{H}_i\) with closable/closed \(\mathscr L\) and \(\mathscr R\) operators "localized" in the interval \(I\). To relate \(\mathscr L,\mathscr R\) with the smeared intertwining operators, one also needs to calculate the \(4\)-point correlation functions defined by \(\mathscr L,\mathscr R\) as in proposition 26. For that purpose, we consider in section 2.3 a nice subspace \(\mathcal{H}_i^w(I)\subset \mathcal{H}_i^\mathrm{pr}(I)\) and compute the \(4\)-point functions for the vectors in this kind of subspaces.

In section 2.4, we introduce the notion of a weak categorical extension, which consists of a list of axioms satisfied by the \(\mathcal{L}\) and \(\mathcal{R}\) operators. These \(\mathcal{L}\) and \(\mathcal{R}\) operators are the abstraction of products of smeared intertwining operators satisfying the polynomial energy bounds and the strong intertwining property. Taking products of smeared intertwining operators is necessary for constructing enough operators that could be related to \(\mathscr L\) and \(\mathscr R\), and for extending the left and right actions from \(\mathcal{F}^V\) (see the statement of theorem 2) to all the objects of \(\mathrm{Rep}^{\mathrm u}(V)\). The process of taking such products is described, in section 2.7 and especially in the proof of theorem 44, as the construction of a weak categorical extension from a weak categorical local extension, in which the \(\mathcal{L}\) and \(\mathcal{R}\) operators describe single smeared intertwining operators. Note that in both sections, polynomial energy bounds are not explicitly mentioned in the axioms; two crucial consequences of energy bounds, the smoothness and the localizability, are required instead. Section 2.7 can be read immediately after section 2.4.

Section 2.5 is the climax of this article: we show that the existence of weak categorical (local) extensions implies the equivalence of braided \(C^*\)-tensor categories and the strong braiding. This proves theorem 2-(2) and part of (3). To finish proving (3), we need to show that polynomial energy bounds imply the strong intertwining property and (hence) strong braiding. This is discussed in section 2.6 where the weak left and right operators \(A,B\) are the abstraction of energy-bounded smeared intertwining operators.

In chapter 3 we prove theorem 2 in the (original) language of VOA by verifying that the smeared intertwining operators satisfy the requirements in chapter 2. A large part of this work has been done in [29] and [3] chapter 4, and is reviewed here for the readers’ convenience. We remark that theorem 55 and proposition 59 are somewhat special cases of the main results of chapter 2. We give independent proofs of these two results, which contain many key ideas in chapter 2. We hope that these proofs are helpful for the readers to understand the arguments in chapter 2. Note that theorem 55 is exactly [28] theorem 8.1, which states that \(V\) is strongly local if sufficiently many quasi-primary fields are. Our proof differs from that of [28] in that we use proposition 98 to prove \(Y(v,f)\Omega\in\mathscr D(S_I)\) for any \(v\in V\), while [28] uses the theory of real Hilbert subspaces developed in [40]. In section 3.6, we prove that tensor products or cosets of VOAs satisfying condition 2 of section 3.4 also satisfy condition 2. Here, condition 2 is equivalent to the assumptions of theorem 2. With the help of these methods, we prove theorem 1 by checking that the examples in this theorem satisfy condition 2. We also prove that all the intertwining operators of type \(ADE\) unitary affine VOAs, discrete series \(W\)-algebras, and unitary parafermion VOAs are energy-bounded and satisfy strong braiding.

Acknowledgment↩︎

I would like to thank Sebastiano Carpi, Yoh Tanimoto, James Tener, and Mihály Weiner for helpful discussions. I also thank the referees for their valuable comments. I am grateful to Zhipeng Yang for his warm hospitality and the support of Yunnan Key Laboratory of Modern Analytical Mathematics and Applications. Special thanks go to my friends Jing An, Yiwang Chen, Ning Guo, Siqi He, Xuelun Hou, Yuhang Liu, Huadi Qu, Shi Wang, Mingchen Xia, Zetian Yan, Zhipeng Yang, Bingyu Zhang, Hong-Wei Zhang, Yu Zhao, Fan Zheng for their long-term support since the pandemic.

This work was supported by NSFC Grant 12401159.

2 General theory↩︎

2.1 Categorical extensions of conformal nets↩︎

In this article, we let \(\mathbb{N}=\{0,1,2,3,\dots\}\) and \(\mathbb{Z}_+=\{1,2,3,\dots\}\). Set \(\mathbf{i}=\sqrt{-1}\). If \(\mathcal{H},\mathcal{K}\) are Hilbert spaces, we say that \(A:\mathcal{H}\rightarrow\mathcal{K}\) is an unbounded operator on \(\mathcal{H}\) if \(A\) is a linear map from a subspace of \(\mathcal{H}\) to \(\mathcal{K}\). We let \(\mathscr D(A)\) denote the domain of \(A\). If \(\mathscr D(A)\) is dense in \(\mathcal{H}\) and \(A\) is closable, we let \(\overline{A}\) be its closure. We say that \(A:\mathcal{H}\rightarrow\mathcal{K}\) is bounded if \(A\) is a continuous unbounded operator which is defined everywhere (i.e. \(\mathscr D(A)=\mathcal{H}\)). A continuous unbounded operator is densely defined but not necessarily everywhere defined.

Let \(\mathcal{J}\) be the set of non-empty non-dense open intervals in the unit circle \(\mathbb{S}^1\). If \(I\in\mathcal{J}\), then \(I'\) denotes the interior of the complement of \(I\), which is also an element in \(\mathcal{J}\). Let \(\mathrm{Diff}^+(\mathbb{S}^1)\) be the group of orientation-preserving diffeomorphisms of \(\mathbb{S}^1\), which contains the subgroup \(\mathrm{PSU}(1,1)\) of Möbius transforms of \(\mathbb{S}^1\). If \(I\in\mathcal{J}\), we let \(\mathrm{Diff}_I(\mathbb{S}^1)\) be the subgroup of all \(g\in\mathrm{Diff}^+(\mathbb{S}^1)\) such that \(gx=x\) whenever \(x\in I'\). Throughout this article, we let \(\mathcal{A}\) be an (irreducible) conformal net. This means that each \(I\in\mathcal{J}\) is associated with a von Neumann algebra \(\mathcal{A}(I)\) acting on a fixed separable Hilbert space \(\mathcal{H}_0\), such that the following conditions hold:
(a) (Isotony) If \(I_1\subset I_2\in\mathcal{J}\), then \(\mathcal{A}(I_1)\) is a von Neumann subalgebra of \(\mathcal{A}(I_2)\).
(b) (Locality) If \(I_1,I_2\in\mathcal{J}\) are disjoint, then \(\mathcal{A}(I_1)\) and \(\mathcal{A}(I_2)\) commute.
(c) (Conformal covariance) We have a strongly continuous projective unitary representation \(U\) of \(\mathrm{Diff}^+(\mathbb{S}^1)\) on \(\mathcal{H}_0\) which restricts to a strongly continuous unitary representation of \(\mathrm{PSU}(1,1)\), such that for any \(g\in \mathrm{Diff}^+(\mathbb{S}^1),I\in\mathcal{J}\), and any representing element \(V\in\mathcal{U}(\mathcal{H}_0)\) of \(U(g)\), \[\begin{align} V\mathcal{A}(I)V^*=\mathcal{A}(gI). \end{align}\] Moreover, if \(g\in\mathrm{Diff}_I(S^1)\), then \(V\in\mathcal{A}(I)\).
(d) (Positivity of energy) The action of the rotation subgroup \(\varrho\) of \(\mathrm{PSU}(1,1)\) on \(\mathcal{H}_0\) has positive generator.
(e) There exists a unique (up to scalar) unit vector \(\Omega\in\mathcal{H}_0\) fixed by \(\mathrm{PSU}(1,1)\). (\(\Omega\) is called the vacuum vector.) Moreover, \(\Omega\) is cyclic under the action of \(\bigvee_{I\in\mathcal{J}}\mathcal{M}(I)\) (the von Neumann algebra generated by all \(\mathcal{M}(I)\)).

\(\mathcal{A}\) satisfies the following well-known properties (cf. for example [41] and the reference therein):
(1) (Additivity) \(\mathcal{A}(I)=\bigvee_\alpha\mathcal{A}(I_\alpha)\) if \(\{I_\alpha\}\) is a set of open intervals whose union is \(I\).
(2) (Haag duality) \(\mathcal{A}(I)'=\mathcal{A}(I')\).
(3) (Reeh-Schlieder theorem) \(\mathcal{A}(I)\Omega\) is dense in \(\mathcal{H}_0\) for any \(I\in\mathcal{J}\)
(4) For each \(I\in\mathcal{J}\), \(\mathcal{A}(I)\) is a type III factor.

A pair \((\pi_i,\mathcal{H}_i)\) (or \(\mathcal{H}_i\) for short) is called a (normal) representation of \(\mathcal{A}\) (or \(\mathcal{A}\)-module) if \(\mathcal{H}_i\) is a separable Hilbert space, and for each \(I\in\mathcal{J}\), \(\pi_{i,I}\) is a (normal \(*\)-) representation of \(\mathcal{A}(I)\) on \(\mathcal{H}_i\) such that \(\pi_{i,I}|_{\mathcal{A}(I_0)}=\pi_{i,I_0}\) when \(I_0\subset I\). If \(x\in\mathcal{A}(I)\), we sometimes write \(\pi_{i,I}(x)\) as \(x\) for brevity. By [42], any \(\mathcal{A}\)-module \(\mathcal{H}_i\) is conformal covariant, which means the following: Let \(\mathscr G\) be the universal cover of \(\mathrm{Diff}^+(\mathbb{S}^1)\), and let \(\mathscr G_{\mathcal{A}}\) be the central extension of \(\mathscr G\) defined by \[\begin{align} \mathscr G_{\mathcal{A}}=\{(g,V)\in\mathscr G\times \mathcal{U}(\mathcal{H}_0)| V \textrm{ is a representing element of } U(g) \}.\label{eq35} \end{align}\tag{1}\] The topology of \(\mathscr G_{\mathcal{A}}\) inherits from that of \(\mathscr G\times\mathcal{U}(\mathcal{H}_0)\). So we have exact sequence \[\begin{align} 1\rightarrow \mathbb{T}\rightarrow\mathscr G_{\mathcal{A}}\rightarrow\mathscr G\rightarrow 1 \end{align}\] where \(\mathbb{T}\) is the subgroup of \(\mathscr G\times\mathcal{U}(\mathcal{H}_0)\) consisting of all \((1,\lambda\mathbf{1}_{\mathcal{H}_0})\) where \(\lambda\in\mathbb{C}\) and \(|\lambda|=1\). Then there exists a (unique) strongly continuous unitary representation \(U_i\) of \(\mathscr G_{\mathcal{A}}\) on \(\mathcal{H}_i\) such that for any \(I\in\mathcal{J}\) and \(g\in\mathscr G_{\mathcal{A}}(I)\), \[\begin{align} U_i(g)=\pi_i(U(g)).\label{eq34} \end{align}\tag{2}\] As a consequence, for any \(x\in\mathcal{A}(I)\) and \(g\in\mathscr G_{\mathcal{A}}\), one has \[\begin{align} U_i(g)\pi_{i,I}(x)U_i(g)^*=\pi_{i,gI}(U(g)xU(g)^*). \end{align}\] See [3] for more details. We write \(U_i(g)\) as \(g\) when no confusion arises. Homomorphisms of \(\mathcal{A}\)-modules intertwine the representations of \(\mathscr G_{\mathcal{A}}\).

Let \(\mathrm{Rep}(\mathcal{A})\) be the \(C^*\)-category of \(\mathcal{A}\)-modules whose objects are denoted by \(\mathcal{H}_i,\mathcal{H}_j,\mathcal{H}_k,\dots\). The vector space of (bounded) homomorphisms between any \(\mathrm{Rep}A\)-modules \(\mathcal{H}_i,\mathcal{H}_j\) is written as \(\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_i,\mathcal{H}_j)\). Then one can equip \(\mathrm{Rep}(\mathcal{A})\) with a structure of braided \(C^*\)-tensor category via Doplicher-Haag-Roberts (DHR) superselection theory [22], [23] or, equivalently, via Connes fusion [3]. The unit of \(\mathrm{Rep}(\mathcal{A})\) is \(\mathcal{H}_0\). We write the tensor (fusion) product of two \(\mathcal{A}\)-modules \(\mathcal{H}_i,\mathcal{H}_j\) as \(\mathcal{H}_i\boxtimes\mathcal{H}_j\). We assume without loss of generality that \(\mathrm{Rep}(\mathcal{A})\) is strict, which means that we will not distinguish between \(\mathcal{H}_0,\mathcal{H}_0\boxtimes\mathcal{H}_i,\mathcal{H}_i\boxtimes\mathcal{H}_0\), or \((\mathcal{H}_i\boxtimes \mathcal{H}_j)\boxtimes\mathcal{H}_k\) and \(\mathcal{H}_i\boxtimes(\mathcal{H}_j\boxtimes\mathcal{H}_k)\) (abbreviated to \(\mathcal{H}_i\boxtimes\mathcal{H}_j\boxtimes \mathcal{H}_k\)). In the following, we review the definition and the basic properties of closed vector-labeled categorical extensions of \(\mathcal{A}\) introduced in [3].

To begin with, if \(\mathcal{H}_i,\mathcal{H}_j\) are \(\mathcal{A}\)-modules and \(I\in\mathcal{J}\), then \(\mathrm{Hom}_{\mathcal{A}(I')}(\mathcal{H}_i,\mathcal{H}_j)\) denotes the vector space of bounded linear operators \(T:\mathcal{H}_i\rightarrow\mathcal{H}_j\) such that \(T\pi_{i,I'}(x)=\pi_{j,I'}(x)T\) for any \(x\in\mathcal{A}(I')\). We then define \(\mathcal{H}_i(I)=\mathrm{Hom}_{\mathcal{A}(I')}(\mathcal{H}_0,\mathcal{H}_i)\Omega\), which is a dense subspace of \(\mathcal{H}_i\). Note that \(I\subset J\) implies \(\mathcal{H}_i(I)\subset\mathcal{H}_i(J)\). Moreover, if \(G\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_i,\mathcal{H}_j)\), then \(G\mathcal{H}_i(I)\subset\mathcal{H}_j(I)\).

If \(I\in\mathcal{J}\), an arg-function of \(I\) is, by definition, a continuous function \(\arg_I:I\rightarrow\mathbb{R}\) such that for any \(e^{\mathbf{i}t}\in I\), \(\arg_I(e^{\mathbf{i}t})-t\in 2\pi\mathbb{Z}\). \(\widetilde{I}=(I,\arg_I)\) is called an arg-valued interval. Equivalently, \(\widetilde{I}\) is a branch of \(I\) in the universal cover of \(\mathbb{S}^1\). We let \(\widetilde{\mathcal{J}}\) be the set of arg-valued intervals. If \(\widetilde{I}=(I,\arg_I)\) and \(\widetilde{J}=(J,\arg_J)\) are in \(\widetilde{\mathcal{J}}\), we say that \(\widetilde{I}\) and \(\widetilde{J}\) are disjoint if \(I\) and \(J\) are so. Suppose moreover that for any \(z\in I,\zeta\in J\) we have \(\arg_J(\zeta)<\arg_I(z)<\arg_J(\zeta)+2\pi\), then we say that \(\widetilde{I}\) is anticlockwise to \(\widetilde{J}\) (equivalently, \(\widetilde{J}\) is clockwise to \(\widetilde{I}\)). We write \(\widetilde{I}\subset\widetilde{J}\) if \(I\subset J\) and \(\arg_J|_I=\arg_I\). Given \(\widetilde{I}\in\widetilde{\mathcal{J}}\), we also define \(\widetilde{I}'=(I',\arg_{I'})\in\widetilde{\mathcal{J}}\) such that \(\widetilde{I}\) is anticlockwise to \(\widetilde{I}'\). We say that \(\widetilde{I}'\) is the clockwise complement of \(\widetilde{I}\). We let \(\widetilde{\mathbb{S}^1_+}\in\widetilde{\mathcal{J}}\) where \(\mathbb{S}^1_+=\{a+\mathbf{i}b\in\mathbb{S}^1:b>0 \}\) is the upper semi-circle, and \(\arg_{\mathbb{S}^1_+}\) takes values in \((0,\pi)\). We set \(\widetilde{\mathbb{S}^1_-}\) to be the clockwise complement of \(\widetilde{\mathbb{S}^1_+}\).

Note that \(\mathbb{B}_{i,j}\) is unique by the density of fusion products. Moreover, \(\mathbb{B}_{i,j}\) commutes with the actions of \(\mathcal{A}\), and is the same as the braid operator of \(\mathrm{Rep}(\mathcal{A})\); see [3] sections 3.2, 3.3. The existence of \(\mathscr E\) was also proved in [3] sections 3.2. A sketch of the construction can be found in [43] section A.

Note that the categorical extensions of \(\mathcal{A}\) are unique up to unitary isomorphisms, cf. Thm. 3.10 and 3.12 of [3]. In Sec. 2.4 of this paper, we will extend the uniqueness to weak categorical extensions. As mentioned in the Introduction of [3] and described more rigorously in [44], \(\mathscr E\) can be viewed as a universal non-local extension of \(\mathcal{A}\) such that any extension of \(\mathcal{A}\) is a subquotient of \(\mathscr E\) described by \(Q\)-systems. Therefore, the uniqueness of \(\mathscr E\) says that \(\mathcal{A}\) has a unique universal non-local extension of \(\mathscr E\). (The relation between \(\mathscr E\) and a non-local extension is similar to that between a free group and a group determined by some relations. Therefore, the \(Q\)-systems play the same role as the group relations.) Since \(\mathscr E\) does not live on a fixed Hilbert space but on a category of Hilbert spaces, we call it a categorical extension.

We collect some useful formulas. By the locality and the state-field correspondence, it is easy to see that \[\begin{align} L(\xi,\widetilde{I})\eta=R(\eta,\widetilde{J})\xi \end{align}\] whenever \(\xi\in\mathcal{H}_i(I)\), \(\eta\in\mathcal{H}_j(J)\), and \(\widetilde{I}\) is anticlockwise to \(\widetilde{J}\). Moreover, if \(F\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_i,\mathcal{H}_{i'})\), \(G\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_j,\mathcal{H}_{j'})\), \(\xi\in\mathcal{H}_i(I)\), and \(\eta\in\mathcal{H}_j\), then \[\begin{align} (F\boxtimes G)L(\xi,\widetilde{I})\eta=L(F\xi,\widetilde{I})G\eta,\qquad (G\boxtimes F)R(\xi,\widetilde{I})\eta=R(F\xi,\widetilde{I})G\eta;\label{eq14} \end{align}\tag{4}\] see [43] section 2. We also recall the following fusion relations proved in [43] proposition 2.3.

As a special case, we see that if \(\xi\in\mathcal{H}_i(I)\) and \(x\in\mathcal{A}(I)\), then \(x\xi\in\mathcal{H}_i(I)\), and \[\begin{gather} L(x\xi,\widetilde{I})=xL(\xi,\widetilde{I}),\qquad R(x\xi,\widetilde{I})=xR(\xi,\widetilde{I}).\label{eq13} \end{gather}\tag{5}\] Set \(\xi=\Omega\) and notice that \(L(\Omega,\widetilde{I})=\mathbf{1}\). Then we have \[\begin{align} L(x\Omega,\widetilde{I})|_{\mathcal{H}_k}= R(x\Omega,\widetilde{I})|_{\mathcal{H}_k}=\pi_{k,I}(x) \end{align}\] for any \(\mathcal{H}_k\in\mathrm{Obj}(\mathrm{Rep}(\mathcal{A}))\).

Next, we discuss the conformal covariance of \(\mathscr E\). For any \(\widetilde{I}=(I,\arg_I)\in\widetilde{\mathcal{J}}\) and \(g\in\mathscr G_{\mathcal{A}}\), we have \(gI\) defined by the action of \(\mathrm{Diff}^+(\mathbb{S}^1)\) on \(\mathbb{S}^1\). We now set \(g\widetilde{I}=(gI,\arg_{gI})\), where \(\arg_{gI}\) is defined as follows. Choose any map \(\gamma:[0,1]\rightarrow\mathscr G_{\mathcal{A}}\) satisfying \(\gamma(0)=1,\gamma(1)=g\) such that \(\gamma\) descends to a (continuous) path in \(\mathscr G\). Then for any \(z\in I\) there is a path \(\gamma_z:[0,1]\rightarrow \mathbb{S}^1\) defined by \(\gamma_z(t)=\gamma(t)z\). The argument \(\arg_I(z)\) of \(z\) changes continuously along the path \(\gamma_z\) to an argument of \(gz\), whose value is denoted by \(\arg_{gI}(gz)\).

The element \(g\xi g^{-1}\) satisfying the requirement of Thm. 3 is uniquely determined by \(\widetilde{I},\xi,g\), since we clearly have \[\begin{align} g\xi g^{-1}=gL(\xi,\widetilde{I})g^{-1}\Omega=gR(\xi,\widetilde{I})g^{-1}\Omega. \end{align}\]

The following density result is easy but useful.

We are going to prove a weak version of additivity property for \(\mathscr E\). First, we need a lemma.

For any \(\xi\in\mathcal{H}_i(I)\), we define \(\lVert L(\xi,\widetilde{I}) \lVert\equiv \lVert R(\xi,\widetilde{I}) \lVert:=\lVert L(\xi,\widetilde{I})|_{\mathcal{H}_0} \lVert=\lVert R(\xi,\widetilde{I})|_{\mathcal{H}_0} \lVert\) to be the norms of \(L(\xi,\widetilde{I})\) and \(R(\xi,\widetilde{I})\). The following proposition is also needed.

We now prove the weak additivity property for \(\mathscr E\). For \(I_0,I\in\mathcal{J}\), we write \(I_0\Subset I\) if the closure of \(I_0\) is a subset of \(I\).

In particular, \(\xi_n\) is converging to \(\xi\) since \(\xi_n=L(\xi_n,\widetilde{I})\Omega\) and \(\xi=L(\xi,\widetilde{I})\Omega\).

2.2 Closable field operators↩︎

For any \(\widetilde{I}\in\widetilde{\mathcal{J}}\), recall that \(\widetilde{I}'\) is the clockwise complement of \(\widetilde{I}\). We define \({{}^\backprime\widetilde{I}}\in\widetilde{\mathcal{J}}\) such that \(({}^\backprime\widetilde{I})'=\widetilde{I}\), and call \({{}^\backprime\widetilde{I}}\) the anticlockwise complement of \(\widetilde{I}\). For any \(\xi\in\mathcal{H}_i\), we let \(\mathscr L(\xi,\widetilde{I})\) (resp. \(\mathscr R(\xi,\widetilde{I})\)) act on any \(\mathcal{H}_j\in\mathrm{Obj}(\mathrm{Rep}(\mathcal{A}))\) as an unbounded operator \(\mathcal{H}_j\rightarrow\mathcal{H}_i\boxtimes\mathcal{H}_j\) (resp. \(\mathcal{H}_j\rightarrow\mathcal{H}_j\boxtimes\mathcal{H}_i\)) with domain \(\mathcal{H}_j(I')\) such that for any \(\eta\in\mathcal{H}_j(I')\), \[\begin{align} \mathscr L(\xi,\widetilde{I})\eta=R(\eta,\widetilde{I}')\xi,\quad \text{resp.} \quad \mathscr R(\xi,\widetilde{I})\eta=L(\eta,{}^\backprime\widetilde{I})\xi.\label{eq7} \end{align}\tag{6}\] It is clear that \(\Omega\) is inside the domains of \(\mathscr L(\xi,\widetilde{I})|_{\mathcal{H}_0}\) and \(\mathscr R(\xi,\widetilde{I})|_{\mathcal{H}_0}\), and the state-field correspondence \[\begin{align} \mathscr L(\xi,\widetilde{I})\Omega=\mathscr R(\xi,\widetilde{I})\Omega=\xi \end{align}\] is satisfied. We also have that \[\begin{align} \mathscr L(\xi,\widetilde{I})|_{\mathcal{H}_0}=\mathscr R(\xi,\widetilde{I})|_{\mathcal{H}_0}, \end{align}\] and that they depend only on \(I\) but not on the choice of \(\arg_I\). Indeed, both operators send any \(y\Omega\in\mathcal{A}(I')\Omega\) to \(y\xi\).

Observe the following easy fact:

The following theorem was proved in [43] section 7. Here we give a different but (hopefully) more conceptual proof.

Note that when \(\xi\in\mathcal{H}_i(I)\), it is clear that our definition of the two closed operators agree with the original bounded ones. The following proposition shows that \(\mathscr L(\xi,\widetilde{I})\) and \(\mathscr R(\xi,\widetilde{I})\) intertwine the actions of \(\mathcal{A}(I')\).

In the above proposition, we have actually used the following definition.

We now state the main result of this section.

Note that in the axiom of Möbius covariance, \(\widetilde{\mathrm{PSU}}(1,1)\) is the universal covering of \(\mathrm{PSU}(1,1)\), which is regarded as a subgroup of \(\mathscr G_{\mathcal{A}}\). This is possible since, by [45], the restriction of any strongly continuous projective representation \(\mathscr G\) to \(\widetilde{\mathrm{PSU}}(1,1)\) can be lifted to a unique strongly continuous unitary representation of \(\widetilde{\mathrm{PSU}}(1,1)\).

Isotony is obvious; note that the inclusion relations are reversed since \(I_1'\supset I_2'\). To check the strong commutativity of the first of ?? , one just need to show that for any \(\eta\in\mathcal{H}_k(I'),\eta'\in\mathcal{H}_{k'}(I')\), we have \(\mathscr L(\xi,\widetilde{I})G\eta=(\mathbf{1}_i\boxtimes G)\mathscr L(\xi,\widetilde{I})\eta\) and \(\mathscr L(\xi,\widetilde{I})G^*\eta'=(\mathbf{1}_i\boxtimes G^*)\mathscr L(\xi,\widetilde{I})\eta'\). Indeed, by 4 , we have \[\begin{align} \mathscr L(\xi,\widetilde{I})G\eta=R(G\eta,\widetilde{I}')\xi=(\mathbf{1}_i\boxtimes G)R(\eta,\widetilde{I}')\xi=(\mathbf{1}_i\boxtimes G)\mathscr L(\xi,\widetilde{I})\eta. \end{align}\] The other equation is proved similarly. For the second diagram of ?? , one uses a similar argument.

To prove the locality, we first prove the following weaker version:

We now want to approximate \(\mathscr L(\xi,\widetilde{I})\) and \(\mathscr R(\xi,\widetilde{I})\) using bounded \(L\) and \(R\) operators. Again we assume \(\xi\in\mathcal{H}_i^\mathrm{pr}(I)\). Consider the polar decomposition \(\mathscr L(\xi,\widetilde{I})=UH\) where \(U\) is a partial isometry from each \(\mathcal{H}_j\) to \(\mathcal{H}_i\boxtimes\mathcal{H}_j\), and \(H\) is a positive closed operator on each \(\mathcal{H}_j\). Consider the spectral decomposition \(H=\int_0^{+\infty}tdq_t\), where for each \(\mathcal{H}_j\in\mathrm{Obj}(\mathrm{Rep}(\mathcal{A}))\), \(q_t\) is a projection on \(\mathcal{H}_j\) defined by \(q_t=\chi_{[0,t]}(H)\), and \(p_t=Uq_tU^*\) is a projection on \(\mathcal{H}_i\boxtimes\mathcal{H}_j\). Then, by the spectral theory, we have the following equation of bounded operators: \[\begin{align} \overline{p_t\mathscr L(\xi,\widetilde{I})}=\mathscr L(\xi,\widetilde{I})q_t, \end{align}\] and for any \(\eta\in\mathcal{H}_j\), \(\eta\) is inside the domain of \(\mathscr L(\xi,\widetilde{I})\) if and only if the limit \[\begin{align} \lim_{t\rightarrow+\infty}\lVert {p_t\mathscr L(\xi,\widetilde{I})}\eta \lVert^2=\lim_{t\rightarrow+\infty}\lVert {\mathscr L(\xi,\widetilde{I})}q_t\eta \lVert^2 \end{align}\] is finite, in which case we have \[\begin{align} \mathscr L(\xi,\widetilde{I})\eta=\lim_{t\rightarrow+\infty} {p_t\mathscr L(\xi,\widetilde{I})}\eta =\lim_{t\rightarrow+\infty}{\mathscr L(\xi,\widetilde{I})}q_t\eta. \end{align}\] We call \(\{p_t:t\geq0\}\) (resp. \(\{q_t:t\geq0\}\)) the left (resp. right) bounding projections of \(\mathscr L(\xi,\widetilde{I})|_{\mathcal{H}_j}\). Similarly, for each \(\mathcal{H}_j\) we have left and right bounding projections \(\{e_t\}\) (on \(\mathcal{H}_j\boxtimes \mathcal{H}_i\)) and \(\{f_t\}\) (on \(\mathcal{H}_j\)) of \(\mathscr R(\xi,\widetilde{I})\). We write \(p_t,q_t,e_t,f_t\) respectively as \(p_t(\xi,\widetilde{I}),q_t(\xi,\widetilde{I}),e_t(\xi,\widetilde{I}),f_t(\xi,\widetilde{I})\) to emphasize the dependence of the projections on the vectors and the arg-valued intervals.

2.3 The dense subspaces \(\mathcal{H}_i^\infty\), \(\mathcal{H}_i^\infty(I)\), and \(\mathcal{H}_i^w(I)\)↩︎

We now introduce for each \(\mathcal{H}_i\in\mathrm{Rep}(\mathcal{A})\) some important dense subspaces. Let \(L_0\) be the generator of the one parameter rotation subgroup \(\varrho\) of \(\widetilde{\mathrm{PSU}}(1,1)\) which is positive by [46]. (\(\varrho\) is the lift of the one parameter rotation subgroup of \(\mathrm{PSU}(1,1)\) to \(\widetilde{\mathrm{PSU}}(1,1)\).) Set \[\begin{align} \mathcal{H}_i^\infty=\bigcap_{n\in\mathbb{Z}_+}\mathscr D(L_0^n). \end{align}\] Vectors in \(\mathcal{H}_i^\infty\) are called smooth. For each \(\xi\in\mathcal{H}_i\), we have \(\xi_h\in\mathcal{H}_i^\infty\) where \(h\in C_c^\infty(\mathbb{R})\) and \[\begin{align} \xi_h=\int_{\mathbb{R}}h(t)\varrho(t)\xi dt. \end{align}\] Since \(\xi_h\rightarrow\xi\) as \(h\) converges to the \(\delta\)-function at \(0\), we conclude that \(\mathcal{H}_i^\infty\) is dense in \(\mathcal{H}_i\).

A closable operator \(T\) from \(\mathcal{H}_i\) to \(\mathcal{H}_j\) is called smooth if \[\begin{gather} \mathcal{H}_i^\infty\subset\mathscr D(T),\qquad \mathcal{H}_j^\infty\subset\mathscr D(T^*),\qquad T\mathcal{H}_i^\infty\subset \mathcal{H}_j^\infty,\qquad T^*\mathcal{H}_j^\infty\subset \mathcal{H}_i^\infty. \end{gather}\]

If \(T:\mathcal{H}_i\rightarrow\mathcal{H}_j\) is bounded, then it is a routine check that for each \(h\in C_c^\infty(\mathbb{R})\) satisfying \(\int_{\mathbb{R}}h(t)dt=1\), \[\begin{align} T_h:=\int_{\mathbb{R}}h(t)\varrho(t)T\varrho(-t)dt \end{align}\] is bounded and smooth, and \(T_h\) converges \(*\)-strongly to \(T\) as \(h\) converges to the \(\delta\)-function at \(0\). See for example [30] Prop. 4.2 for details.

Let \(\mathcal{A}^\infty(I)\) be the set of all \(x\in\mathcal{A}(I)\) such that \(\pi_{i,I}(x)\) is smooth for any \(\mathcal{H}_i\in\mathrm{Obj}(\mathrm{Rep}(\mathcal{A}))\). Then clearly \(\mathcal{H}^\infty_0(I)=\mathcal{A}^\infty(I)\Omega\). As a special case of proposition 21, we have:

The following relations are obvious: \[\begin{align} \mathcal{H}_i^\infty(I)\subset&\mathcal{H}_i^w(I)\subset\mathcal{H}_i^\infty\\ &~~~\cap\\ &\mathcal{H}_i^\mathrm{pr}(I) \end{align}\] Therefore, \(\mathcal{H}_i^w(I)\) is a dense subspace of \(\mathcal{H}_i\).

Using the results in section 2.2 and chapter 4, we are able to prove many important results, for example:

2.4 Weak categorical extensions↩︎

Let \(\mathscr C\) be a full (\(C^*\)-)subcategory of \(\mathrm{Rep}(\mathcal{A})\) containing \(\mathcal{H}_0\) and closed under taking submodules and finite (orthogonal) direct sums. Equip \(\mathscr C\) with a tensor \(*\)-bifunctor \(\boxdot\), natural unitary associativity isomorphisms \((\mathcal{H}_i\boxdot\mathcal{H}_j)\boxdot H_k\xrightarrow\simeq\mathcal{H}_i\boxdot(\mathcal{H}_j\boxdot\mathcal{H}_k)\), and unitary isomorphisms \(\mathcal{H}_i\boxdot\mathcal{H}_0\xrightarrow\simeq\mathcal{H}_i,\mathcal{H}_0\boxdot\mathcal{H}_i\xrightarrow\simeq \mathcal{H}_i\), so that \(\mathscr C\) becomes a \(C^*\)-tensor category with unit \(\mathcal{H}_0\). We treat \(\mathscr C\) as if it is strict by identifying \((\mathcal{H}_i\boxdot\mathcal{H}_j)\boxdot H_k\), \(\mathcal{H}_i\boxdot(\mathcal{H}_j\boxdot\mathcal{H}_k)\) as \(\mathcal{H}_i\boxdot\mathcal{H}_j\boxdot \mathcal{H}_k\), and \(\mathcal{H}_i\boxdot \mathcal{H}_0\), \(\mathcal{H}_0\boxdot\mathcal{H}_i\) as \(\mathcal{H}_i\). Our main example of \(\mathscr C\) arises from VOAs, cf. Def. 65.

Assume that \(T:\mathcal{H}_i\rightarrow\mathcal{H}_j\) is a closable and smooth operator with domain \(\mathcal{H}_i^\infty\). Unlike previous sections, we will differentiate between \(T\) and its closure \(\overline{T}\). We define \(T^\dagger\) to be \(T^*|_{\mathcal{H}_j^\infty}\), called the formal adjoint of \(T\). Then \(T^\dagger:\mathcal{H}_j\rightarrow\mathcal{H}_i\) has dense domain \(\mathcal{H}_j^\infty\). A dense subspace \(\mathscr D_0\) of \(\mathcal{H}_i^\infty\) is called quasi-rotation invariant (QRI for short), if there exists \(\delta>0\) and a dense subspace \(\mathscr D_\delta\subset\mathscr D_0\) such that \(\varrho(t)\mathscr D_\delta\subset\mathscr D_0\) for any \(t\in(-\delta,\delta)\). For example, \(\mathcal{H}_i^\infty(I)\) is QRI for any \(I\in\mathcal{J}\). We say that \(T\) is localizable if any dense and QRI subspace \(\mathscr D_0\) of \(\mathcal{H}_i^\infty\) is a core for \(T\). By saying that \(T:\mathcal{H}_i\rightarrow\mathcal{H}_j\) is smooth and localizable, we always assume that \(T\) is closable and has domain \(\mathcal{H}_i^\infty\).

Suppose that in the following diagrams, \(A,B,C,D\) are smooth. (Namely, the formal adjoints of these operators should also map smooth spaces to smooth ones.) \[\begin{align} \begin{CD} \mathcal{H}_i^\infty @>C>> \mathcal{H}_j^\infty\\ @V A VV @V B VV\\ \mathcal{H}_k^\infty @>D>> \mathcal{\mathcal{H}}_l^\infty. \end{CD} \end{align}\] We say that this diagram commutes adjointly if \(DA=BC\) on \(\mathcal{H}_i^\infty\), and if \(CA^\dagger=B^\dagger D\) on \(\mathcal{H}_k^\infty\) (equivalently, \(AC^\dagger=D^\dagger B\) on \(\mathcal{H}_j^\infty\)).

The axiom of weak locality explains the adjective “weak" in the notion of weak categorical extensions. However, as we will see Cor. 36, a weak categorical extension \(\mathscr E^w\) actually satisfies strong locality. Moreover, if we let \(\mathscr E=(\mathcal{A},\mathrm{Rep}(\mathcal{A}),\boxtimes,\mathcal{H})\) be a closed and vector-labeled categorical extension, then Thm. 35 will imply that \(\mathscr E^w\) can be embedded into the”unbounded version of \(\mathscr E\)" (i.e., \(\mathscr E\) together with the closed operators \(\mathscr L,\mathscr R\) as described in Sec. 2.2) so that the latter can be viewed as the maximal extension of \(\mathscr E^w\).

In the above discussion, we have actually showed:

Let us write the closure of \(\mathcal{L}(\xi,\widetilde{I})|_{\mathcal{H}_j^\infty}\) as \(\overline{\mathcal{L}(\xi,\widetilde{I})}|_{\mathcal{H}_j}=\overline{\mathcal{L}(\xi,\widetilde{I})|_{\mathcal{H}_j^\infty}}\). We close this section with the following important observation.

2.5 From weak to strong categorical extensions↩︎

Let \(\mathscr C\) be as in the last section. Let \(\widehat{\mathscr C}\) be the \(C^*\)-category of all \(\mathcal{A}\)-modules \(\mathcal{H}_i\) such that \(\mathcal{H}_i\) is unitarily equivalent to some object in \(\mathscr C\).

2.6 Weak left and right operators↩︎

Let \(\mathscr C\) be as in section 2.4, and let \(\mathscr E^w=(\mathcal{A},\mathscr C,\boxdot,\mathfrak H)\) be a weak categorical extension of \(\mathcal{A}\).

Recall the natural unitary isomorphism \(\Psi=\Psi_{i,j}:\mathcal{H}_i\boxdot\mathcal{H}_j\rightarrow\mathcal{H}_i\boxtimes\mathcal{H}_j\) defined in theorem 35. We first state the main result of this section.

Since the proofs of (a) and (b) are similar, we shall only prove (a).

To prepare for the proof of \(\xi\in\mathcal{H}_i^\mathrm{pr}(I)\), we recall the geometric modular theory of rigid categorical extensions proved in [43]. Consider the one parameter rotation subgroup \(\rho(t)=\left( \begin{array}{cc} e^{\frac{\mathbf{i}t}{2}} & 0 \\ 0 &e^{\frac{-\mathbf{i}t}{2}} \end{array} \right)\) and dilation subgroup \(\delta(t)=\left( \begin{array}{cc} \cosh\frac{t}{2} & -\sinh\frac{t}{2} \\ -\sinh\frac{t}{2} &\cosh\frac{t}{2} \end{array} \right)\) of \(\mathrm{PSU}(1,1)\). For each \(I\in\mathcal{J}\), define \(\delta_I(t)=g\delta(t)g^{-1}\) where \(g\in\mathrm{PSU}(1,1)\) and \(g\mathbb{S}^1_+=I\). Then \(\delta_I\) is well defined, \(\delta_I(t)(I)=I\), and \(\delta(t)=\delta_{\mathbb{S}^1_+}(t)\). We lift \(\varrho\) and \(\delta\) to one-parameter subgroups of \(\widetilde{\mathrm{PSU}}(1,1)\) and denote them by the same symbols \(\varrho\) and \(\delta_I\).

Let \(\mathrm{Rep}^{\mathrm f}(\mathcal{A})\) be the braided \(C^*\)-tensor subcategory of \(\mathrm{Rep}(\mathcal{A})\) of all dualizable \(\mathcal{A}\)-modules. Then for any \(\mathcal{H}_i\in\mathrm{Obj}(\mathrm{Rep}^{\mathrm f}(\mathcal{A}))\), one can find a dual object \(\mathcal{H}_{\overline{i}}\in\mathrm{Obj}(\mathrm{Rep}^{\mathrm f}(\mathcal{A}))\) and \(\mathrm{ev}_{i,\overline{i}}\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_i\boxtimes\mathcal{H}_{\overline{i}},\mathcal{H}_0),\mathrm{ev}_{\overline{i},i}\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_{\overline{i}}\boxtimes\mathcal{H}_i,\mathcal{H}_0)\) satisfying the conjugate equations \[\begin{gather} (\mathrm{ev}_{i,\overline{i}}\boxtimes\mathbf{1}_i)(\mathbf{1}_i\boxtimes\mathrm{coev}_{\overline{i},i})=\mathbf{1}_i=(\mathbf{1}_i\boxtimes\mathrm{ev}_{\overline{i},i})(\mathrm{coev}_{i,\overline{i}}\boxtimes\mathbf{1}_i),\\ (\mathrm{ev}_{\overline{i},i}\boxtimes\mathbf{1}_{\overline{i}})(\mathbf{1}_{\overline{i}}\boxtimes\mathrm{coev}_{i,\overline{i}})=\mathbf{1}_{\overline{i}}=(\mathbf{1}_{\overline{i}}\boxtimes\mathrm{ev}_{i,\overline{i}})(\mathrm{coev}_{\overline{i},i}\boxtimes\mathbf{1}_{\overline{i}}), \end{gather}\] where we set \(\mathrm{coev}_{i,\overline{i}}=\mathrm{ev}_{i,\overline{i}}^*,\mathrm{coev}_{\overline{i},i}=\mathrm{ev}_{\overline{i},i}^*\). Moreover, by [47] (see also [48], [49]), we may assume that the \(\mathrm{ev}\) and \(\mathrm{coev}\) to be standard, which means that the following is also satisfied for any \(F\in\mathrm{End}_{\mathcal{A}}(\mathcal{H}_i)\): \[\begin{align} \mathrm{ev}_{i,\overline{i}}(F\boxtimes\mathbf{1}_{\overline{i}})\mathrm{coev}_{i,\overline{i}}=\mathrm{ev}_{\overline{i},i}(\mathbf{1}_{\overline{i}}\boxtimes F)\mathrm{coev}_{\overline{i},i}. \end{align}\]

Now, for each \(\widetilde{I}\in\widetilde{\mathcal{J}}\) and \(\mathcal{H}_i\in\mathrm{Obj}(\mathrm{Rep}^{\mathrm f}(\mathcal{A}))\), we define unbounded operators \(S_{\widetilde{I}},F_{\widetilde{I}}:\mathcal{H}_i\rightarrow\mathcal{H}_{\overline{i}}\) with common domain \(\mathcal{H}_i(I)\) by setting \[S_{\widetilde{I}}\xi=L(\xi,\widetilde{I})^*\mathrm{coev}_{i,\overline{i}}\Omega,\qquad F_{\widetilde{I}}\xi=R(\xi,\widetilde{I})^*\mathrm{coev}_{\overline{i},i}\Omega\] for any \(\xi\in\mathcal{H}_i(I)\). By [43] section 5, \(S_{\widetilde{I}}\) and \(F_{\widetilde{I}}\) are closable, whose closures are denoted by the same symbols in the following. Then we have \[\begin{gather} F_{\widetilde{I}}=\varrho(2\pi)S_{\widetilde{I}},\\ F_{\widetilde{I}'}=S_{\widetilde{I}}^*. \end{gather}\] Thus \(\Delta_{\widetilde{I}}:=S_{\widetilde{I}}^*S_{\widetilde{I}}\) equals \(F_{\widetilde{I}}^*F_{\widetilde{I}}\). Moreover, by [43] section 6, \(\Delta_{\widetilde{I}}\) is independent of \(\arg_I\) and hence can be written as \(\Delta_I\), and \[\begin{align} \Delta_I^{\mathbf{i}t}=\delta_I(-2\pi t). \end{align}\] By [43], if \(\mathcal{H}_i\in\mathrm{Obj}(\mathrm{Rep}^{\mathrm f}(\mathcal{A}))\) and \(\xi\in\mathcal{H}_i\), then the following three conditions are equivalent.

(a) \(\mathrm{coev}_{i,\overline{i}}\Omega\) is in the domain of \((\mathscr L(\xi,\widetilde{I})|_{\mathcal{H}_{\overline{i}}})^*\).

(a’) \(\mathrm{coev}_{\overline{i},i}\Omega\) is in the domain of \((\mathscr R(\xi,\widetilde{I})|_{\mathcal{H}_{\overline{i}}})^*\).

(b) \(\xi\) is in the domain of \(\Delta_I^{\frac{1}{2}}\) (which are also the domains of \(S_{\widetilde{I}}\) and \(F_{\widetilde{I}}\)).
Moreover, if any of them are true, then \(\xi\in\mathcal{H}_i^\mathrm{pr}(I)\), and \[\begin{align} S_{\widetilde{I}}\xi=\mathscr L(\xi,\widetilde{I})^*\mathrm{coev}_{i,\overline{i}}\Omega,\qquad F_{\widetilde{I}}\xi=\mathscr R(\xi,\widetilde{I})^*\mathrm{coev}_{\overline{i},i}\Omega.\label{eq32} \end{align}\tag{17}\]

2.7 Weak categorical partial extensions↩︎

In this section, we assume that \(\mathscr C\) is equipped with a braid operator \(\ss\) so that \((\mathscr C,\boxdot,\ss)\) is a braided \(C^*\)-tensor category. In other words, we have a natural \(\ss\), where for each \(\mathcal{H}_i,\mathcal{H}_j\in\mathrm{Obj}(\mathscr R)\), \(\ss_{i,j}:\mathcal{H}_i\boxdot\mathcal{H}_j\rightarrow\mathcal{H}_j\boxdot\mathcal{H}_i\) is a unitary isomorphism, and \(\ss\) satisfies the Hexagon axioms. We also assume that \(\mathscr C\) is semisimple, i.e., any module \(\mathcal{H}_i\in\mathrm{Obj}(\mathscr C)\) is unitarily equivalent to a finite direct sum of irreducible \(\mathcal{A}\)-modules in \(\mathscr C\). If \(\mathcal{F}\) is a set of \(\mathcal{A}\)-modules in \(\mathscr C\), we say that \(\mathcal{F}\) generates \(\mathscr C\) if for any irreducible \(\mathcal{H}_i\in\mathrm{Obj}(\mathscr C)\), there exist \(\mathcal{H}_{i_1},\dots,\mathcal{H}_{i_n}\in\mathcal{F}\) such that \(\mathcal{H}_i\) is equivalent to an (irreducible) \(\mathcal{A}\)-submodule of \(\mathcal{H}_{i_1}\boxdot\mathcal{H}_{i_2}\boxdot\cdots\boxdot\mathcal{H}_{i_n}\).

From the construction of \(\mathscr E^w\) from \(\mathscr E^w_\mathrm{par}\), it is clear that we have:

3 Application to vertex operator algebras↩︎

We assume that the readers are familiar with the basics of unitary vertex operator algebras (VOAs) and their representations (modules) [28], [50], and the intertwining operators of VOAs [39]. We follow basically the notations in [29] chapter 1, in which all the relevant definitions and properties are presented.

Let \(V\) be a unitary simple (equivalently, CFT-type [28]) VOA \(V\). Then the pre-Hilbert space \(V\) (with inner product \(\langle {\cdot|\cdot}\rangle\)⁹) has orthogonal grading \(V=\bigoplus_{n\in\mathbb{N}}V(n)\) where \(V(0)\) is spanned by the vacuum vector \(\Omega\), and \(\dim V(n)<+\infty\). For any \(v\in V\), we write the vertex operator as \(Y(v,z)=\sum_{n\in\mathbb{Z}}Y(v)_nz^{-n-1}\) where \(z\in\mathbb{C}^\times=\mathbb{C}-\{0\}\), and \(Y(v)_n\) is a linear map on \(V\), called the \(n\)-th mode of \(Y(v,z)\). Let \(\nu\in V(2)\) be the conformal vector. Then \(\{L_n=Y(\nu)_{n+1}:n\in\mathbb{Z}\}\) form a Virasoro algebra with central charge \(c\geq0\). We have \(L_0|_{V(n)}=n\mathbf{1}_{V(n)}\).

A \(V\)-module \(W_i\) has grading \(W_i=\bigoplus_{n\in\mathbb{R}}W_i(n)\) where each \(W_i(n)\) is finite-dimensional, and \(\dim W_i(n)=0\) when \(n\) is small enough. For each \(v\in V\) we also have a vertex operator \(Y_i(v,z)=\sum_{n\in\mathbb{Z}}Y_i(v)_nz^{-n-1}\) where each \(Y_i(v)_n\in\mathrm{End}(W_i)\). \(\{L_n=Y_i(\nu)_{n+1}:n\in\mathbb{Z}\}\) form a representation of the Virasoro algebra on \(W_i\) with the same central charge \(c\). We have \(L_0|_{W_i(n)}=n\mathbf{1}_{W_i(n)}\) for each \(n\in\mathbb{R}\). A vector \(w^{(i)}\in W_i\) is called homogeneous if \(w^{(i)}\in W_i(s)\) for some \(s\in\mathbb{R}\). In this case we say that \(s\) is the conformal weight of \(w^{(i)}\). If moreover \(L_1w^{(i)}=0\), then \(w^{(i)}\) is called quasi-primary; if \(L_nw^{(i)}=0\) for all \(n>0\), then \(w^{(i)}\) is called primary. When \(W_i\) is a unitary \(V\)-module, the grading \(W_i=\bigoplus_{n\in\mathbb{R}}W_i(n)\) is orthogonal, \(\dim W_i(n)=0\) when \(n<0\), and \(L_n^\dagger=L_{-n}\).¹⁰ \(V\) itself is a \(V\)-module, called the vacuum \(V\)-module, and is sometimes denoted by \(W_0\).

We assume that any irreducible \(V\)-module \(W_i\) is unitarizable, i.e., the vector space \(W_i\) admits a (unique up to scalar multiple) inner product under which \(W_i\) becomes a unitary \(V\)-module. Such \(V\) is called strongly unitary.

For any \(V\)-modules \(W_i,W_j,W_k\), we let \(\mathcal{V}{k\choose i~j}\) be the vector space of type \({W_k\choose W_iW_j}={k\choose i~j}\) intertwining operators of \(V\). Then for any \(\mathcal{Y}\in\mathcal{V}{k\choose i~j}\) and any \(w^{(i)}\in W_i\), we have \(\mathcal{Y}(w^{(i)},z)=\sum_{s\in\mathbb{R}}\mathcal{Y}(w^{(i)})_sz^{-s-1}\) where each \(\mathcal{Y}(w^{(i)})_s\) is a linear map from \(W_j\) to \(W_k\). We say that \(W_i,W_j,W_k\) are respectively the charge space, the source space, and the target space of \(\mathcal{Y}\). We say that \(\mathcal{Y}\) is a unitary (resp. irreducible) intertwining operator if all \(W_i,W_j,W_k\) are unitary (resp. irreducible) \(V\)-modules. Note that \(Y_i\in\mathcal{V}{i\choose 0~i}=\mathcal{V}{W_i\choose W_0W_i}\) and, in particular, \(Y\in\mathcal{V}{0\choose 0~0}\).

3.1 Polynomial energy bounds↩︎

Suppose that \(\mathcal{Y}\in\mathcal{V}{k\choose i~j}\) is unitary. Let \(w^{(i)}\in W_i\) be homogeneous. Given \(a\geq 0\), we say that \(\mathcal{Y}(w^{(i)},z)\)¹¹ satisfies \(a\)-th order energy bounds, if there exist \(M,b\geq0\), such that for any \(s\in\mathbb{R},w^{(j)}\in W_j\), \[\begin{align} \lVert \mathcal{Y}(w^{(i)})_sw^{(j)} \lVert\leq M(1+|s|)^b\lVert (1+L_0)^aw^{(j)}\lVert. \end{align}\] \(1\)-st order energy bounds are called linear energy bounds. We say that \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded if satisfies \(a\)-th order energy bounds for some \(a\), that \(\mathcal{Y}\) is energy-bounded if \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded for any homogeneous \(w^{(i)}\in W_i\), that a unitary \(V\)-module \(W_i\) is energy-bounded if \(Y_i\) is an energy-bounded intertwining operator, that \(V\) is energy-bounded if the vacuum module \(W_0\) is energy-bounded. We say that \(V\) is strongly energy-bounded if (the vertex operator of) any irreducible unitary \(V\)-module is energy-bounded. We say that \(V\) is completely energy-bounded if any irreducible unitary intertwining operator of \(V\) is energy-bounded.

The following is [29] corollary 3.7-(b).

We shall define smeared intertwining operators and recall some basic properties. See [29] chapter 3 and [3] section 4.4 for details. For each unitary \(V\)-module \(W_i\) we let \(\mathcal{H}_i\) be the Hilbert space completion of the pre-Hilbert space \(W_i\). We set \[\begin{align} \mathcal{H}_i^\infty=\bigcap_{n\in\mathbb{Z}_+}\mathscr D(\overline{L_0}^n). \end{align}\] This notation will coincide with the previous one defined for conformal net modules after \(\mathcal{H}_i\) is equipped with a natural structure of conformal net module. For any \(\widetilde{I}=(I,\arg_I)\in\widetilde{\mathcal{J}}\) and \(f\in C^\infty_c(I)\) , we call \(\widetilde{f}=(f,\arg_I)\) a (smooth) arg-valued function on \(S^1\) with support inside \(\widetilde{I}\), and let \(C^\infty_c(\widetilde{I})\) be the set of all such \(\widetilde{f}\). If \(\widetilde{I}\subset\widetilde{J}\in\widetilde{\mathcal{J}}\), then \(C^\infty_c(\widetilde{I})\) is naturally a subspace of \(C^\infty_c(\widetilde{J})\) by identifying each \((f,\arg I)\in C^\infty_c(\widetilde{I})\) with \((f,\arg J)\). Now, assume that \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded. \(\widetilde{I}=(I,\arg_I)\in\widetilde{\mathcal{J}}\), and \(\widetilde{f}=(f,\arg_I)\in C^\infty_c(\widetilde{I})\) , we define the smeared intertwining operator \(\mathcal{Y}(w^{(i)},\widetilde{f})\) \[\begin{align} \mathcal{Y}(w^{(i)},\widetilde{f})=\int_{\arg_I(I)}\mathcal{Y}(w^{(i)},e^{i\theta})f(e^{i\theta})\cdot\frac{e^{i\theta}}{2\pi}d\theta \end{align}\] which indeed maps \(W_j\) into \(\mathcal{H}_k^\infty\). Regarding \(\mathcal{Y}(w^{(i)},\widetilde{f})\) as an unbounded operator from \(\mathcal{H}_j\) to \(\mathcal{H}_k\) with domain \(W_j\), then \(\mathcal{Y}(w^{(i)},\widetilde{f})\) is closable, and its closure contains \(\mathcal{H}_j^\infty\). Moreover, we have \[\begin{gather} \overline{\mathcal{Y}(w^{(i)},\widetilde{f})}\mathcal{H}_j^\infty\subset\mathcal{H}_k^\infty,\qquad \overline{\mathcal{Y}(w^{(i)},\widetilde{f})}^*\mathcal{H}_k^\infty\subset\mathcal{H}_j^\infty, \end{gather}\] i.e., \(\overline{\mathcal{Y}(w^{(i)},\widetilde{f})}\) is smooth. In the following, we will always denote by \(\mathcal{Y}(w^{(i)},\widetilde{f})\) the restriction of the closed operator \(\overline{\mathcal{Y}(w^{(i)},\widetilde{f})}\) to the core \(\mathcal{H}_j^\infty\). Then the formal adjoint \(\mathcal{Y}(w^{(i)},\widetilde{f})^\dagger\) exists, which is the restriction of \(\mathcal{Y}(w^{(i)},\widetilde{f})^*\) to \(\mathcal{H}_k^\infty\).

It was proved in [29] lemma 3.8 that if \(\mathcal{Y}(w^{(i)},z)\) satisfies \(a\)-th order energy bounds, then for any \(p\geq0\), there exists \(M\geq0\) such that for any \(\widetilde{f}\) and \(\eta\in\mathcal{H}_j^\infty\), \[\begin{align} \big\lVert (1+\overline{L_0}^p) \mathcal{Y}(w^{(i)},\widetilde{f})\eta\big\lVert \leq M \big\lVert (1+\overline{L_0}^{(a+p)}) \eta\big\lVert. \end{align}\] As a consequence, any product of (finitely many) energy-bounded smeared intertwining operators is bounded by a scalar multiple of \(1+\overline{L_0}^r\) for some \(r\geq0\). By [28] lemma 7.2, \(\overline{L_0}^r\) is localizable (if the action of rotation group is defined by \(\varrho(t)=e^{\mathbf{i}t\overline{L_0}}\)). Thus we have:

If \(v\in V\) is homogeneous, and \(Y_i(v,z)\) is energy-bounded, then the smeared vertex operator \(Y_i(v,\widetilde{f})\) is independent of the argument \(\arg_I\). So we will write it as \(Y_i(v,f)\) instead.

3.2 Möbius and conformal covariance↩︎

We discuss the Möbius and conformal covariance of smeared intertwining operators. First, by [51], for any unitary \(V\)-module \(W_i\), the action of Virasoro algebra on \(W_i\) can be integrated to a unitary representation of \(\mathscr G_{\mathcal{A}_V}\) on \(\mathcal{H}_i\). More precisely, consider the Lie algebra \(\mathrm{Vec}(\mathbb{S}^1)\) of \(\mathrm{Diff}^+(\mathbb{S}^1)\) and its universal cover \(\mathscr G\), which is the Lie algebra of (global and smooth) vector fields on \(\mathbb{S}^1\). Let \(\mathrm{Vec}^{\mathbb{C}}(\mathbb{S}^1)\) be the complexification of \(\mathrm{Vec}(\mathbb{S}^1)\), which has a natural \(*\)-structure (involution) preserving the elements in \(\mathbf{i}\mathrm{Vec}(\mathbb{S}^1)\). Let \(e_n\in C^\infty(\mathbb{S}^1)\) be defined by \(e_n(e^{\mathbf{i}\theta})=e^{\mathbf{i}n\theta}\) for any \(n\in\mathbb{Z}\). Define for each \(n\in\mathbb{Z}\), \[\begin{align} l_n=-\mathbf{i}e_n\frac{\partial}{\partial\theta}\quad \in\mathrm{Vec}^{\mathbb{C}}(\mathbb{S}^1). \end{align}\] Then \(l_n^*=l_{-n}\). If \(f=\sum_{n\in\mathbb{Z}}\widehat f_ne_n\in C_c^\infty(\mathbb{S}^1)\) (where each \(\widehat f_n\in\mathbb{C}\)), we define \[\begin{align} T(f)=\sum_{n\in\mathbb{Z}}\widehat f_nl_{n-1}\quad\in\mathrm{Vec}^{\mathbb{C}}(\mathbb{S}^1). \end{align}\] Then \(\mathbf{i}T(f)\in\mathrm{Vec}(\mathbb{S}^1)\) if and only if \(e_{-1}f\) is real. In that case, we can exponentiate \(\mathbf{i}T(f)\) to an element \(\exp(\mathbf{i}T(f))\) in \(\mathscr G\). For instance, we have \[\begin{gather} \varrho(t)=\exp(\mathbf{i}tl_0),\qquad \delta(t)=\exp(t\frac{l_1-l_{-1}}{2}). \end{gather}\]

By [51] section 2, if \(Y_i(v,z)\) satisfies linear energy bounds and \(Y_i(f)^\dagger=Y_i(f)\), then \(Y_i(v,f)\) is essentially self-adjoint (i.e. \(\overline{Y_i(f)}\) is self-adjoint). For the conformal vector \(\nu\), the vertex operator \(Y_i(\nu,z)\) satisfies linear energy bounds by [52] section 2. Moreover, when \(e_1f\) is real, \(Y_i(\nu,f)^\dagger=Y_i(\nu,f)\). So \(\overline{Y_i(\nu,f)}\) is self-adjoint. By [51] theorem 5.2.1, there exists a unique strongly continuous projective representation \(U_i\) of \(\mathscr G\) on \(\mathcal{H}_i\) such that for any \(f\in C^\infty(\mathbb{S}^1)\) where \(e_1f\) is real, \(e^{\mathbf{i}\overline{Y_i(\nu,f)}}\) is a representing element of \(U_i(\exp(\mathbf{i}T(f)))\). We write \(U_0\) as \(U\).¹² Then, as in section 2.1, one can use 1 to define a central extension \(\mathscr G_{\mathcal{A}_V}\) of \(\mathscr G\) associated \(U:\mathscr G\curvearrowright\mathcal{H}_0\). The natural action of \(\mathscr G_{\mathcal{A}_V}\) on \(\mathcal{H}_0\) is also denoted by \(U\). By (the proof of) [3] proposition 4.9, we have:

Let \(g\in\mathscr G_{\mathcal{A}_V}\). If \(\widetilde{I}\in\widetilde{\mathcal{J}}\) and \(\widetilde{f}\in C_c^\infty(\widetilde{I})\), we define \[\begin{align} \widetilde{f}\circ g^{-1}=(f\circ g^{-1},\arg_{gI})\quad \in C_c^\infty(g\widetilde{I}) \end{align}\] where \(\arg_{gI}\) is defined by \(g\widetilde{I}=(gI,\arg_{gI})\). Recall that \(\mathscr G_{\mathcal{A}_V}\) is acting on \(\mathbb{S}^1\). So \(f\circ g^{-1}(z)=f(g^{-1}z)\) for any \(z\in\mathbb{S}^1\). Define \(\partial_zg^{-1}\) to be a smooth function on \(\mathbb{S}^1\) by setting \[\begin{align} (\partial_zg^{-1})(e^{\mathbf{i}\theta})=\frac{\partial (g^{-1}e^{\mathbf{i}\theta})}{\mathbf{i}e^{\mathbf{i}\theta}\partial\theta}. \end{align}\] This function depends only on the image \(\pi(g)\) of \(g\) in \(\mathrm{Diff}^+(\mathbb{S}^1)\). (In the case that \(\pi(g^{-1})\) is analytic near \(\mathbb{S}^1\), \(\partial_zg^{-1}\) is just the ordinary derivative of the analytic function \(\pi(g^{-1})\).) We also define \(\arg \big((\partial_zg^{-1})(e^{\mathbf{i}\theta})\big)\) as follows: Choose any map \(\gamma:[0,1]\rightarrow\mathscr G_{\mathcal{A}_V}\) satisfying \(\gamma(0)=1,\gamma(1)=g\) such that \(\gamma\) descends to a (continuous) path in \(\mathrm{Diff}^+(\mathbb{S}^1)\). Let \(\arg \big((\partial_z\gamma(0)^{-1})(e^{\mathbf{i}\theta})\big)\) be \(0\) and let the argument of \((\partial_z\gamma(t)^{-1})(e^{\mathbf{i}\theta})\) change continuously as \(t\) varies from \(0\) to \(1\). Then \(\arg \big((\partial_zg^{-1})(e^{\mathbf{i}\theta})\big)\) is defined to be the argument of \((\partial_z\gamma(1)^{-1})(e^{\mathbf{i}\theta})\). Using this arg-value, one can define a smooth function \((\partial_zg^{-1})^\lambda\) for all \(\lambda\in\mathbb{C}\), where \((\partial_zg^{-1})^\lambda(e^{\mathbf{i}\theta})=\big((\partial_zg^{-1})(e^{\mathbf{i}\theta})\big)^\lambda\). We understand \[\begin{align} (\partial_zg^{-1})^\lambda\cdot (\widetilde{f}\circ g^{-1})=((\partial_zg^{-1})^\lambda\cdot(f\circ g^{-1}),\arg_{gI}) \end{align}\] to be in \(C_c^\infty(g\widetilde{I})\).

The following can be proved in essentially the same way as [28] proposition 6.4. Note that by the fact that \(\mathscr G\) is generated by elements of the form \(g=\exp(\mathbf{i}T(f))\) (see for instance [3] proposition 2.4), and that for any such \(g\), \(U_i(g)\) is smooth (cf. [51] proposition 2.1), we conclude that \[\begin{align} \label{eq51} \text{U_i(g) is smooth for any g\in\mathscr G_{\mathcal{A}_V}}. \end{align}\tag{19}\]

In this article, the above proposition will be used only when \(g\) is an element inside the rotation subgroup \(\varrho\) or dilation subgroup \(\delta\). (Note that these two subgroups generate \(\widetilde{\mathrm{PSU}}(1,1)\).) When \(g=\varrho(t)\), this proposition is also proved in [29] proposition 3.15. The calculations there can be easily adapted to prove the case \(g=\delta(t)\).

3.3 Strong locality and intertwining property↩︎

Assume that \(V\) is strongly energy-bounded. The following theorem follows from [29] propositions 2.13 and 3.13. (See also the proof of [29] proposition 3.16.)

Let \(E\) be a set of homogenous vectors in \(V\). Given a homogeneous \(w^{(i)}\in W_i\) such that \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded, we say that \(\mathcal{Y}(w^{(i)},z)\) satisfies the \(E\)-strong intertwining property, if for any \(\widetilde{I}\in\widetilde{\mathcal{J}}\), \(J\in\mathcal{J}\) disjoint from \(\widetilde{I}\), \(f\in C_c^\infty(\widetilde{I})\), \(g\in C_c^\infty(J)\), and \(v\in E\), diagram ?? commutes strongly. It is called the strong intertwining property if \(E\) is the set of all homogeneous vectors of \(V\). We say that \(\mathcal{Y}\) satisfies the strong intertwining property if \(\mathcal{Y}(w^{(i)},z)\) is so for any homogeneous \(w^{(i)}\in W_i\). We say that \(V\) is \(E\)-strongly local [28] if \(Y(v,z)\) (the vertex operator for the vacuum module \(V\)) satisfies the strong intertwining property for any homogeneous \(v\in E\); if moreover \(E\) consists of all homogeneous vectors, we say that \(V\) is strongly local.

We now review the construction of conformal nets from VOAs. We say that \(E\) generates \(V\) if the smallest subspace of \(V\) containing \(E\) and invariant under all \(Y(v)_n\) (where \(v\in E,n\in\mathbb{Z}\)) is \(V\). We say that \(E\) is quasi-primary if all the vectors in \(E\) are quasi-primary. A generating and quasi-primary set of vectors always exists:

To construct a conformal net from \(V\), we let the vacuum Hilbert space be \(\mathcal{H}_0\) (the completion of \(V=W_0\)), and let \(\Omega\in V\) be the vacuum vector. For each \(I\in\mathcal{J}\), define \(\mathcal{A}_E(I)\) to be the von Neumann algebra on \(\mathcal{H}_0\) generated by all \(\overline{Y(v,f)}\) where \(v\in E\) and \(f\in C_c^\infty(I)\). Let \(\mathcal{A}_E\) be the family \(I\in\mathcal{J}\mapsto \mathcal{A}_E(I)\). The following two theorems are due to [28].

We now discuss the relation between unitary \(V\)-modules and \(\mathcal{A}_V\)-modules. To begin with, suppose that \(\mathcal{M}\) is a von Neumann algebra on a Hilbert space \(\mathcal{K}\), and \((\pi,\mathcal{K})\) is a normal representation of \(\mathcal{M}\) on another Hilbert space \(\mathcal{H}\). If \(A\) is a closed operator on \(\mathcal{H}\) affiliated with \(\mathcal{M}\), then we can define \(\pi(A)\) to be a closable operator on \(\mathcal{K}\) affiliated with \(\pi(\mathcal{M})\) as follows. Let \(A=UH\) be the polar decomposition of \(A\) with \(U\) unitary and \(H\) positive. Let \(\pi(H)\) be the generator of the one parameter group \(\pi(e^{\mathbf{i}t H})\). Then we set \(\pi(A)=\pi(U)\pi(H)\). \(\pi(A)\) is described by the fact that \(TA\subset\pi(A)T\) for any bounded linear operator \(T:\mathcal{H}\rightarrow\mathcal{K}\) intertwining the actions of \(\mathcal{M}\). We note that the ranges of all such \(T\) span a dense subspace of \(\mathcal{K}\); equivalently, the projection of \(\mathcal{H}\oplus\mathcal{K}\) onto \(\mathcal{H}\) has central support \(1\) in the commutant of \(\mathcal{M}\curvearrowright(\mathcal{H}\oplus\mathcal{K})\). If \(\mathcal{M}\) is type III and \(\mathcal{H},\mathcal{K}\) are separable, there exists a unitary \(T\).

Following [53], when \(V\) is strongly local, we say that a unitary energy-bounded \(V\)-module \(W_i\) is strongly integrable if there exists a (necessarily unique) representation \(\pi_i\) of \(\mathcal{A}_V\) on \(\mathcal{H}_i\), such that for any homogeneous \(v\in V\) and any \(I\in\mathcal{J},f\in C_c^\infty(I)\), \[\begin{align} \pi_{i,I}(\overline{Y(v,f)})=\overline{Y_i(v,f)}.\label{eq36} \end{align}\tag{20}\] Of course, the vacuum module \(W_0\) is strongly integrable. Clearly we have:

The following was proved in [3] proposition 4.9.

We are also interested in a weaker notion: Suppose that \(E\) is generating and quasi-primary, then \(W_i\) is called \(E\)-strongly integrable if there exists a (necessarily unique) \(\pi_i\) such that 20 holds only when \(v\in E\). The following proposition can be proved in the same way as [30] theorem 4.8.¹⁴ The main idea is to use the phases of (the closures of) the smeared intertwining operators \(\mathcal{Y}(w^{(i)},\widetilde{f})\) to construct many bounded linear operators \(T_\alpha\) satisfying the commutativity relations in [30] proposition 4.7 for any \(v\in E\).

The following proposition is a variant of theorem 55.

3.4 The rigid braided \(C^*\)-tensor categories \(\mathrm{Rep}^{\mathrm u}(V)\) and \(\mathscr C\)↩︎

We assume that \(V\) is regular, or equivalently, rational and \(C_2\)-cofinite. For the definitions of these notions and the proof of equivalence, see [38]. In this case, there are only finitely many inequivalent irreducible \(V\)-modules ([54] Cor. 6.6). Consequently, any \(V\)-module is a finite direct sum of irreducible ones. Moreover, any \(\mathcal{V}{k\choose i~j}\) is finite dimensional ([25] theorem 3.1 and remark 3.8).

By [25], [55]–[58] or (under a slightly stronger condition) [26], the category \(\mathrm{Rep}(V)\) of \(V\)-modules form a ribbon braided tensor category. A sketch of the construction of braided tensor structure can be found in [29] section 2.4 or [3] section 4.1. We let \(\boxdot\) denote the tensor (fusion) bifunctor, and let \(\ss\) denote the braiding. \(\boxdot\) is defined in such a way that for any \(W_i,W_j,W_k\in\mathrm{Obj}(\mathrm{Rep}(V))\) there is a natural isomorphism of vector spaces \[\begin{align} \mathcal{V}{k\choose i~j}\simeq\mathrm{Hom}_V(W_i\boxdot W_j,W_k)\label{eq47} \end{align}\tag{22}\] Moreover, by [27], [54], [59], \(\mathrm{Rep}(V)\) is a modular tensor category.

Let \(\mathrm{Rep}^{\mathrm u}(V)\) be the \(C^*\)-category of unitary \(V\)-modules. Since \(V\) is assumed to be strongly unitary, \(\mathrm{Rep}^{\mathrm u}(V)\) is equivalent to \(\mathrm{Rep}(V)\) as linear categories. If \(W_i,W_j\) are unitary, then \(W_i\boxdot W_j\) is a unitarizable \(V\)-module. One needs to choose a correct unitary \(V\)-module structure on each \(W_i\boxdot W_j\) making \(\mathrm{Rep}^{\mathrm u}(V)\) a unitary (i.e. \(C^*\)-) tensor category. This amounts to finding the correct inner product on each \(\mathcal{V}{k\choose i~j}\). This was achieved in [29], [30]. In these two papers, we defined a (necessarily non-degenerate and Hermitian) sesquilinear form \(\Lambda\) on each \(\mathcal{V}{k\choose i~j}\), called the invariant sesquilinear form. We say that a strongly unitary regular VOA \(V\) is completely unitary if \(\Lambda\) is positive on any vector space of intertwining operator of \(V\). We proved in [30] theorem 7.9 that if \(V\) is completely unitary, then \(\mathrm{Rep}^{\mathrm u}(V)\) is a unitary modular tensor category. In particular, \(\mathrm{Rep}^{\mathrm u}(V)\) is a rigid braided \(C^*\)-tensor category.

As in section 2.7, we say that a set \(\mathcal{F}^V\) of irreducible unitary \(V\)-modules generates \(\mathrm{Rep}(V)\) if any irreducible \(V\)-module \(W_i\) is equivalent to a submodule of a (finite) fusion product of irreducible \(V\)-modules in \(\mathcal{F}^V\). By 22 , this is equivalent to saying that for any irreducible unitary \(V\)-module \(W_k\), there is a chain of non-zero unitary irreducible intertwining operators \(\mathcal{Y}_1,\mathcal{Y}_2,\dots,\mathcal{Y}_n\) with charge spaces in \(\mathcal{F}^V\) such that \(\mathcal{Y}_1\) has source space \(W_0=V\), \(\mathcal{Y}_n\) has target space \(W_k\), and the source space of \(\mathcal{Y}_m\) equals the target space of \(\mathcal{Y}_{m-1}\) for any \(m=2,3,\dots,n\).

Consider the following two conditions:

In the remaining part of this article, we shall always assume condition 1. The following is essentially proved in [30]. (See [3] remark 4.21 and theorem 4.22 for a detailed explanation.)

Moreover, by proposition 59 and 60, we have:

Thus, each \(W_i\in\mathrm{Rep}^{\mathrm u}(V)\) is associated with an \(\mathcal{A}_V\)-module \(\mathcal{H}_i\). We define a \(*\)-functor¹⁶ \(\mathfrak F:\mathrm{Rep}^{\mathrm u}(V)\rightarrow \mathrm{Rep}(\mathcal{A}_V)\) sending each \(W_i\) to \(\mathcal{H}_i\). If \(G\in\mathrm{Hom}_V(W_i,W_j)\), then \(\mathfrak F(G):\mathrm{Hom}_{\mathcal{A}_V}(\mathcal{H}_i,\mathcal{H}_j)\) is the closure of the densely defined continuous map \(G\).

Let \(\mathscr C=\mathfrak F(\mathrm{Rep}^{\mathrm u}(V))\). Then one can define the inverse \(*\)-functor \(\mathfrak F^{-1}\) as follows. Choose any \(\mathcal{H}_i\in\mathrm{Obj}(\mathscr C)\). Then there exists a unitary \(W_i\) such that \(\mathfrak F(W_i)=\mathcal{H}_i\). We let \(\mathfrak F^{-1}(\mathcal{H}_i)=W_i\). This is well defined. Indeed, if \(W_{i'}\) is another unitary \(V\)-module such that \(\mathfrak F(W_{i'})=\mathcal{H}_i\), then \(W_i\) and \(W_{i'}\) are both dense subspaces of \(\mathcal{H}_i\). By theorem 64, the morphism \(\mathbf{1}_{\mathcal{H}_i}:\mathfrak F(W_i)\rightarrow\mathfrak F(W_{i'})\) arises from a morphism \(G\in\mathrm{Hom}_V(W_i,W_{i'})\). Since \(\mathbf{1}_{\mathcal{H}_i}=\mathfrak F (G)\), and since \(\mathfrak F(G)\) is the closure of \(G\), \(G\) must be \(\mathbf{1}_{W_i}\). This proves that \(W_i=W_{i'}\) both as pre-Hilbert spaces and as unitary \(V\)-modules.

3.5 The weak vertex categorical partial extension \(\mathscr E^w_\mathrm{par}\)↩︎

The most important results of this section are corollaries 70 and 73.

\(\mathscr C=\mathfrak F(\mathrm{Rep}^{\mathrm u}(V))\) is a semi-simple \(C^*\)-subcategory of \(\mathrm{Rep}(\mathcal{A}_V)\). We shall show that \(\mathscr C\) is a braided \(C^*\)-tensor subcategory of \(\mathrm{Rep}(\mathcal{A}_V)\) by constructing a weak categorical partial extension \(\mathscr E^w_\mathrm{par}=(\mathcal{A}_V,\mathcal{F},\boxdot,\mathfrak H)\) where \(\mathcal{F}=\mathfrak F(\mathcal{F}^V)\) clearly generates \(\mathscr C\).

For each unitary \(W_i,W_j\), we write \(W_{ij}=W_i\boxdot W_j\). Thus, by our notations, \(\mathcal{H}_{ij}\) is the associated \(\mathcal{A}_V\)-module, and \(\mathcal{H}_{ij}^\infty\) is the dense subspace of smooth vectors. By [3] section 4.2, we have \(\mathcal{L}^V\) associates to each unitary \(W_i,W_k\) an intertwining operator \(\mathcal{L}^V\) of type \({ik\choose i~k}={W_i\boxdot W_k\choose W_i~W_k}\) such that for any unitary \(W_l\) and any \(\mathcal{Y}\in\mathcal{V}{l\choose i~k}\), there exists a unique \(T\in\mathrm{Hom}_V(W_i\boxdot W_k,W_l)\) such that \(\mathcal{Y}(w^{(i)},z)w^{(k)}=T\mathcal{L}^V(w^{(i)},z)w^{(k)}\) for any \(w^{(i)}\in W_i,w^{(k)}\in W_k\). Moreover, we define \(\mathcal{R}^V\) to be a type \({ki\choose i~k}={W_k\boxdot W_i\choose W_i~W_k}\) intertwining operator by setting \[\begin{align} \mathcal{R}^V(w^{(i)},z)w^{(k)}=\ss_{i,k}\mathcal{L}^V(w^{(i)},z)w^{(k)}, \end{align}\] We understand \(\mathcal{L}^V(w^{(i)},z)\) and \(\mathcal{R}^V(w^{(i)},z)\) (and their smeared intertwining operators) as categorical operators acting on any possible unitary \(W_k\). We write them as \(\mathcal{L}^V(w^{(i)},z)|_{W_k}\) and \(\mathcal{R}^V(w^{(i)},z)|_{W_k}\) if we want to emphasize that they are acting on \(W_k\). (Note that in [3], these two operators are written as \(\mathcal{L}_i(w^{(i)},z)\) and \(\mathcal{R}_i(w^{(i)},z)\).) For any \(v\in W_0=V\) we have \[\begin{align} \mathcal{L}^V(v,z)|_{W_k}=\mathcal{R}^V(v,z)|_{W_k}=Y_k(v,z). \end{align}\] where we have identified \(W_{0k}\) and \(W_{k0}\) with \(W_k\).

We recall some important properties of \(\mathcal{L}^V\) and \(\mathcal{R}^V\) proved in [3] chapter 4.

Note that by the semisimpleness of (unitary) \(V\)-modules, if \(W_i\) is irreducible and \(w^{(i)}\in W_i\) is homogeneous, then \(\mathcal{L}^V(w^{(i)},z)\) is energy-bounded (when acting on any unitary \(W_k\)) if and only if for any irreducible unitary intertwining operator \(\mathcal{Y}\) with charge space \(W_i\), \(\mathcal{Y}(w^{(i)},z)\) is energy-bounded.

Note that by proposition 51, the dense subspace in the above proposition is also QRI. The following weak locality of smeared \(\mathcal{L}^V\) and \(\mathcal{R}^V\) proved in [3] theorem 4.8 is a generalization of theorem 52.

We now construct the weak vertex categorical partial extension. Note that \(\mathcal{F}:=\mathfrak F(\mathcal{F}^V)\) generates the \(C^*\)-tensor category \(\mathscr C=\mathfrak F(\mathrm{Rep}^{\mathrm u}(V))\). Recall that \(\mathcal{H}_i=\mathfrak F(W_i)\).

Combine the above theorem with theorems 44 and 35 and corollary 36, we obtain the following corollaries.

From the construction of \(\mathscr E^w_\mathrm{par}\), it is clear that we have:

By proposition 72 and corollary 47, we have:

Note that the assumption on \(W_i,W_j\) hold if we choose \(W_i,W_j\in\mathcal{F}^V\).

By corollary 73 and the above remark, we immediately have:

Recall that we say that \(V\) is completely energy-bounded if any unitary irreducible intertwining operator of \(V\) is energy-bounded. If \(V\) is completely unitary and completely energy-bounded, we say that the unitary intertwining operators of \(V\) satisfy strong braiding, if for any \(W_i,W_j,W_k\in\mathrm{Obj}(\mathrm{Rep}^{\mathrm u}(V))\), \(w^{(i)}\in W_i,w^{(j)}\in W_j\) being homogeneous, \(\widetilde{J}\) clockwise to \(\widetilde{I}\), and \(\widetilde{f}\in C_c^\infty(\widetilde{I}),\widetilde{g}\in C_c^\infty(\widetilde{J})\), the diagram ?? of closable operators commutes strongly. The following corollary follows immediately from corollary 73.

3.6 Proving conditions 1 and 2↩︎

In this section, we give two useful methods of proving conditions 1 and 2. The first one concerns sub-VOAs and coset VOAs, the second one is about tensor product VOAs.

3.6.1 Compression principle↩︎

The goal of this subsection is to show that compressing intertwining operators preserves the polynomial energy bounds and the strong intertwining property. Let \(U\) be a unitary simple VOA with vertex operator \(Y^U\), and let \(V\) a unitary sub-VOA of \(U\) with vertex operator \(Y\). This means that \(V\) is a graded subspace of \(U\), that \(U\) and \(V\) share the same vacuum vector \(\Omega\), that \(Y^U(v_1,z)v_2=Y(v_1,z)v_2\) for all \(v_1,v_2\in V\), and that there is \(\nu\in V\) such that \(V\) (with the inner product \(\langle {\cdot|\cdot}\rangle\) inherited from \(U\)) is a unitary VOA with conformal vector \(\nu\). Let \(V^c\) be the set of all \(u\in U\) satisfying \(Y(v)_nu=0\) for any \(n\in\mathbb{N}\). Then by [28] proposition 5.31, \(V^c\) is a unitary VOA (the coset VOA) with conformal vector \(\nu'\) such that \(\nu+\nu'\) is the conformal vector of \(U\). Moreover, let \(Y'\) be the vertex operator of \(V^c\). Then the linear map \[\begin{align} V\otimes V^c\rightarrow U,\qquad v\otimes v'\mapsto Y(v)_{-1}Y(v')_{-1}\Omega \end{align}\] is an isometry and a homomorphism of \(V\otimes V^c\)-modules. Thus one can regard \(V\otimes V^c\) as a (conformal) sub-VOA of \(U\) with the same conformal vector \(\nu+\nu'=\nu\otimes\Omega+\Omega\otimes\nu'\). Note that we have the identification \(v=v\otimes\Omega,v'=\Omega\otimes v'\) (where \(v\in V,v'\in V^c\)). We let \(L_n=Y(\nu)_{n+1},L'_n=Y(\nu')_{n+1}\). Note also that \(V\) and \(V^c\) are clearly CFT-type. Thus they are simple.

Let \((W_{\mathbb{I}},Y_{\mathbb{I}})\) be a unitary \(U\)-module. Then \(W_{\mathbb{I}}\) restricts to a weak \(V\)-module. We say that \((W_i,Y_i)\) is a graded irreducible \(V\)-submodule of \(W_{\mathbb{I}}\) if \(W_i\) is a \(L_0^U\)-graded subspace of \(W_{\mathbb{I}}\) (equivalently, \(W_i\) is \(L_0^U\)-invariant subspace), \(Y_i(v,z)w^{(i)}=Y_{\mathbb{I}}(v,z)w^{(i)}\) for each \(v\in V,w^{(i)}\in W_i\), and \((W_i, Y_i)\) is an irreducible (ordinary) \(V\)-module.

Recall that \(\mathcal{H}_{\mathbb{I}}\) is the Hilbert space completion of \(W_{\mathbb{I}}\), and \(\mathcal{H}^\infty_{\mathbb{I}}\) is the subspace of smooth vectors (defined by \(L_0^U\)). By proposition 77, \(\mathcal{H}_i^\infty\) can be defined unambiguously using either \(L_0\) or \(L_0^U\). One can thus regard \(\mathcal{H}_i^\infty\) as a subspace of \(\mathcal{H}_i\) and also of \(\mathcal{H}_{\mathbb{I}}^\infty\). If \(p_i\) is the projection of \(\mathcal{H}_{\mathbb{I}}\) onto \(\mathcal{H}_i\), then, as each \(e^{\mathbf{i}t\overline{L^U_0}}\) leaves \(\mathcal{H}_i\) invariant, \(e^{\mathbf{i}t\overline{L^U_0}}\) commutes (adjointly) with \(p_i\), which shows that \(L^U_0\) commutes strongly with \(p_i\). Thus \(p_i\) is smooth. In the following, we shall always consider \(p_i\) as a densely defined continuous operator with domain \(\mathcal{H}_{\mathbb{I}}^\infty\).

Let \(W_{\mathbb{J}},W_{\mathbb{K}}\) be unitary \(U\)-modules, and let \(W_j\) and \(W_k\) be respectively graded irreducible \(V\)-submodules of \(W_{\mathbb{J}},W_{\mathbb{K}}\). Let \(\mathcal{V}{k\choose i~j}\) and \(\mathcal{V}^U{\mathbb{K}\choose \mathbb{I}~\mathbb{J}}\) be the vector spaces of type \(k\choose i~j\) intertwining operators of \(V\) and type \(\mathbb{K}\choose \mathbb{I}~\mathbb{J}\) intertwining operators of \(U\) respectively. Define projections \(p_j\) and \(p_k\) in a similar way. If \(\mathcal{Y}\in\mathcal{V}{k\choose i~j}\) and \(\mathcal{Y}^U\in\mathcal{V}^U{\mathbb{K}\choose \mathbb{I}~\mathbb{J}}\), we say that \(\mathcal{Y}\) is a compression of \(\mathcal{Y}^U\) if there exists \(\lambda\in\mathbb{R}\) such that for any \(w^{(i)}\in W_i,w^{(j)}\in W_j,s\in\mathbb{R}\), \[\begin{align} \mathcal{Y}(w^{(i)})_sw^{(j)}=p_k\mathcal{Y}^U(w^{(i)})_{s+\lambda}w^{(j)}.\label{eq50} \end{align}\tag{25}\] By proposition 77, it is clear that \(\mathcal{Y}(w^{(i)},z)\) is (\(L_0\)-) energy-bounded if \(\mathcal{Y}^U(w^{(i)},z)\) is (\(L_0^U\)-) energy-bounded. Thus we have:

It is clear that \(Y_i\) is a compression of \(Y^U_{\mathbb{I}}\) (the vertex operator of the \(U\)-module \(W_{\mathbb{I}}\)). Thus, as a special case of the above proposition, we see that \(W_i\) is energy-bounded if \(W_{\mathbb{I}}\) is so.

We now show that the strong intertwining property is preserved by compressing intertwining operators. Let \(\widetilde{f}=(f,\arg_I)\in C_c^\infty(\widetilde{I})\). By 25 , we have the following natural identification \[\begin{align} \mathcal{Y}(w^{(i)},\widetilde{f})=p_k\mathcal{Y}^U(w^{(i)},\widetilde{f}_\lambda)p_j \end{align}\] where \(\widetilde{f}_\lambda=(f_\lambda,\arg_I)\) and \(f_\lambda(e^{\mathbf{i}t})=e^{\mathbf{i}\lambda t}f(e^{\mathbf{i}t})\).

3.6.2 Coset VOAs↩︎

Assume that \(V\subset U\) is as above. Assume also that \(U,V,V^c\) are all regular. The goal of this subsection is to show that the coset VOA \(V^c\) satisfies condition 2 if \(U\) and \(V\) do. We first need a preliminary result.

Note that the condition \(V^{cc}=V\) is equivalent to that \(V\) is a coset of another unitary sub-VOA, i.e. there is a unitary sub-VOA \(\widetilde{V}\subset U\) such that \(V=\widetilde{V}^c\).

The following is the main result of this section.

3.6.3 Tensor product VOAs↩︎

3.7 Examples↩︎

In [3], we proved the equivalence of braided \(C^*\)-tensor categories and the strong braiding for the following examples: \(c<1\) unitary Virasoro VOAs, Heisenberg VOAs¹⁷, even lattice VOAs (except showing the essential surjectivity), and unitary affine VOAs of type \(A,C,G_2\). The goal of this subsection is to prove theorem 1 in the introduction, which greatly expands above list. We will see that all the examples in theorem 1 (including their tensor products and regular cosets) satisfy condition 2, and some of them are completely energy-bounded. We first give criteria on the complete rationality of \(\mathcal{A}_V\) and the essential surjectivity of \(\mathfrak F\).

3.7.1 Complete rationality and numbers of irreducibles↩︎

Recall that if \(U\) is a unitary VOA, a conformal unitary sub-VOA \(V\) has the same conformal vector as that of \(U\). We let \(\mathcal{A}_U,\mathcal{A}_V\) be the conformal nets of \(U,V\) respectively.

Recall that \(\mathrm{Rep}^{\mathrm f}(\mathcal{A})\) for a conformal net \(\mathcal{A}\) is the braided \(C^*\)-tensor category of dualizable representations of \(\mathcal{A}\).

Next, we prove theorem 1 for the four classes of VOAs mentioned in that theorem. For some examples, the strong braiding of all intertwining operators is also discussed.

3.7.2 WZW models and related coset VOAs↩︎

Let \(\mathfrak g\) be a finite dimensional (unitary) complex simple Lie algebra with dual Coxeter number \(h^\vee\). Let \(l\in\mathbb{Z}_+\), and set \(l'=\frac{l+h^\vee}{l+h^\vee+1}-h^\vee\). Denote by \(L_l(\mathfrak g)\) the level \(l\) unitary affine VOA and \(\mathcal{W}_{l'}(\mathfrak g)\) the discrete series \(W\)-algebra (see [69] for the definition), both of which are simple. \(L_l(\mathfrak g)\) is naturally a unitary VOA, and any \(L_l(\mathfrak g)\)-module is unitarizable, i.e., \(L_l(\mathfrak g)\) is strongly unitary. (Cf. [28], [31], [50], [70].) Moreover, \(L_l(\mathfrak g)\) and \(\mathcal{W}_{l'}(\mathfrak g)\) are regular [63], [71], [72].

We prove these two theorems simultaneously.

Let \(L_l(\mathfrak g)\) be any unitary affine VOA. Let \(\mathfrak h\) be the Cartan subalgebra of \(\mathfrak g\). Then the (unitary) Heisenberg VOA \(L(\mathfrak h)\) is a unitary sub-VOA of \(L_l(\mathfrak g)\). The commutant of \(L(\mathfrak h)\) in \(L_l(\mathfrak g)\), which we denote by \(K_l(\mathfrak g)\), is called a parafermion VOA. \(K_l(\mathfrak g)\) is regular [84]–[86], and is also the commutant in \(L_l(\mathfrak g)\) of \(V_{\sqrt l\Lambda_\mathfrak g}\) where \(\Lambda_\mathfrak g\) is the even lattice generated by the long roots of \(\mathfrak g\) [87]. We have seen that \(L_l(\mathfrak g)\) satisfies condition 2. By [3], \(V_{\sqrt l\Lambda_\mathfrak g}\) also satisfies condition 2. So does \(K_l(\mathfrak g)\) by theorem 82. So (a) (b) (c) of theorem 1 hold for \(K_l(\mathfrak g)\). In particular, the argument in the last paragraph of the proof of theorem 87 (and theorem 88) proves the complete rationality and the essential surjectivity for \(K_l(\mathfrak g)\). In the cases that \(\mathfrak g\) is of type \(ADE\), the complete energy bounds and the strong braiding hold by corollary 83. This proves the following theorem.

3.7.3 Lattice VOAs↩︎

Let \(\Lambda\subset \mathbb{R}^n\) be a rank \(n\) even lattice. This means that \(\Lambda\simeq\mathbb{Z}^n\) as abelian groups, and that \(\lVert\alpha\lVert^2:=(\alpha|\alpha)\) is an even number for any \(\alpha\in\Lambda\), where \((\cdot|\cdot)\) is the inner product inherited from that of \(\mathbb{R}^n\). Then the lattice VOA \(V_\Lambda\) is simple, unitary [50], regular [63], and satisfy condition 2 [3]. Moreover, it was shown in [3] that \(V_\Lambda\) is completely energy-bounded, and the unitary intertwining operators satisfy strong braiding. Thus, it remains to prove the complete rationality of \(\mathcal{A}_{V_\Lambda}\) and the essential surjectivity of \(\mathfrak F\).

We say that a rank \(n\) even lattice is orthogonal if it is generated by \(n\) mutually orthogonal vectors.

The method of proving theorem 93 allows us to prove the following result conjectured in [28] section 8.

We remark that the equivalence \(\mathrm{Rep}^{\mathrm u}(V_\Lambda)\simeq \mathrm{Rep}^{\mathrm f}(\mathcal{B}_\Lambda)\) was proved in [91] section 3.3. This result, together with the above theorem, gives another proof of the equivalence \(\mathrm{Rep}^{\mathrm u}(V_\Lambda)\simeq \mathrm{Rep}^{\mathrm f}(\mathcal{A}_{V_\Lambda})\).

By the results proved so far, the claims for \(U\) and \(V\) in theorem 1 hold. Similar to the proof for \(W_{l'}(\mathfrak g)\) and \(K_l(\mathfrak g)\), we may use corollary 70 and theorems 82, 85, 86 to show that the commutant \(V^c\) also satisfies condition 2 and (hence) (a) (b) (c). The proof of theorem 1 is now complete.

4 Strongly commuting closed operators↩︎

In this section, we prove some useful properties for strongly commuting closed operators. We fix a Hilbert space \(\mathcal{H}\). Recall that if \(A\) is a closed operator (with dense domain) on \(\mathcal{H}\) with polar decomposition \(A=UH\) (\(H\) is positive and \(U\) is a partial isometry) then we have left and right bounding projections \(\{p_s\},\{q_s\}\) defined by \(q_s=\chi_{[0,s]}(H)\) and \(p_s=Uq_sU^*\). Note that \(\overline{p_sA}=Aq_s\) is bounded, and \(\overline{p_sA^*A}=p_s A^*Ap_s\) is bounded and positive. The von Neumann algebra generated by \(A\) is the one generated by \(U\) and all \(\{p_s\}\) (equivalently, all \(\{q_s\}\)). Two closable operators \(A,B\) are said to commute strongly if the von Neumann algebras generated by \(\overline{A}\) and by \(\overline{B}\) commute. If \(A\) and \(B\) are bounded, then they commute strongly if and only if \([A,B]=[A^*,B]=0\), namely, \(A\) commutes adjointly with \(B\).

In the case that one of the two strongly commuting closed operators is bounded, we have the following well known properties:

We prove the following result as an interesting application of the above propositions. The special case \(H=K\) is well-known, but we will use this result in the case that \(H=\log\Delta\) and \(K=\Delta^{\pm\frac{1}{2}}\) (where \(\Delta\) is the modular operator).

We now study the case where neither of the two strongly commuting operators are assumed to be bounded.

As shown below, strong commutativity implies weak commutativity.

The following was proved in [3] lemma 4.17.

: binguimath@gmail.combingui@tsinghua.edu.cn

References↩︎

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1. This fact, together with the weak locality mentioned below, relies only on the fact that the \(n\)-point correlation functions formed by products of vertex operators are rational functions. I thank Sebastiano Carpi for pointing out this to me.↩︎

2. We have only proved this when the intertwining operators are energy-bounded (see theorem 68 or [3] theorem 4.8). Although all intertwining operators of unitary regular VOAs are expected to be energy-bounded, so far we do not have a general proof of this fact. However, we will prove the weak braiding in a future work without using the energy bounds condition.↩︎

3. For general (i.e., non-necessarily closed or vector-labeled) categorical extensions, this axiom is replaced by the neutrality and the Reeh-Schlieder property; see [3] section 3.1.↩︎

4. A sequence of bounded operators \(x_n\) is said to converge \(*\)-strongly to \(x\) if \(x_n\) and \(x_n^*\) converge strongly to \(x^*\) and \(x_n^*\) respectively.↩︎

5. Note that they also commute adjointly and hence strongly since we have \((F\boxdot G)^*=F^*\boxdot G^*\) for any morphisms \(F,G\) in a \(C^*\)-tensor category.↩︎

6. In [3], we only assume \(\mathfrak H_i(\widetilde{I})\) to be a subset of \(\mathcal{H}_i\).↩︎

7. Naturality means that \(\Psi_{i',j'}(F\boxdot G)=(F\boxtimes G)\Psi_{i,j}\) when \(F\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_i,\mathcal{H}_{i'})\) and \(G\in\mathrm{Hom}_{\mathcal{A}}(\mathcal{H}_j,\mathcal{H}_{j'})\).↩︎

8. Note that we do not assume \(\mathcal{H}_i\) or \(\mathcal{H}_j\) to be in \(\mathcal{F}\).↩︎

9. All inner products are assumed to be non-degenerate.↩︎

10. In this chapter, if \(A\) is a linear map from a pre-Hilbert space \(M\) to another one \(N\), then \(A^\dagger\) (if exists) is the unique linear map from \(N\) to M satisfying \(\langle {A\mu|\nu}\rangle=\langle {\mu|A^\dagger\nu}\rangle\) for any \(\mu\in M,\nu\in N\).↩︎

11. Here, \(z\) is understood as a variable but not as a concrete complex number.↩︎

12. After constructing a conformal net \(\mathcal{A}_V\) from \(V\), \(U\) will be the projective representation making \(\mathcal{A}_V\) conformal covariant.↩︎

13. More precisely, it is the restriction of \(U\) from \(\mathscr G\) to \(\mathrm{Diff}^+(\mathbb{S}^1)\).↩︎

14. The first paragraph of step 2 of the proof of [30] theorem 4.8. checks that the linearly energy-bounded intertwining operator satisfies the strong intertwining property. Here we already assume the strong intertwining property. So that paragraph can be skipped when proving proposition 58.↩︎

15. It follows from proposition 48 that \(\mathcal{Y}\) is energy-bounded.↩︎

16. A functor \(\mathfrak F\) is called a \(*\)-functor if \(\mathfrak F(G^*)=\mathfrak F(G)^*\) for any morphism \(G\).↩︎

17. If \(V\) is a Heisenberg VOA then \(V\) is not regular. Nevertheless, if we take \(\mathrm{Rep}^{\mathrm u}(V)\) to be the category of semisimple unitary \(V\)-modules, then the results in the last section still hold for \(V\).↩︎