1 Introduction↩︎

Conformal nets are one of the most developed mathematical approaches to 2-dimensional chiral conformal field theory (CFT). A conformal net consists of a vacuum Hilbert space \(H_0\) together with a net of von Neumann algebras of observables \(I\subset S^1\mapsto \A(I)\subset B(H_0)\) parametrized by intervals of \(S^1\). The vacuum \(H_0\) is also equipped with a projective action of the group \(\Diff^+(S^1)\) of orientation preserving diffeomorphisms of \(S^1\) in a way that is compatible with the net \(\A\). See Definition 2 for a precise definition and [3]–[9] for an introduction to the theory of conformal nets.

Given a conformal net \(\A\), one can consider actions of \(\A\) on Hilbert spaces other than the vacuum \(H_0\). More precisely, a representation of \(\A\) consists of a Hilbert space \(H\) together with \(*\)-actions of all the von Neumann algebras \(\A(I)\) which are compatible with the inclusions \(\A(J)\subset \A(I)\) for \(J\subset I\subset S^1\). We write \(\Rep(\A)\) for the category of representations of \(\A\) and bounded linear maps intertwining the actions of all the algebras \(\A(I)\). It is well known that \(\Rep(\A)\) is a braided tensor category. We refer the reader to [10], [11] and [6] for the definition of the tensor product of representations and the braiding, and [9], [12], [13] for a more modern treatment. The algebras of endomorphisms of \(\Rep(\A)\) are von Neumann algebras, and \(\Rep(\A)\) is furthermore a braided \(\mathrm{W}^*\)-tensor category. See [14] for an introduction to \(\mathrm{W}^*\)-tensor categories.

Most of the literature on conformal nets focuses on the subclass of rational conformal nets (we note that rationality and complete rationality are equivalent by [15]). One of the main reasons for this restriction is that, whenever \(\A\) is a rational conformal net, the braided tensor category \(\Rep(\A)\) is a modular tensor category [16]. In particular, \(\Rep(\A)\) is a finite semisimple rigid tensor category, and it carries a balance (or categorical twist). Recall that a balance on a tensor category \(\mathcal{C}\) with a braiding \(\beta\) consists of a natural family of isomorphisms \[\theta_X: X\to X\] satisfying that \(\theta_{X\otimes Y} = (\theta_X\otimes\theta_Y)\circ \beta_{Y,X}\circ\beta_{X,Y}\) for all \(X,Y\in\mathcal{C}\). If \(\mathcal{C}\) is furthermore rigid and \(\theta\) is compatible with the evaluation and coevaluation maps, the balance is called a ribbon structure.

When \(\A\) is not rational, the category \(\Rep(\A)\) is in general not finite, semisimple, or rigid: it may have infinitely (or continuously) many simple objects, and a generic representation of \(\A\) is a direct integral of irreducible representations as opposed to a direct sum. Whilst there is a vast literature on finite semisimple rigid tensor categories, much less is known about categories with direct integrals and continuously many simple objects. For example, the fact that \(\Rep(\A)\) is a balanced tensor category even when \(\A\) is not rational only appeared in the literature in [17]. In this paper, we do not assume rationality at any point.

Even when non-rational conformal nets are considered, most of the literature approaches the representation theory of conformal nets through the language of localized endomorphisms [18]. Fix a point \(\mathrm{p}\in S^1\) and let \(\A_\infty\) be the \(*\)-subalgebra of \(B(H_0)\) constructed as the union of the algebras \(\A(I)\) for \(I\subset S^1\) not containing \(\mathrm{p}\). The endomorphisms of \(\A_\infty\) are the objects of a category \(\End\,\A_\infty\) whose morphisms are elements of \(\A_\infty\) intertwining the actions of the endomorphisms. An endomorphism \(\rho\in \End\,\A_\infty\) is said to be localized if there is an interval \(\mathrm{p}\notin I\subset S^1\) such that \(\rho\) acts like the identity outside of \(I\), and it is transportable if it is equivalent to an endomorphism localized in \(J\) for every \(\mathrm{p}\notin J\subset S^1\). The full subcategory \(\DHR(\A)\subset \End\,\A_\infty\) (for Doplicher, Roberts, Haag) on the localized and transportable endomorphisms is a braided tensor category equivalent to \(\Rep(\A)\), see [2] and [13].

Fixed-points conformal nets and orbifolding. Given a CFT and a compact subgroup \(G\) of symmetries of the CFT, there exists a process, known as orbifolding, which constructs a new theory by modding out by the symmetries \(G\). The study of orbifolding in CFTs began in [19] in the context of rational CFTs and finite subgroups of symmetries. Mathematically, if the CFT is encoded as a conformal net \(\A\), one considers a compact group \(G\) and a continuous group homomorphism from \(G\) into the group of automorphisms of \(\A\). An automorphism \(\varphi\) of \(\A\) is a collection \(\{\varphi_I\}_{I\subset S^1}\) of automorphisms of the von Neumann algebras \(\A(I)\) which is implemented in the vacuum Hilbert space in the sense that there exists a unitary \(V_\varphi\in U(H_0)\) such that \(\varphi(x) = V_\varphi x V_\varphi^*\) for all intervals \(I\subset S^1\) and all elements \(x\in \A(I)\). We write \(\Aut(\A)\) for the group of automorphisms of \(\A\), which inherits a topology from that of \(U(H_0)\). Given such an action \(G\to \Aut(\A)\), orbifolding amounts to taking the fixed-points conformal net \(\A^G\), defined as follows.

Let us identify an element of \(G\) with the automorphism of \(\A\) it produces. The fixed-points conformal net \(\A^G\) is the net that assigns to an interval \(I\subset S^1\) the von Neumann algebra \(\A^G(I) := \A(I)^G:=\{x\in\A(I)\,|\,gx = x\,\text{for all g\in G}\}\) acting on the fixed-points Hilbert space \(H_0^G:=\{\xi\in H_0\,|\,\text{V_g\xi = \xi for all g\in G}\}\). It is natural to ask whether one can understand the category of representations of \(\A^G\) in terms of representations of \(\A\). It is known that \(\Rep(\A)\) does not contain all the necessary information to compute \(\Rep(\A^G)\), but one has to, in addition, consider twisted representations of \(\A\).

Twisted representations of conformal nets. Given an automorphism \(\varphi\in \Aut(\A)\), a \(\varphi\)-twisted representation of \(\A\) is, roughly speaking, a Hilbert space \(H\) together with a family of \(*\)-actions of the algebras \(\A(I)\) compatible with the inclusions \(\A(J)\subset \A(I)\) for \(J\subset I\) everywhere away from a fixed point \(\mathrm{p}\in S^1\), and only up to an application of \(\varphi\) at the point \(\mathrm{p}\), see Definition 4. We write \(\Rep^\varphi(\A)\) for the \(\mathrm{W}^*\)-category of \(\varphi\)-twisted representations of \(\A\). Twisted representations were introduced, in the context of localized endomorphisms, in [2] precisely to study representations of fixed-points conformal nets.

Given an automorphism \(\varphi\in\Aut(\A)\), Müger constructs in [2] a category \(\varphi-\Loc(\A)\subset \End\,\A_\infty\) of endomorphisms which are \(\varphi\)-localized in some interval \(\mathrm{p}\notin I\subset S^1\), meaning that they act by the identity on the left of \(I\) and by \(\varphi\) on the right, and are transportable. Note that, if \(\varphi = \id\) is the identity automorphism, then \(\id-\Loc(\A) = \DHR(\A)\). Given an action \(G\to \Aut(\A)\) of a discrete group on \(\A\) and identifying an element of \(g\) with the automorphisms it produces, Müger showed that the category \(G-\Loc(\A) : = \bigoplus\limits_{g\in G}g-\Loc(\A)\) has the structure of a \(G\)-crossed braided tensor category. In particular, \(G-\Loc(\A)\) admits a tensor product (given by composition of endomorphisms) covering the multiplication of \(G\), as well as an action of \(G\) by tensor automorphisms and a version of a braiding in the context of a group acting on a graded tensor category. In [1], we produced explicitly a \(G\)-crossed braided tensor structure on the \(\mathrm{W}^*\)-category \(\Rep^G(\A):=\bigoplus\limits_{g\in G}\Rep^g(\A)\) and provided an equivalence of \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories \(\Rep^G(\A)\cong G-\Loc(\A)\). Working in the language of representations, we also provided a \(G\)-crossed balance on \(\Rep^G(\A)\), that is, a natural family of unitary isomorphisms \(\theta_H: H\to T_g(H)\), for \(g\in G\) and \(H\in\Rep^g(\A)\) compatible with the \(G\)-crossed braiding, where \(T_g(H)\) is the image of \(H\) under the action of \(g\).

The \(G\)-equivariantization of a \(G\)-crossed braided (balanced) \(\mathrm{W}^*\)-tensor category is canonically a braided (resp. balanced) \(\mathrm{W}^*\)-tensor category. Hence, the \(G\)-equivariantization \((G-\Loc(\A))^G\) of \(G-\Loc(\A)\) is a braided \(\mathrm{W}^*\)-tensor category, and the \(G\)-equivariantization \((\Rep^G(\A))^G\) of \(\Rep^G(\A)\) is a balanced tensor category.

The category of representations of the fixed-points conformal net. Assume now that the group \(G\) is finite and that it is acting faithfully on \(\A\). Then, one of the main results in [2] is that, if \(\A\) is a rational conformal net, \(\DHR(\A^G)\) is completely controlled by \(G-\Loc(\A)\), as follows.

In the language of representations, we obtain an equivalence of braided \((\mathrm{W}^*-)\)tensor categories \(\Rep(\A^G)\cong (\Rep^G(\A))^G\) for a faithful action of a finite group \(G\) on a rational conformal net \(\A\). The purpose of the current paper is to generalize the result above in two directions. Firstly, we remove the hypothesis of \(\A\) being rational (while keeping \(G\) finite), and secondly we upgrade the equivalence to an equivalence of balanced \(\mathrm{W}^*\)-tensor categories. The main result of the paper is Theorem 9.

The underlying functor of this equivalence is the functor \(\mathfrak{R}: (\Rep^G(\A))^G\to \Rep(\A^G)\) that restricts a \(G\)-equivariant twisted representation of \(\A\) on a Hilbert space \(H\) to an honest \(\A^G\)-representation on the subspace of \(G\)-invariant vectors of \(H\).

Our proof strategy is as follows. The vacuum \(H_0\), seen as an \(\A^G\)-representation by restriction along \(\A^G\subset\A\), has the canonical commutative \(\mathrm{C}^*\)-Frobenius algebra (or \(Q\)-system) in \(\Rep(\A^G)\). Denoting this algebra by \(\Gamma\), we argue that the category \(\M\) of unitary modules over \(\Gamma\) is a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category equivalent to \(\Rep^G(\A)\), using similar arguments to those in [20]. This requires us to introduce \(G\)-crossed categorical extensions of conformal nets in Section 3, following [13]. The fact that \(\M\) and \(\Rep^G(\A)\) are equivalent as \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories implies that their equivariantizations \(\M^G\) and \((\Rep^G(\A))^G\) are equivalent as braided \(\mathrm{W}^*\)-tensor categories. Then, we construct a braided \(\mathrm{W}^*\)-tensor functor \(\Rep(\A^G)\to \M^G\) and show that the composition \(\Rep(\A^G)\to \M^G\cong (\Rep^G(\A))^G\) is an equivalence of braided \(\mathrm{W}^*\)-tensor categories with inverse the functor \(\mathfrak{R}: (\Rep^G(\A))^G\to \Rep(\A^G).\) We borrow some of the arguments from [21], which proves a similar equivalence to 1 in the context of vertex operator algebras. It is then easy to show that \(\mathfrak{R}: (\Rep^G(\A))^G\to \Rep(\A^G)\) is compatible with the balances on both sides.

In [1], we constructed an involutive structure on \(\Rep^G(\A)\) compatible with the \(G\)-crossed balance. Then, both the \(G\)-equivariantization \((\Rep^G(\A))^G\) and the category \(\Rep(A^G)\) are involutive \(\mathrm{W}^*\)-tensor categories. In Appendix 5, we upgrade the equivalence \(\Rep(\A^G)\cong (\Rep^G(\A))^G\) to also be compatible with these involutive structures.

These results are used in [22] to produce the first explicit computation of a category of representations of a conformal net with irreducible representations that tensor to a direct integral of irreducible representations, as opposed to a direct sum.

Acknowledgements↩︎

I am grateful to André Henriques and Bin Gui for their comments, and to Josep Fontana McNally for proofreading this paper. This work has been funded by the EPSRC grant EP/W524311/1.

For the purpose of Open Access, the author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.

2 Preliminaries↩︎

2.1 Conformal nets↩︎

We denote by \(S^1:=\{z\in \C\;|\;|z| = 1\}\) the standard circle, and define an interval of \(S^1\) to be an open, connected, non-empty, non-dense subset of \(S^1\). We write \(\Jcal\) for the collection of intervals of \(S^1\). The group \(\Mob\) is the group of Möbius transformations of \(S^1\), that is, transformations of the form \[z\mapsto \frac{az+b}{\bar{b}z+\bar{a}}\] with \(a,b\in \C\), \(|a|^2-|b|^2 = 1\). Then, \(\Mob\) is a Lie group isomorphic to \(PSL(2, \R).\) We denote by \(\Rot\) the subgroup of rotations of \(\Mob\), and an anticlockwise rotation of angle \(t\) by \(R_t\). In this paper, all Hilbert spaces and \(\mathrm{C}^*\)-algebras are assumed to be separable.

Let \(\Diff^+(S^1)\) denote the group of orientation-preserving diffeomorphisms of \(S^1\). A strongly continuous projective unitary representation \(V\) of \(\Diff^+(S^1)\) on a Hilbert space \(H\) is a strongly continuous homomorphism \(V: \Diff^+(S^1)\to \mathcal{P}U(H)\). Note that \(\Mob\subset\Diff^+(S^1)\). We say that \(V\) is an extension of a unitary representation \(U\) of \(\Mob\) on \(H\) if, for every \(\varphi\in \Mob\), it holds that \(V(\varphi) = [U(\varphi)]\), where \([-]: U(H)\to \mathcal{P}U(H)\) denotes the projection map.

The following are well-known properties of any conformal net \((H_0, \A, U, \Omega)\).

1. Haag duality: the commutant of \(\A(I)\) in \(B(H_0)\) is \(\A(I^c)\), where \(I^c\) denotes the interior of \(S^1\setminus I\) [6];

2. The Reeh-Schlieder Theorem: the vacuum vector \(\Omega\) is cyclic and separating for every von Neumann algebra \(\A(I)\) [6];

3. Each von Neumann algebra \(\A(I)\) is a type \(\mathrm{III}_1\) factor [23];

4. Additivity: for any interval \(I\in \Jcal\) and any collection of intervals \(\{I_\alpha\}_{\alpha \in A}\) with \(I_\alpha\in \Jcal\) and \(\bigcup\limits_{\alpha\in A}I_\alpha = I\), it holds that \(\A(I) = \bigvee\limits_{\alpha\in A}\A(I_\alpha)\) [24].

Fix a conformal net \((H_0,\A,U, \Omega)\), which we will simply denote by \(\A\) from now on.

The conformal net \(\A\) has a canonical representation on its vacuum Hilbert space \(H_0\) by definition, which we call the vacuum representation of \(\A\). Given \(I\in\Jcal\) and \(x\in \A(I)\), we write \(\pi_{0,I}(x)\) or \(\pi_0(x)\) for the action of \(x\) on the vacuum representation \(H_0\).

The category \(\Rep(\A)\) is a braided tensor category, the structure of which can be produced in different equivalent ways [6], [9], [13], [25].

We wish to consider more general representations of \(\A\), those which are twisted by an automorphism of the conformal net. An automorphism \(\varphi\) of \(\A\) consists of a von Neumann algebra automorphism \(\varphi_I: \A(I)\to\A(I)\) for every \(I\in\Jcal\) such that for every inclusion \(J\subset I\) of intervals, the following diagram commutes \[\begin{tikzcd} {\A(I)} & {\A(I)} \\ {\A(J)} & {\A(J)}. \arrow["{\varphi_I}", from=1-1, to=1-2] \arrow[hook, from=2-1, to=1-1] \arrow["{\varphi_J}"', from=2-1, to=2-2] \arrow[hook, from=2-2, to=1-2] \end{tikzcd}\] Automorphisms are furthermore required to be unitarily implemented in the vacuum representation in the sense that they come equipped with a unitary \(V_\varphi: H_0\to H_0\) such that, for every \(I\in \Jcal\), it holds that \[\pi_{0,I}\circ\varphi_I(-) = V_\varphi\circ \pi_{0,I}(-)\circ V_\varphi^{-1},\] and \(V_\varphi(\Omega) = \Omega\). We denote by \(\Aut(\A)\) the group of automorphisms of \(\A\). Given \(\varphi\in \Aut(\A)\), an interval \(I\in \Jcal\) and \(x\in\A(I)\), we will write \(\varphi x:=\varphi_I x\).

Automorphisms of \(\A\) allow us to define twisted representations of the conformal net, as follows. Let us pick \(\mathrm{p}:=1\in S^1\), and fix it for the rest of the paper.

The condition in Definition 4 can also be formulated as follows. Let \(q: \R\to S^1\) be the map \(t\mapsto e^{2\pi i t}\). Given an interval \(I\in \Jcal\), we shall write \(\hat{I}\subset \R_{>0}\) for the lift of \(I\) to \(\R_{> 0}\) as close to zero as possible so that \(\mathrm{cl}(\hat{I})\subset\R_{>0}\). We call an inclusion \(J\subset I\) in \(S^1\) standard if \(\hat{J}\subset \hat{I}\), and special if it is not standard. For an inclusion \(J\subset I\), we shall write \(\delta^{\varphi^{-1}}_{J\subset I}: \A(J)\to \A(I)\) for the inclusion \(\A(J)\hookrightarrow \A(I)\) if \(J\subset I\) is standard, and for \(\A(J)\hookrightarrow\A(I)\xrightarrow{\varphi^{-1}_I}\A(I)\) if \(J\subset I\) is special. Then, the condition in Definition 4 is equivalent to the diagram \[\begin{tikzcd} {\A(I)} & {B(H)} \\ {\A(J)} \arrow["{\pi^H_I}", from=1-1, to=1-2] \arrow["{\delta_{J\subset I}^{\varphi^{-1}}}", hook, from=2-1, to=1-1] \arrow["{\pi^H_J}"', from=2-1, to=1-2] \end{tikzcd}\] commuting for all inclusions \(J\subset I\) of intervals in \(\Jcal\).

In what follows, it will be useful to have the following characterization of special inclusions. Let \(\Jcal_\R := \{\tilde{I}\subset \R\;|\;\tilde{I} \text{ is an open interval such that q\tilde{I} \in \Jcal}\}\). For an interval \(\tilde{I}\in \Jcal_\R\), we denote \(I : =q\tilde{I}\in\Jcal\). Also, let \[\epsilon(\tilde{I}):=\begin{cases} |(0, \partial_-\tilde{I})\cap \Z|, & \text{if \partial_{-} \tilde{I}} > 0\\ -|[\partial_-\tilde{I}, 0]\cap \Z|, & \text{if \partial_-\tilde{I}\leq0}. \end{cases}\] Given an inclusion \(\tilde{J}\subset \tilde{I}\) of intervals in \(\Jcal_\R\), the inclusion \(J\subset I\) is standard if and only if \(\epsilon(\tilde{J}) = \epsilon(\tilde{I})\). In addition, if the inclusion \(J\subset I\) is special, then \(\epsilon(\tilde{I}) = \epsilon(\tilde{J}) - 1\), see [1].

Given a conformal net \(\A\), we can produce a partially defined net \(\A_\R\) on the real line by assigning to every interval \(\tilde{I}\in \Jcal_\R\) the von Neumann algebra \(\A_\R(\tilde{I}) := \A(I)\). The net \(\A_\R\) satisfies the weaker version of locality that \(\A_\R(\tilde{I})\) and \(\A_\R(\tilde{J})\) commute if \(I\cap J = \emptyset\). Given a \(\varphi\)-twisted representation \((H^\varphi, \pi^H)\) of \(\A\), we obtain an action \(\pi^H_{\tilde{I}}\) of every von Neumann algebra \(\A_\R(\tilde{I})\) on \(H^\varphi\) by setting \[\pi^H_{\tilde{I}}(x) := \pi^H_I(\varphi^{\epsilon(\tilde{I})}x)\] for every \(\tilde{I}\in \Jcal_\R\) and \(x\in A_\R(\tilde{I}) = \A(I)\). Note that it holds that, for every inclusion \(\tilde{J}\subset \tilde{I}\) in \(\Jcal_\R\) and \(x\in \A_\R(\tilde{J}) = \A(J)\), \[\pi^H_{\tilde{J}}(x) = \pi^H_J(\varphi^{\epsilon(\tilde{J})}x) = \pi^H_I(\delta^{\varphi^{-1}}_{J\subset I}\varphi^{\epsilon(\tilde{J})}x) = \pi^H_I(\varphi^{\epsilon(\tilde{I})}x) = \pi_{\tilde{I}}^H(x).\] In addition, for all \(x\in \A_\R(\tilde{I}) = \A(I) = \A_\R(\tilde{I} + 1)\), we have \[\label{eq:32TwistedRepOnR} \pi_{\tilde{I}}^H(\varphi x ) = \pi_{\tilde{I} + 1}^H( x)\tag{2}\] coming from the fact that \(\epsilon(\tilde{I} + 1) = \epsilon(\tilde{I}) + 1\). Actually, given a Hilbert space \(K\) with a \(*\)-action \(\pi_{\tilde{I}}\) of the von Neumann algebra \(\A_\R(\tilde{I})\) for every \(\tilde{I}\in\Jcal_\R\), compatible with the inclusion of intervals in \(\Jcal_\R\) and satisfying 2 , we obtain a \(\varphi\)-twisted representation \(\pi\) of \(\A\) on \(K\) by setting \[\pi_I(x) := \pi_{\hat{I}}(x)\] for every \(I\in\Jcal\) and \(x\in \A(I)\). We will freely move between the notion of a \(\varphi\)-twisted representation of \(\A\) on a Hilbert space \(H\) in the sense of Definition 4 and in the sense of a compatible family of \(*\)-actions of the von Neumann algebras \(\A_\R(\tilde{I})\) on \(H\) satisfying 2 .

2.2 Equivariantization of \(G\)-crossed balanced categories↩︎

In this section, we discuss equivariantizations of braided and balanced \(\mathrm{W}^*\)-tensor categories. Let us briefly recall the notions of crossed-braided and crossed-balanced tensor categories. For a more extensive introduction the reader may read [1]. Let \(\mathcal{C}\) be a \(\mathrm{W}^*\)-tensor category. We write \(\underline{\text{Aut}}_\otimes(\mathcal{C})\) for the monoidal category of \(\mathrm{W}^*\)-tensor automorphisms and unitary natural transformations of \(\mathcal{C}\).

Let \(G\) be a discrete group and write \(\underline{G}\) for the tensor category whose objects are elements of \(G\) and whose morphisms are only identities. A \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category structure on \(\mathcal{C}\) consists of a grading \(\mathcal{C} = \bigoplus\limits_{g\in G}\mathcal{C}_g\) such that the tensor product covers the group multiplication, an action \(T: \underline{G}\to \underline{\text{Aut}}_\otimes(\mathcal{C})\) of \(G\) on \(\mathcal{C}\) covering the conjugation action of \(G\) on itself, and a family of unitary isomorphisms \(\beta_{X, Y}: X\otimes Y\to T_g(Y)\otimes X\) for \(g\in G\), \(X\in \mathcal{C}_g\) and \(Y\in \mathcal{C}\). The family \(\beta\) is called the \(G\)-crossed braiding and is required to satisfy the obvious versions of the hexagon diagrams. Note that the action \(T\) comes, by definition, with unitary equivalences \(i_g: \1\to T_g(\1)\), for \(\1\in\mathcal{C}\) the unit object, as well as unitary natural isomorphisms \(s_g:T_g(-)\otimes T_g(-)\to T_g(-\otimes -)\) and \(n_{g,h}: T_g\circ T_h\to T_{gh}\), for \(g,h\in G\).

Given \((\mathcal{C}, \otimes^\Ccal, T^\Ccal, \beta^\Ccal)\) and \((\mathcal{D}, \otimes^\Dcal, T^\Dcal, \beta^\Dcal )\) two \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories, a functor between them consists of a \(\mathrm{W}^*\)-tensor functor \((F,\Psi): (\Ccal,\otimes^\Ccal)\to (\Dcal, \otimes^\Dcal)\) preserving the \(G\)-grading, together with unitary natural isomorphisms \[\Phi_g(X): T_g^\Dcal(F(X)) \xrightarrow{\cong } F(T^\Ccal_g(X))\] for all \(g\in G\) and \(X\in \Ccal\). These are required to be compatible with the \(G\)-action and the \(G\)-crossed braiding in the obvious way, see [1]. We shall call such a triple \((F,\Psi,\Phi)\) a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor functor. A \(G\)-crossed braided \(\mathrm{W}^*\)-tensor functor is an equivalence of \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories if its underlying \(\mathrm{W}^*\)-functor is an equivalence of \(\mathrm{W}^*\)-categories.

Given a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category \(\mathcal{C}\), a \(G\)-crossed balance on \(\mathcal{C}\) is a family of natural unitary isomorphisms \(\theta_X: X\to T_g(X)\) for every \(g\in G\) and \(X\in \mathcal{C}_g\) such that \(T_g(\theta_X) = \theta_{T_g(X)}\) for every \(g\in G\) and \(X\in \mathcal{C}_g\) and the following diagram commutes \[\label{eq:32Xtwist} \begin{tikzcd} {X\otimes Y} && {T_{gh}(X\otimes Y)} \\ && {T_{gh}(X)\otimes T_{gh}(Y)} \\ && {T_{ghgh^{-1}g^{-1}}T_{ghg^{-1}}(X)\otimes T_{ghg^{-1}}T_g(Y)} \\ {T_g(Y)\otimes X} && {T_{ghg^{-1}}(X)\otimes T_g(Y)} \arrow["{\theta_{X\otimes Y}}", from=1-1, to=1-3] \arrow["{\beta_{X,Y}}"', from=1-1, to=4-1] \arrow["\cong"', from=2-3, to=1-3] \arrow["\cong"', from=3-3, to=2-3] \arrow["{\beta_{T_g(Y), X}}"', from=4-1, to=4-3] \arrow["{\theta_{T_{ghg^{-1}}(X)}\otimes\theta_{T_g(Y)}}"', from=4-3, to=3-3] \end{tikzcd}\tag{3}\] for all \(g,h\in G\), \(X\in \mathcal{C}_g\), \(Y\in\mathcal{C}_h\). A \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor category is a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category with a \(G\)-crossed balance.

Given \(\Ccal\) and \(\Dcal\) two \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories with \(G\)-crossed balances \(\theta^\Ccal\) and \(\theta^\Dcal\) respectively, a \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor functor \((F,\Psi,\Phi):\Ccal\to \Dcal\) is a functor of \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories such that \(F(\theta^\mathcal{C}_X) = \Phi_g(X)\circ \theta^\mathcal{D}_{F(X)}\) for all \(g\in G\) and \(X\in\Ccal_g\).

It is well-known that, given a \(G\)-crossed braided tensor category \(\mathcal{C}\) one can canonically obtain a braided category \(\mathcal{C}^G\) by means of an equivariantization procedure [26]. The equivariantization construction extends to an assignment from \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor categories to balanced \(\mathrm{W}^*\)-tensor categories. We spell it out here for completeness.

The equivariantized category \(\mathcal{C}^G\) has objects pairs \((X,u)\) where \(X\in \mathcal{C}\) and \(u = \{u\}_{g\in G}\) is a family of unitary isomorphisms \(u_g: T_g(X)\cong X\) such that the diagram \[\begin{tikzcd} {T_gT_h(X)} && {T_g(X)} \\ {T_{gh}(X)} && X \arrow["{T_g(u_h)}", from=1-1, to=1-3] \arrow["\cong"', from=1-1, to=2-1] \arrow["{u_g}", from=1-3, to=2-3] \arrow["{u_{gh}}"', from=2-1, to=2-3] \end{tikzcd}\] commutes for every \(g,h\in G\). We also require that \(u_e\) is induced by the trivialization \(\text{Id}_\mathcal{C}\cong T_e\). A morphism between two objects \((X,u)\) and \((Y,v)\) in \(\mathcal{C}^G\) is a morphism from \(X\) to \(Y\) in \(\mathcal{C}\) compatible with \(u_g\) and \(v_g\) for all \(g\in G\). Given \((X, u)\in \mathcal{C}^G\), its algebra of endomorphisms is the subalgebra of \(\End_\mathcal{C}(X)\) of morphisms \(f:X\to X\) such that \(u_g\circ T_g(f)\circ u_g^* = f\) for all \(g\in G\). Since \(T_g\) is a \(\mathrm{W}^*\)-functor and \(u_g\) is unitary for each \(g\in G\), the map \(f\mapsto u_g\circ T_g(f)\circ u_g^*\) is a normal isometry from \(\End_{\mathcal{C}}(X)\) to itself. Hence, \(\End_{\mathcal{C}^G}((X, u)) = \bigcap_{g\in G}\ker(u_gT_g(-)u_g^*-\id)\) is a von Neumann algebra. Therefore, \(\mathcal{C}^G\) is a \(\mathrm{W}^*\)-category. The \(\mathrm{W}^*\)-category \(\mathcal{C}^G\) inherits a tensor structure \[(X,u)\otimes (Y,v):=(X\otimes Y,u\otimes v)\] from that of \(\mathcal{C}\), where \(u\otimes v = \{s_g^{-1}(X,Y)\circ(u_g\otimes v_g)\}_{g\in G}\). The unit is the object \((\1, \{i^{-1}_g\}_{g\in G})\in\mathcal{C}^G\). The fact that the action of \(G\) on \(\mathcal{C}\) is by tensor automorphisms implies that the associators and unitors of \(\mathcal{C}\) descend to data on \(\mathcal{C}^G\), providing its tensor structure. The \(G\)-crossed braiding \(\beta\) on \(\mathcal{C}\) induces an honest braiding on \(\mathcal{C}^G\) as follows. Let \((X,u), (Y,v)\in \mathcal{C}^G\). Then, we can write \(X = \bigoplus\limits_{g\in G} X_g\) with \(X_g\in \Ccal_g\), and similarly for \(Y = \bigoplus\limits_{h\in G}Y_h\), so that \(X\otimes Y = \bigoplus\limits_{(g,h)\in G^2}X_g\otimes Y_h\). Note that for every \(g\in G\), the isomorphism \(v: T_g(Y)\cong Y\) breaks as a direct sum \(\bigoplus_{h\in G}v_g^h\) with \(v_g^h: T_g(Y_h)\xrightarrow{\cong} Y_{ghg^{-1}}\). On the \((g,h)\)-component of \(\bigoplus\limits_{(g,h)^2\in G}X_g\otimes Y_h\), we define \(\beta^G_{(X,u), (Y,v)}\) as \[X_g\otimes Y_h\xrightarrow{\beta_{X_g, Y_h}} T_g(Y_h)\otimes X_g\xrightarrow{v^h_g\otimes \id} Y_{ghg^{-1}}\otimes X_g\hookrightarrow Y\otimes X.\] This map is extended linearly to produce \(\beta^G_{(X,u), (Y,v)}\). The hexagon axioms follow from the definition of a \(G\)-crossed braiding. Finally, the \(G\)-crossed balance \(\theta\) on \(\mathcal{C}\) provides an honest balance on \(\mathcal{C}^G\). Let us write \(u_g = \bigoplus\limits_{h\in G} u_g^h\) with \(u_g^h: T_g(X_h)\xrightarrow{\cong } X_{ghg^{-1}}\) for all \(g,h\in G\). Then, we define the balance \(\theta^G\) on \((X,u)\) to be the direct sum of the maps \[X_g\xrightarrow{\theta_{X_g}} T_g(X_g)\xrightarrow{u_{g}^g} X_g\hookrightarrow X\] for all \(g\in G\). Diagram 3 implies that \({\theta}^G\) is a balance on the braided \(\mathrm{W}^*\)-tensor category \((\mathcal{C}^G,{\beta}^G)\) as follows. The balance condition on \((X,u)\) and \((Y, v)\) reads \[\label{eq:32BalanceEquivariantization} \theta^G_{(X,u)\otimes(Y,v)} = (\theta_{(X,u)}^G\otimes \theta^G_{(Y,v)})\circ \beta^G_{(Y,v), (X,u)}\circ \beta^G_{(X,u), (Y, v)}.\tag{4}\] Given \(g,h\in G\), the equality 4 on the component \(X_g\otimes Y_h\) of \(X\otimes Y\) is equivalent to the commutativity of the outer diagram of

\[\begin{tikzcd} {X_g\otimes Y_h} & {T_{gh}(X_g\otimes Y_h)} & {T_{gh}(X_g)\otimes T_{gh}(Y_h)} \\ {T_g(Y_h)\otimes X_g} & {T_{ghg^{-1}}(X_g)\otimes T_g(Y_h)} \\ {Y_{ghg^{-1}}\otimes X_g} && {X_{ghgh^{-1}g^{-1}}\otimes Y_{ghg^{-1}}} \\ {T_{ghg^{-1}}(X_g)\otimes Y_{ghg^{-1}}} \\ & {T_{ghgh^{-1}g^{-1}}T_{ghg^{-1}}(X_g)\otimes T_{ghg^{-1}}T_g(Y_h)} \\ {X_{ghgh^{-1}g^{-1}}\otimes Y_{ghg^{-1}}} && {T_{ghgh^{-1}g^{-1}}(X_{ghgh^{-1}g^{-1}})\otimes T_{ghg^{-1}}(Y_{ghg^{-1}})}. \arrow["{\theta_{X_g\otimes Y_h}}", from=1-1, to=1-2] \arrow["{\beta_{X_g, Y_h}}"', from=1-1, to=2-1] \arrow["\cong", from=1-2, to=1-3] \arrow["{u_{gh}\otimes v_{gh}}", from=1-3, to=3-3] \arrow["{\beta_{T_g(Y_h), X_g}}", from=2-1, to=2-2] \arrow["{v_g^h\otimes\id}"', from=2-1, to=3-1] \arrow["{\id\otimes v_{g}^h}"', from=2-2, to=4-1] \arrow["{\theta_{T_{ghg^{-1}}(X_g)}\otimes\theta_{T_g(Y_h)}}"{pos=0.2}, from=2-2, to=5-2] \arrow["{u^{g}_{ghg^{-1}}\otimes v^h_g}"{pos=0.4}, from=2-2, to=6-1] \arrow["{\beta_{Y_{ghg^{-1}}, X_g}}"', from=3-1, to=4-1] \arrow["{u^g_{ghg^{-1}}\otimes\id}"', from=4-1, to=6-1] \arrow["\cong"', from=5-2, to=1-3] \arrow["\begin{array}{c} T_{ghgh^{-1}g^{-1}}(u^g_{ghg^{-1}})\otimes\\ T_{ghg^{-1}}(v^h_g) \end{array}", shift right, from=5-2, to=6-3] \arrow["{\theta_{X_{ghgh^{-1}g^{-1}}}\otimes \theta_{Y_{ghg^{-1}}}}"', from=6-1, to=6-3] \arrow["{u^{ghgh^{-1}g^{-1}}_{ghgh^{-1}g^{-1}}\otimes v_{ghg^{-1}}^{ghg^{-1}}}"', from=6-3, to=3-3] \end{tikzcd}\] The top diagram commutes by definition of a \(G\)-crossed balance. The uppermost left triangle commutes by naturality of the \(G\)-crossed braiding, and the one immediately below commutes trivially. The bottom diagram commutes by naturality of the \(G\)-crossed balance. The right-most inner diagram commutes by hypothesis on the structure morphisms \(u\) and \(v\). Hence, the outer diagram commutes and, since \(g,h\in G\) were arbitrary, \(\theta^G\) is indeed a balance on the braided tensor category \(\mathcal{C}^G\).

Given two \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories \(\Ccal\) and \(\Dcal\), it is straightforward to check that a \(G\)-crossed braided \(\mathrm{W}^*\)-functor from \(\Ccal\) to \(\Dcal\) defines a braided \(\mathrm{W}^*\)-tensor functor between the equivariantizations \(\Ccal^G\) and \(\Dcal^G\). If \(\Ccal\) and \(\Dcal\) are furthermore \(G\)-crossed balanced, and the functor is compatible with the crossed balance, then the induced functor \(\Ccal^G\to \Dcal^G\) is compatible with the balances.

2.3 The \(G\)-crossed balanced category of representations of a conformal net↩︎

Let \(G\) be a discrete group acting on a conformal net \(\A\) by a group homomorphism \(G\to \Aut(\A)\). Throughout, we identify an element \(g\in G\) with the automorphism of \(\A\) it induces. In this section, we recall from [1] the construction of the \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor structure of \[\Rep^G(\A):=\bigoplus\limits_{g\in G}\Rep^g(\A).\] Actually, we shall produce an \(\Aut(\A)\)-crossed balanced \(\mathrm{W}^*\)-tensor structure on \(\Rep^{\Aut(\A)}(\A)\). The \(G\)-crossed braided \(\mathrm{W}^*\)-tensor structure on \(\Rep^G(\A)\) can be obtained by pulling the latter back along \(\Phi: G\to \Aut(\A)\).

2.3.1 Connes fusion of twisted representations↩︎

Recall that we write \(q:\R\to S^1\), \(t\mapsto e^{2\pi i t}\) for the exponential map, and we denote by \(\Jcal_\R\) the collection of connected open intervals in \(\R\) whose image under \(q\) is an interval of \(S^1\). Given \(\tilde{I}\in \Jcal_\R\), we write \(\tilde{I}^{c+} := (\partial_+\tilde{I}, \partial_-\tilde{I} + 1)\) and \(\tilde{I}^{c-}:= (\partial_+\tilde{I} - 1, \partial_-\tilde{I})\) for the positive and the negative complements of \(\tilde{I}\) respectively, which are intervals in \(\Jcal_\R\). Note that \(q(\tilde{I}^{c\pm}) = I^{c}\). Fix an automorphism \(\varphi\in \Aut(\A)\) and let \(H^\varphi = (H^\varphi, \pi^H)\in\Rep^\varphi(\A)\) be a \(\varphi\)-twisted representation. We denote by \(\Hom_{\A_\R(\tilde{I})}(H_0, H^\varphi)\) the space of bounded linear operators \(T:H_0\to H^\varphi\) intertwining the actions of \(\A_\R(\tilde{I})\). Since \(\A_\R(\tilde{I})\) is a type \(\mathrm{III}\) factor, there are unitary operators in \(\Hom_{\A_\R(\tilde{I})}(H_0, H^\varphi)\).

We say that a vector \(\xi\in H^\varphi\) is \((\tilde{I}, +)\)-bounded if there exists \(T\in \Hom_{\A_\R(\tilde{I}^{c+})}(H_0, H^\varphi)\) such that \(\xi = T\Omega\). We write \(H_+^\varphi(\tilde{I})\) for the subspace of \(H^\varphi\) of \((\tilde{I},+)\)-bounded vectors. Given \(\xi\in H_+^\varphi(\tilde{I})\), there exists a unique operator \(T\in \Hom_{\A_\R(\tilde{I}^{c+})}(H_0, H^\varphi)\) such that \(\xi = T\Omega\), as \(\A(I^{c})\Omega\) is dense in \(H_0\) by the Reeh-Schlieder Theorem. We denote this operator by \(T = Z^+(\xi, \tilde{I})\). We define similarly the space of \((\tilde{I},-)\)-bounded vectors \(H^\varphi_-(\tilde{I})\), using the negative complement \(\tilde{I}^{c-}\) of \(\tilde{I}\), and \(Z^-(\xi, \tilde{I}): H_0\to H^\varphi\) for every \(\xi\in H^\varphi_-(\tilde{I})\).

For an untwisted representation \(K\in \Rep(\A)\), there is no distinction between \((\tilde{I},+)\)- and \((\tilde{I}, -)\)-bounded vectors of \(K\), and these furthermore only depend on \(I\). Also, given \(\xi\in K_{\pm}(\tilde{I})\), we have that \(Z^+(\xi, \tilde{I}) = Z^-(\xi,\tilde{I})\), and these also only depend on \(I\). We may therefore write \(K_{\pm}(\tilde{I})\) as \(K(I)\). In particular, for the vacuum representation we have that \(H_0(I) = \A(I)\Omega\), by Haag duality. This implies that \(H_0(I)\subset H_0\) is dense for all \(I\in\Jcal\). Since there exist unitary operators in \(\Hom_{\A_\R(\tilde{I}^{c\pm})}(H_0, H^\varphi)\), we obtain that \(H_\pm^\varphi(\tilde{I})\subset H^\varphi\) are dense inclusions for all \(\tilde{I}\in\Jcal_\R\). In addition, if \(\tilde{I_1}\subset \tilde{I}\) is an inclusion of intervals in \(\Jcal_\R\), we have inclusions \(H_\pm^\varphi(\tilde{I_1})\subset H_\pm^\varphi(\tilde{I})\), which are also dense.

Let \(\mu\in \Aut(\A)\) be another automorphism and \(K^\mu = (K^\mu, \pi^K)\in \Rep^\mu(\A)\) be a \(\mu\)-twisted representation. Fix intervals \(\tilde{I}, \tilde{J}\in\Jcal_\R\) satisfying \(\tilde{J}\subset\tilde{I}^{c+}\).

Note that \(Z^+(\xi', \tilde{I})^*Z^+(\xi, \tilde{I})\in \Hom_{\A(I^c)}(H_0, H_0) \cong \A(I) = \A_\R(\tilde{I})\) and \(Z^-(\eta', \tilde{J})^*Z^-(\eta, \tilde{J})\in \Hom_{\A(J^c)}(H_0, H_0) \cong \A(J) = \A_\R(\tilde{J})\), hence the inner product ?? can be expressed in the following equivalent ways \[\begin{align} \label{eq:32DiffConnesFroms} \langle \xi\otimes\eta|\xi'\otimes\eta'\rangle&=\langle Z^-(\eta', \tilde{J})^*Z^-(\eta, \tilde{J})Z^+(\xi', \tilde{I})^*Z^+(\xi, \tilde{I})\Omega, \Omega\rangle\\\nonumber&=\langle Z^+(\xi', \tilde{I})^*Z^+(\xi, \tilde{I})Z^-(\eta', \tilde{J})^*Z^-(\eta, \tilde{J})\Omega|\Omega\rangle\\\nonumber & = \langle \pi^K_{\tilde{I}}(Z^+(\xi', \tilde{I})^*Z^+(\xi, \tilde{I}))\eta|\eta'\rangle\\\nonumber & =\langle \pi^H_{\tilde{J}}(Z^-(\eta', \tilde{J})^*Z^-(\eta, \tilde{J}))\xi|\xi'\rangle. \end{align}\tag{5}\]

Given twisted representations \(\hat{H}^\varphi\in \Rep^\varphi(\A)\) and \(\hat{K}^\mu\in \Rep^\mu(\A)\), and morphisms \(F: H^\varphi\to \hat{H}^\varphi\) and \(G: K^\mu\to \hat{K}^\mu\), the map \(F\otimes G: H^\varphi_+(\tilde{I})\otimes K^\mu_-(\tilde{J})\to \hat{H}^\varphi_+(\tilde{I})\otimes \hat{K}^\mu_-(\tilde{J})\) induces a bounded linear map \(F\boxtimes G: H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J})\to \hat{H}^\varphi_+(\tilde{I})\boxtimes \hat{K}^\mu_-(\tilde{J})\), which we call the Connes fusion of \(F\) and \(G\).

Given \(\tilde{I_1},\tilde{J_1}\in\Jcal_\R\) such that \(\tilde{J_1}\subset\tilde{I_1}^{c+}\) and \(\tilde{I_1}\subset\tilde{I}\), \(\tilde{J_1}\subset \tilde{J}\), there is a canonical equivalence \[\label{eq:32CanonicalEquivalenceInclusion} H^\varphi_+(\tilde{I_1})\boxtimes K_-^\mu(\tilde{J_1})\xrightarrow{\cong} H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\tag{6}\] induced by the inclusion \(H_+^\varphi(\tilde{I_1})\otimes K_-^\mu(\tilde{J_1})\hookrightarrow H_+^\varphi(\tilde{I})\otimes K_-^\mu(\tilde{J})\).

Now, given \(z, \zeta\in \R\) such that \(\zeta\in (z,z+1)\), we define \[H_+^\varphi(z)\boxtimes K_-^\mu(\zeta):=\varinjlim\limits_{(z,\zeta)\in\tilde{I}\times \tilde{J}}H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J}) = \Bigg(\bigsqcup\limits_{(z,\zeta)\in\tilde{I}\times \tilde{J}}H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\Bigg)\Big/\cong,\] where the limit runs over pairs \(\tilde{I}, \tilde{J}\in \Jcal_\R\) containing \(z\) and \(\zeta\) respectively and satisfying \(\tilde{J}\subset\tilde{I}^{c+}\), and \(\cong\) denotes the equivalence obtained by identifying \(H^\varphi_+(\tilde{I_1})\boxtimes K^\mu_-(\tilde{J_1})\) with \(H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J})\) by the unitary 6 whenever \(\tilde{I_1}\subset \tilde{I}\) and \(\tilde{J_1}\subset \tilde{J}\). Given intervals \(\tilde{I}, \tilde{J}\in \Jcal_\R\) containing \(z\) and \(\zeta\) respectively and such that \(\tilde{J}\subset\tilde{I}^{c+}\), the canonical map \[H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\to H_+^\varphi(z)\boxtimes K_-^\mu(\zeta)\] is a unitary equivalence.

We next relate the Connes fusion over different pairs of intervals. Consider the topological space \((\R\times\R)\setminus (\R\times_{S^1}\R) = \{(z,\zeta)\in \R\times\R\;|\; qz\neq q\zeta\}\), which is a disconnected space with contractible connected components. We write \(\ConfR\) for the connected component of \((\R\times\R)\setminus (\R\times_{S^1}\R)\) given by the points \((z,\zeta)\) with \(\zeta\in (z,z+1).\) Then, \(\ConfR\) is the universal cover of \(\text{Conf}_2(S^1)\), the space of configurations of two points in \(S^1\). Let \(\gamma = (\alpha, \beta):[0,1]\to \ConfR\) be a path in \(\ConfR\) from \(\gamma(0) =: (z_0, \zeta_0)\) to \(\gamma(1) =: (z_1, \zeta_1)\). Assume that \(\gamma\) is small enough so that \(\gamma([0,1])\subset \tilde{I}\times \tilde{J}\), for some \(\tilde{I}, \tilde{J}\in \Jcal_\R\) satisfying \(\tilde{J}\subset \tilde{I}^{c+}\). We define the unitary equivalence \(\gamma^\bullet: H_+^\varphi(z_0)\boxtimes K_-^\mu(\zeta_0)\xrightarrow{\cong}H_+^\varphi(z_1)\boxtimes K_-^\mu(\zeta_1)\) as the composition \[H_+^\varphi(z_0)\boxtimes K_-^\mu(\zeta_0)\xrightarrow{\cong}H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\xrightarrow{\cong}H_+^\varphi(z_1)\boxtimes K_-^\mu(\zeta_1).\] For a general path \(\gamma\), we choose \(0 = t_0<t_1<\ldots <t_k = 1\) such that each \(\gamma_n:=\gamma|_{[t_n, t_{n+1}]}\) is small enough in the sense above, and define \(\gamma^\bullet: H_+^\varphi(z_0)\boxtimes K_-^\mu(\zeta_0)\xrightarrow{\cong}H_+^\varphi(z_1)\boxtimes K_-^\mu(\zeta_1)\) as \[\gamma^\bullet = \gamma_{k-1}^\bullet\ldots\gamma_1^\bullet\gamma_0^\bullet.\] Since finer partitions give the same result, the map \(\gamma^\bullet\) is independent of the choice of partition. We call \(\gamma^\bullet\) the path-continuation induced by \(\gamma\).

Let \(\tilde{I_0}, \tilde{I_1}, \tilde{J_0}, \tilde{J_1}\in \Jcal_\R\) be intervals such that \(\tilde{J_0}\subset\tilde{I_0}^{c+}\) and \(\tilde{J_1}\subset\tilde{I_1}^{c+}\). Let \(\gamma\) be a path in \(\ConfR\) from \(\gamma(0) =: (z_0, \zeta_0)\in \tilde{I_0}\times\tilde{J_0}\) to \(\gamma(1) =: (z_1, \zeta_1)\in \tilde{I_1}\times \tilde{J_1}\). We define the path continuation \(\gamma^\bullet : H_+^\varphi(\tilde{I_0})\boxtimes K_-^\mu(\tilde{J_0})\xrightarrow{\cong} H_+^\varphi(\tilde{I_1})\boxtimes K_-^\mu(\tilde{J_1})\) as the composition \[H_+^\varphi(\tilde{I_0})\boxtimes K_-^\mu(\tilde{J_0})\xrightarrow{\cong}H_+^\varphi(z_0)\boxtimes K_-^\mu(\zeta_0)\xrightarrow{\gamma^\bullet} H_+^\varphi(z_1)\boxtimes K_-^\mu(\zeta_1)\xrightarrow{\cong}H_+^\varphi(\tilde{I_1})\boxtimes K_-^\mu(\tilde{J_1}).\] Homotopic paths produce the same path continuation.

We next introduce Connes fusion over a single interval. By the equalities in Equation 5 , the Connes fusion \(H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J})\) can also be obtained as a completion of the vector space \(H^\varphi_+(\tilde{I})\otimes K^\mu\) with respect to \[\langle \xi\otimes\eta\,|\,\xi'\otimes \eta'\rangle := \langle\pi^K_{\tilde{I}}\big(Z^+(\xi', \tilde{I})^*Z^+(\xi, \tilde{I})\big)(\eta)\,|\, \eta'\rangle\] for \(\xi, \xi'\in H_+^\varphi(\tilde{I})\) and \(\eta, \eta'\in K^\mu\). We call this completion the Connes fusion of \(H^\varphi\) and \(K^\mu\) over \(\tilde{I}\) on the left and denote it \(H^\varphi_+(\tilde{I})\boxtimes K^\mu\). Then, if \(\tilde{J}\in \Jcal_\R\) is an interval such that \(\tilde{J}\subset \tilde{I}^{c+}\), the inclusion \(H_+^\varphi(\tilde{I})\otimes K_-^\mu(\tilde{J})\hookrightarrow H^\varphi_+(\tilde{I})\otimes K^\mu\) induces a canonical equivalence \(H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\xrightarrow{\cong} H_+^\varphi(\tilde{I})\boxtimes K^\mu\). For any \(z\in \R\), we define \(H^\varphi_+(z)\boxtimes K^\mu := \varinjlim\limits_{z\in\tilde{I}}\, H^\varphi_+(\tilde{I})\boxtimes K^\mu\). Given a path \(\alpha\) in \(\R\) from \(z_0\) to \(z_1\), we also obtain a path continuation \(\alpha^\bullet: H^\varphi_+(z_0)\boxtimes K^\mu\xrightarrow{\cong} H^\varphi_+(z_1)\boxtimes K^\mu\) as follows. If \(\alpha\) is small enough so that \(\alpha([0,1])\) can be covered by an interval \(\tilde{I}\in \Jcal_\R\), we define \[\alpha^\bullet: H^\varphi_+(z_0)\boxtimes K^\mu\xrightarrow{\cong} H^\varphi_+(\tilde{I})\boxtimes K^\mu\xrightarrow{\cong} H^\varphi_+(z_1)\boxtimes K^\mu.\] For a generic path \(\alpha\), we pick a partition \(0 = t_0<t_1<\ldots<t_k = 1\) of \([0,1]\) so that each \(\alpha_n:=\alpha|_{[t_n, t_{n+1}]}\) is small enough in the sense above, and define \(\alpha^\bullet = \alpha_{k-1}^\bullet\ldots\alpha_0^\bullet\). Given two intervals \(\tilde{I_0},\tilde{I_1}\in \Jcal_{\R}\) and a path \(\alpha\) in \(\R\) from a point in \(\tilde{I_0}\) to a point in \(\tilde{I_1}\), we define \[\alpha^\bullet:H^\varphi_+(\tilde{I_0})\boxtimes K^\mu\xrightarrow{\cong } H_+^\varphi(\alpha(0))\boxtimes K^\mu\xrightarrow{\alpha^\bullet} H^\varphi_+(\alpha(1))\boxtimes K^\mu\xrightarrow{\cong} H^\varphi_+(\tilde{I_1})\boxtimes K^\mu.\] Homotopic paths induce the same unitary equivalence. Similar properties hold for \(H^\varphi\boxtimes K^\mu_-(\tilde{J})\), the Connes fusion of \(H^\varphi\) and \(K^\mu\) over \(\tilde{J}\) on the right, defined in the obvious analogous way.

2.3.2 Endowing the Connes fusion with a twisted action↩︎

From now on, we fix intervals \(\tilde{I},\tilde{J}\in \Jcal_\R\) such that \(\tilde{J}\subset \tilde{I}^{c+}\), as well as automorphisms \(\varphi,\mu\in\Aut(\A)\) and twisted representations \(H^\varphi = (H^\varphi, \pi^H)\in \Rep^\varphi(\A)\) and \(K^\mu = (K^\mu, \pi^K)\in \Rep^\mu(\A)\). We recall how to endow the Hilbert space \(H^\varphi(\tilde{I})\boxtimes K^\mu(\tilde{J})\) with a \((\varphi\circ\mu)\)-twisted action of \(\A\). We will do this by endowing \(H^\varphi(\tilde{I})\boxtimes K^\mu(\tilde{J})\) with compatible actions \(\pi^{H\boxtimes K}_{\tilde{L}}\) of the \(*\)-algebras \(\A_\R(\tilde{L})\) such that \(\pi^{H\boxtimes K}_{\tilde{L}} \circ (\varphi\circ\mu) = \pi^{H\boxtimes K}_{\tilde{L}+1}\) for all \(\tilde{L}\in \Jcal_\R\). Note that we have natural actions of \(\A_\R(\tilde{I} )\) and \(\A_\R(\tilde{J})\) on \(H^\varphi(\tilde{I})\boxtimes K^\mu(\tilde{J})\). Indeed, given \(x\in \A(I) = \A_\R(\tilde{I})\) and \(y\in \A(J) = \A_\R(\tilde{J})\), we define \[\pi^{H \boxtimes K}_{\tilde{I}}(x)(\xi\otimes \eta) = \Big(\pi_{\tilde{I}}^H(x)(\xi)\Big)\otimes \eta \pi^{H\boxtimes K}_{\tilde{J} }(y)(\xi\otimes\eta) = \xi\otimes \Big(\pi_{\tilde{J}}^K(y)(\eta)\Big)\] for \(\xi\otimes\eta \in H_+^\varphi(\tilde{I})\otimes K_-^\mu(\tilde{J})\subset H_+^\varphi(\tilde{I})\boxtimes K_-^\mu(\tilde{J})\). This also provides canonical actions of all \(\A_\R(\tilde{L})\) for \(\tilde{L}\subset \tilde{I}\) or \(\tilde{L}\subset\tilde{J}\).

The Connes fusion of two morphisms of twisted \(\A\)-representations is again a morphism of twisted \(\A\)-representations.

We can define similarly twisted actions of \(\A\) on fusions over single intervals. Fix \(\tilde{I}\in \Jcal_\R\). If \(\tilde{L}\in \Jcal_\R\) is a subinterval \(\tilde{L}\subset \tilde{I}\), we define the action of \(x\in\A_\R(\tilde{L})\) on \(H_+^\varphi(\tilde{I})\boxtimes K^\mu\) by \(\pi_{\tilde{L}}^{H\boxtimes K}(\xi\otimes \eta) = \pi_{\tilde{L}}^H(x)(\xi)\otimes \eta\) for all \(\xi\in H_+^\varphi(\tilde{I})\) and \(\eta\in K^\mu\). For a generic interval \(\tilde{L}\in\Jcal_\R\), we choose a path \(\alpha\) in \(\R\) from \(\tilde{I}\) to \(\tilde{L}\), and define \(\pi_{\tilde{L}}^{ H\boxtimes K}(x) = (\alpha^\bullet)^{-1}\circ \pi^{H\boxtimes K}_{\tilde{L}}(x)\circ \alpha^\bullet\). This is independent of the path chosen. Alternatively, one can define the action of \(\A_\R\) on \(H^\varphi_+(\tilde{I})\boxtimes K^\mu\) via the canonical equivalence \(H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J})\cong H^\varphi_+(\tilde{I})\boxtimes K^\mu\) for some \(\tilde{J}\in \Jcal_\R\) with \(\tilde{J}\subset\tilde{I}^{c+}\) and the action of \(\A_\R\) on \(H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J})\) from Theorem 1. These two actions agree.

2.3.3 The crossed balanced \(\mathrm{W}^*\)-structure↩︎

In this section, we recall how to equip the category \[\Rep^{\Aut(\A)}(\A):= \bigoplus\limits_{\varphi\in \Aut(\A)}\Rep^\varphi(\A)\] with the structure of an \(\Aut(\A)\)-crossed balanced \(\mathrm{W}^*\)-tensor category.

Let \(\widetilde{S_-^1}:=(-\frac{1}{2}, 0)\) and \(\widetilde{S_+^1}:=(0,\frac{1}{2})\). Given automorphisms \(\varphi,\mu\in \Aut(A)\), for any twisted representations \(H^\varphi\in \Rep^\varphi(\A)\) and \(K^\mu\in \Rep^\mu(\A)\), we define their tensor product as \[H^\varphi\boxtimes K^\mu:=H^\varphi_+(\widetilde{S^1_-})\boxtimes K^\mu_-(\widetilde{S^1_+}).\] We also identify \(H^\varphi\boxtimes K^\mu\) with \(H_+^\varphi(\widetilde{S^1_-})\boxtimes K^\mu\) and \(H^\varphi\boxtimes K_-^\mu(\widetilde{S^1_+})\) using the canonical equivalences from the previous section. We extend this tensor product linearly to obtain a functor \[\boxtimes: \Rep^{\Aut(\A)}(\A)\times \Rep^{\Aut(\A)}(\A)\to \Rep^{\Aut(\A)}(\A).\] Let us denote \(\widetilde{S^1_-}\) and \(\widetilde{S^1_+}\) by \(-\) and \(+\) in the Connes fusions. Given three automorphisms \(\varphi,\mu,\nu\in \Aut(\A)\) and representations \(H^\varphi,K^\mu,R^\nu\) of \(\A\) twisted by \(\varphi,\mu\) and \(\nu\) respectively, the map \((H^\varphi\otimes K^\mu)\otimes R^\nu\xrightarrow{\cong}H^\varphi\otimes (K^\mu\otimes R^\nu)\) given by \((\xi\otimes\eta)\otimes \chi\mapsto \xi\otimes(\eta\otimes\chi)\) induces a unitary equivalence of twisted \(\A\)-representations \((H^\varphi\boxtimes K^\mu)\boxtimes R^\nu\xrightarrow{\cong}H^\varphi\boxtimes (K^\mu\boxtimes R^\nu)\) which produces an associator for \(\Rep^G(\A)\). The unit object is \(H_0\), for the unitors see [1].

The action of \(\Aut(\A)\) on \(\Rep^G(\A)\) is given as follows. Given \(\varphi,\nu\in \Aut(\A)\) and a twisted representation \(H^\varphi\in\Rep^\varphi(\A)\), we define \(T_\nu(H^\varphi)\) to be the \((\nu\varphi\nu^{-1})\)-twisted representation of \(\A\) on the underlying Hilbert space \(H\) of \(H^\varphi\) given by \[\pi^{\nu\ast H}_{\tilde{I}}(x)(\xi) = \pi_{\tilde{I}}^H(\nu^{-1}x)(\xi)\] for all \(\tilde{I}\in\Jcal_\R\), \(x\in\A_\R(\tilde{I})\) and \(\xi\in H\). On morphisms, the action of \(\nu\) is defined to be trivial. Let us write \(\Gamma_\nu: H^\varphi\to T_\nu(H^\varphi)\) for the identity map of Hilbert spaces, which is not a morphism of twisted \(\A\)-representations. It is easy to see that \(\Gamma_\nu\) induces equivalences \(H^\varphi_-(\tilde{I}) \cong T_\nu(H^\varphi)_-(\tilde{I})\) and \(H^\varphi_+(\tilde{I}) \cong T_\nu(H^\varphi)_+(\tilde{I})\). In addition, given an automorphism \(\mu\in \Aut(\A)\), a twisted representation \(K^\mu\in\Rep^\mu(\A)\) and an interval \(\tilde{J}\in\Jcal_\R\) with \(\tilde{J}\subset\tilde{I}^{c+}\), the map \(H^\varphi_+(\tilde{I})\otimes K^\mu_-(\tilde{J})\to T_\nu(H^\varphi)_+(\tilde{I})\otimes T_\nu(K^\mu)_-(\tilde{J})\) given by \(\xi\otimes\eta\to \Gamma_\nu\xi\otimes\Gamma_\nu\eta\) induces a unitary equivalence of twisted \(\A\)-representations \[\label{eq:32MonoidalT} T_\nu(H^\varphi_+(\tilde{I})\boxtimes K^\mu_-(\tilde{J}))\cong T_\nu(H^\varphi)_+(\tilde{I})\boxtimes T_\nu(K^\mu)_-(\tilde{J}).\tag{7}\]

The functors \(T_\nu\) for \(\nu\in\Aut(\A)\) provide an action \[T: \underline{\Aut(\A)}\to \underline{\Aut}_\otimes(\Rep^{\Aut(\A)}(\A)).\] Given \(\nu\in \Aut(\A)\), the monoidal structure of \(T_\nu\) is given by \[T_\nu(H^\varphi)\boxtimes T_\nu(K^\mu) = T_\nu\big(H^\varphi)_+(-)\boxtimes T_\nu\big(K^\mu\big)_-(+)\cong T_\nu\big(H^\varphi_+(-)\boxtimes K^\mu_-(+)\big) = T_\nu(H^\varphi\boxtimes K^\mu),\] using the unitary equivalence from Equation 7 . The monoidal structure of the functor \(T\) is given, on \(H^\varphi\in\Rep^\varphi(\A)\), by the identity map \(T_\mu\circ T_\nu(H^\varphi) = T_{\mu\nu}(H^\varphi).\)

We define next the \(\Aut(\A)\)-crossed braiding on \(\Rep^{\Aut(\A)}(\A)\). Let \(\varrho:[0,1]\to \R\) be the path \(\varrho(t) = -\frac{1}{4} + \frac{t}{2}\), which goes from \(\widetilde{S_-^1}\) to \(\widetilde{S_+^1}\). Given twisted representations \(H^\varphi\in\Rep^\varphi(\A)\) and \(K^\mu\in \Rep^\mu(\A)\), we define the crossed-braid operator \(\B_{H,K}: H^\varphi\boxtimes K^\mu\xrightarrow{\cong} T_\varphi(K^\mu)\boxtimes H^\varphi\) by \[\B_{H, K}: H^\varphi\boxtimes K^\mu = H^\varphi_+(-)\boxtimes K^\mu\xrightarrow{\varrho^\bullet} H^\varphi_+(+)\boxtimes K^\mu\cong T_\varphi(K^\mu)\boxtimes H^\varphi_-(+) = T_\varphi(K^\mu)\boxtimes H^\varphi,\] where the equivalence in the middle is induced by the map \(H^\varphi_+(+)\otimes K^\mu\cong T_\varphi(K^\mu)\otimes H^\varphi_-(+)\) given by \(\xi\otimes\eta\mapsto \Gamma_\varphi\eta\otimes \xi\), see [1].

Let us finally describe the \(\Aut(\A)\)-crossed balance on \(\Rep^{\Aut(\A)}(\A)\). By [1], any twisted representation \(H^\varphi\in\Rep^\varphi(\A)\) has an honest strongly continuous action of the universal cover \(\widetilde{\Mob}\) of the Möbius group \(\Mob\), constructed from the unitary projective action of \(\Diff^+(S^1)\) on \(H_0\). The group \(\widetilde{\Mob}\) sits inside the universal cover \(\widetilde{\Diff^+}(S^1)\) of \(\Diff^+(S^1)\), which is an infinite dimensional Lie group whose Lie algebra is the Lie algebra \(\text{Vec}(S^1)\) of vector fields on \(S^1\) with the negative of the Lie bracket of vector fields. Let \(\text{Vec}_\mathbb{C}(S^1)\) denote the complexification of \(\text{Vec}(S^1)\) and write \(\mathscr{W}\subset \text{Vec}_\mathbb{C}(S^1)\) for the Witt algebra, which is generated by the complex vector fields \(L_n(e^{i\theta}):=-e^{in\theta}\frac{d}{d\theta}\) for all \(n\in \Z\). The group \(\widetilde{\Mob}\) is generated by the exponential of vectors of the form \(i(\overline{a_1}L_{-1}+a_0L_0+a_1L_1)\in\text{Vec}(S^1)\) for \(a_0\in\R\) and \(a_1\in \C\). We write \(\exp(iX)\in \widetilde{\Mob}\) for the exponential of \(X = \overline{a_1}L_{-1}+a_0L_0+a_1L_1\in \text{Vec}_{\mathbb{C}}(S^1)\), and denote the action of \(\exp(iX)\) on \(H^\varphi\) by \(e^{iX}\). Then, the balance on \(H^\varphi\) is defined to be \[\theta_H:=e^{-2\pi i L_0}: H^\varphi\to T_\varphi(H^\varphi).\] The fact that \(\theta_H\) is a well-defined unitary isomorphism of twisted \(\A\)-representations is the content of [1].

If we have a discrete group \(G\) acting on \(\A\) by a group homomorphism \(\Phi:G\to \Aut(\A)\), we can pull back the \(\Aut(\A)-\)crossed balanced structure on \(\Rep^{\Aut(\A)}(\A)\) along \(\Phi\) to obtain a \(G\)-crossed balanced structure on \(\Rep^G(\A)\).

The \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category \(\Rep^G(\A)\) can be equivalently obtained via localized endomorphisms, as introduced in [2]. We recall Müger’s construction here and its relationship to \(\Rep^G(\A)\) as it will be used in Section 4. Recall that we have fixed the point \(\mathrm{p} = 1\in S^1\). We write \(\mathcal{J}_\mathrm{p} : =\{I\in\Jcal\,|\,\mathrm{p}\notin\text{cl}(I)\}\) for the poset of intervals in \(S^1\) not containing \(\mathrm{p}\) in their closure, and \(\A_\infty = \bigcup\limits_{I\in\Jcal_\mathrm{p}}\A(I)\subset B(H_0)\) for the sub \(*\)-algebra of \(B(H_0)\) which is the union of all the algebras \(\A(I)\) with \(I\in \Jcal_\mathrm{p}\). We define \(\End\,\A_\infty\) as the \(\mathbb{C}\)-linear category whose objects are endomorphisms of the \(*\)-algebra \(\A_\infty\) and such that \(\Hom_{\End\,\A_\infty}(\rho, \sigma) =\{s\in \A_\infty\,|\, s\rho(x) = \sigma(x)s\,\text{ for all x\in \A_\infty}\}\) for \(\rho, \sigma\in \End\,\A_\infty\). Composition is given by multiplication in \(\A_\infty\). Furthermore, \(\End\,\A_\infty\) is a strict tensor category when equipped with the tensor product \(\rho\otimes\sigma = \rho\circ\sigma\) and \(s\otimes t:=s\rho(t) = \rho'(t)s\), where \(\rho,\rho', \sigma, \sigma'\in \End\,\A_\infty\) and \(s\in\Hom_{\End\,\A_\infty}(\rho, \rho')\) and \(t\in \Hom_{\End\,\A_\infty}(\sigma, \sigma')\).

Given an interval \(I\in \Jcal_\mathrm{p}\) and an element \(g\in G\), we say that an endomorphism \(\rho\in\End\,\A_\infty\) is \(\rho\)-localized in \(I\) if \(\rho(x) = x\) for all \(x\in \A(J)\) with \(J\in \Jcal_\mathrm{p}\) an interval clockwise to \(I\) and \(\rho(x) = gx\) for all \(x\in \A(J)\) with \(J\in \Jcal_\mathrm{p}\) an interval counter-clockwise to \(I\). Such an endomorphism \(\rho\) is called transportable if, for every interval \(J\in\Jcal_\mathrm{p}\), there is an endomorphism \(\rho'\), unitarily isomorphic to \(\rho\), which is \(g\)-localized in \(J\). An endomorphism is called \(g\)-localized if it is \(g\)-localized in \(I\) for some \(I\in\Jcal_\mathrm{p}.\) The subcategory of \(\End\,\A_\infty\) on the \(g\)-localized transportable endomorphisms of \(\A\) is denoted \(g-\Loc(\A)\), and the subcategory of the \(g\)-localized-in-\(I\) transportable endomorphisms of \(\A\) is denoted \(g-\Loc_I(\A)\), for any \(I\in \Jcal_\mathrm{p}\). The inclusion \(g-\Loc_I(\A)\subset g-\Loc(\A)\) is an equivalence of categories. Similarly, we write \(G-\Loc(\A)\) for the full subcategory of \(\End\,\A_\infty\) on the objects which are finite direct sums of \(g\)-localized, transportable endomorphisms of \(\A\), for possibly different elements \(g\in G\). That is, an endomorphism \(\rho\in\End\,\A_\infty\) is in \(G-\Loc(\A)\) if there exists a finite set \(\Delta\) and, for all \(i\in \Delta\), an element \(g_i\in G\), an endomorphism \(\rho_i\in \End\,\A_\infty\) which is \(g_i\)-localized and transportable, and \(v_i\in\Hom_{\End\,\A_\infty}(\rho_i, \rho)\) such that \(v_i^*v_j =\delta_{ij}\) and \[\rho(-) = \sum_{i\in\Delta} v_i\rho(-)v_i^*.\] Given \(I\in\Jcal_\mathrm{p}\), the category \(G-\Loc_I(\A)\) of finite sums of \(G\)-localized-in-\(I\) transportable endomorphisms of \(\A_\infty\) is defined analogously. The inclusion \(G-\Loc_I(\A)\subset G-\Loc(\A)\) is an equivalence of categories. In [2] it is shown that \(G-\Loc_I(\A)\) admits the structure of a \(G\)-crossed braided tensor category, and in [1] we showed that the \(G\)-crossed braided tensor category \(G-\Loc_I(\A)\) is equivalent to \(\Rep^G(\A)\). We recall both of these results here.

Fix an interval \(I\in\Jcal_\mathrm{p}.\) Then, the category \(G-\Loc_I(\A)\) is a tensor subcategory of \(\End\,\A_\infty\), since the tensor product induces morphisms \[g-\Loc_I(\A)\times h-\Loc_I(\A)\to gh-\Loc_I(\A)\] for all \(g,h\in G\). The tensor product turns \(G-\Loc_I(\A)\) into a \(\mathrm{W}^*\)-tensor category, as \(\End_{\End\,\A_\infty}(\rho)\) is a von Neumann algebra for any \(\rho\in G-\Loc_I(\A)\) and the tensor product is a \(\mathrm{W}^*\)-functor. In addition, the group \(G\) acts on \(G-\Loc_I(\A)\) by sending \(\rho\in G-\Loc_I(\A)\) to \(\gamma_g(\rho):=g\rho g^{-1}\) for all \(g\in G\). On morphisms, the element \(g\) acts on \(s\in\Hom_{\End\,\A_\infty}(\rho, \rho')\) by \(s\mapsto \gamma_g(s):=g(s)\). It is easy to see that \(g\to \gamma_g\) can be upgraded to an action of \(G\) on \(G-\Loc_I(\A)\) by tensor automorphisms which covers the conjugation action of \(G\) on itself. Finally, the \(G\)-crossed braiding is defined as follows. Let \(\rho,\sigma\in G-\Loc_I(\A)\) be endomorphisms with \(\rho\) \(g\)-localized in \(I_1\subset I\) and \(\sigma\) \(G\)-localized in \(I_2\subset I\), where \((\partial_-I_1, \partial_+I_2)\) can be covered by \(I\). Here, \(\partial_-\) and \(\partial_+\) are defined using the counter-clockwise orientation of \(S^1\). Then, it holds that \(\rho\otimes \sigma = \gamma_g(\sigma)\otimes \rho\). If \(\rho\) is \(g\)-localized in \(I\) and \(\sigma\) is \(G\)-localized in \(I\), we may find \(\rho'\) and \(\sigma'\) localized as above, and unitary isomorphisms \(u: \rho\cong \rho'\) and \(v: \sigma\cong \sigma'\), and define the crossed-braiding \(c_{\rho, \sigma}\) to be given by \[c_{\rho, \sigma}:\rho\otimes \sigma\xrightarrow{u\otimes v}\rho'\otimes \sigma' = \gamma_g(\sigma')\otimes \rho'\xrightarrow{\gamma_g(v^{-1})\otimes u^{-1}} \gamma_g(\sigma)\otimes \rho.\] This morphism is independent of the particular choice of \(\rho', \sigma', u\) and \(v\), and provides a \(G\)-crossed braiding on \(G-\Loc_I(\A)\). The following is the statement of [2] for the category \(G-\Loc_I(\A)\) as opposed to \(G-\Loc(\A)\).

The \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category \(G-\Loc_I(\A)\) is canonically equivalent to \(\Rep^G(\A)\), as it was discussed in [1]. The underlying functor \(\mathfrak{E}: G-\Loc_I(\A)\to \Rep^G(\A)\) of this equivalence is defined as follows. Given \(g\in G\) and \(\rho\in g-\Loc_I(\A)\), we define \(\mathfrak{E}(g)\) to be the \(g\)-twisted representation of \(\A\) on \(H_0\) given by \[\pi_I(x) = \pi_0(\rho(x))\] for all \(J\in\Jcal_\mathrm{p}\) and \(x\in \A(J)\). Since \(\rho\) is \(g\)-localized in \(I\), the formula above extends to a unique \(g\)-twisted representation of \(\A\) on \(H_0.\) We denote this representation by \((H_0, \pi_0\circ\rho)\). On morphisms, it holds by [2] that \(\Hom_{g-\Loc_I(\A)}(\rho, \sigma)\subset \A(I)\) for all \(\rho, \sigma\in g-\Loc_I(\A)\) and hence we may define \(\mathfrak{E}(s) = \pi_0(s)\in B(H_0)\) for all \(s\in \Hom_{g-\Loc_I(\A)}(\rho, \sigma)\). Then, \(\mathfrak{E}\) can be extended linearly to provide a \(\mathrm{W}^*\)-functor \[\mathfrak{E}: G-\Loc_I(\A)\to \Rep^G(\A).\] The following result follows from [1].

3 \(G\)-crossed categorical extensions of conformal nets↩︎

In this section, we extend the definition of a categorical extension of a conformal net \(\A\) from [13] to the context of a group \(G\) acting on \(\A\). We call these \(G\)-crossed categorical extensions of \(\A\). Actually, we will show that, up to equivalence, there is a unique \(G\)-crossed categorical extension of \(\A\), induced by the Connes fusion of representations and the braiding on \(\Rep^G(\A)\) described in Section [Sec:32CatOfTwistedReps]. This uniqueness result will be relevant in the next section when discussing the category of representations of the fixed points conformal net \(\A^G\) (see Proposition [prop:32MisEquivalenceOFG-X-BraidedCats]).

Before defining \(G\)-crossed categorical extensions, we need the following two pieces of notation. The first is the notion of adjoint commutativity. Let \(H_1, H_2, H_3, H_4\) be Hilbert spaces and let \(A: H_1\to H_2\), \(B:H_2\to H_4\), \(C: H_1\to H_3\) and \(D: H_3\to H_4\) be bounded linear maps. Then, we say that the diagram \[\label{eq:32Adjoint1}\begin{tikzcd} {H_1} & {H_3} \\ {H_2} & {H_4} \arrow["C", from=1-1, to=1-2] \arrow["A"', from=1-1, to=2-1] \arrow["D", from=1-2, to=2-2] \arrow["B"', from=2-1, to=2-2] \end{tikzcd}\tag{8}\] commutes adjointly if both the diagram and \[\label{eq:32Adjoint2}\begin{tikzcd} {H_1} & {H_3} \\ {H_2} & {H_4} \arrow["C", from=1-1, to=1-2] \arrow["{A^*}", from=2-1, to=1-1] \arrow["B"', from=2-1, to=2-2] \arrow["{D^*}"', from=2-2, to=1-2] \end{tikzcd}\tag{9}\] commute. Note that the commutativity of 9 is equivalent to the commutativity of \[\begin{tikzcd} {H_1} & {H_3} \\ {H_2} & {H_4}. \arrow["A"', from=1-1, to=2-1] \arrow["{C^*}"', from=1-2, to=1-1] \arrow["D", from=1-2, to=2-2] \arrow["{B^*}", from=2-2, to=2-1] \end{tikzcd}\] In addition, if either of \(A,B\) or \(C,D\) are unitary, the adjoint commutativity of 8 is equivalent to its commutativity in the usual sense.

Let \(\A\) be a conformal net. In [1], we introduced two families of operators, denoted \(L\) and \(R\), in order to prove the hexagon conditions for the crossed braiding on \(\Rep^{\Aut(\A)}(\A)\), analogously to [13]. We recall their definition here. Fix an automorphism \(\varphi\in\Aut(\A)\) and let \(H^\varphi\) be a \(\varphi\)-twisted representation of \(\A\) and \(\tilde{I}\in \Jcal_\R\) be an interval. For any \(K^\mu\in \Rep^\mu(\A)\) and \(\xi\in H_+^\varphi(\tilde{I})\), we denote by \(Z^+(\xi, \tilde{I})\) the bounded linear operator \(K^\mu\to H^\varphi_+(\tilde{I})\boxtimes K^\mu\) induced by \[\begin{array}{cccc} Z^+(\xi, \tilde{I}):& K^\mu &\to &H_+^\varphi(\tilde{I})\otimes K^\mu \\ & \eta&\mapsto &\xi\otimes \eta. \end{array}\] It is clear that \(Z^+(\xi, \tilde{I})\) commutes with the actions of \(\A_\R(\tilde{I}^{c+})\). We will similarly write \(Z^-(\xi',\tilde{I}):K^\mu\to K^\mu\boxtimes H^\varphi_-(\tilde{I})\) for the map induced by \(\eta\mapsto \eta\otimes\xi'\) for \(\xi'\in H_-^\varphi(\tilde{I})\). Let \(\alpha_{\tilde{I}}:[0,1]\to \R\) be a path in \(\R\) from \(\tilde{I}\) to \(-\frac{1}{4}\in \widetilde{S^1_-}\). We write \(L(\xi, \tilde{I}): K^\mu\to H^\varphi\boxtimes K^\mu\) for the map \[\label{eq:32OperatorL} L(\xi,\tilde{I}): K^\mu\xrightarrow{Z^+(\xi, \tilde{I})} H_+^\varphi(\tilde{I})\boxtimes K^\mu\xrightarrow{\alpha_{\tilde{I}}^\bullet} H^\varphi_+(-)\boxtimes K^\mu = H^\varphi\boxtimes K^\mu.\tag{10}\] Given \(\xi'\in H^\varphi_-(\tilde{I})\), we write \(R(\xi', \tilde{I}): K^\mu\to K^\mu\boxtimes H^\varphi\) for the map \[\label{eq:32OperatorR} R(\xi', \tilde{I}): K^\mu\xrightarrow{Z^-(\xi', \tilde{I})}K^\mu\boxtimes H_-^\varphi(\tilde{I})\xrightarrow{(\varrho*\alpha_{\tilde{I}})^\bullet} K^\mu\boxtimes H_-^\varphi(+) = K^\mu\boxtimes H^\varphi.\tag{11}\] Since path continuations intertwine the actions of \(\A_\R\), we have that \(L(\xi, \tilde{I})\in \Hom_{\A_\R(\tilde{I}^{c+})}(K^\mu, H^\varphi\boxtimes K^\mu)\) and \(R(\xi', \tilde{I})\in \Hom_{\A_\R(\tilde{I}^{c-})}(K^\mu, K^\mu\boxtimes H^\varphi)\).

Let \(G\) be a discrete group acting on \(\A\), and recall that we write \(\Rep^G(\A)\) for the \(\mathrm{W}^*\)-category of \(G\)-twisted representations of \(\A\). In Section [Sec:32CatOfTwistedReps] we have constructed an action \(T\) of \(G\) on the \(\mathrm{W}^*\)-category \(\Rep^G(\A)\) compatible with the \(G\)-grading.

The results in [1] show, except for the locality axiom, that the tensor structure \(\boxtimes\) on \(\Rep^G(\A)\), together with the families of operators \(L\) and \(R\) in Equations 10 and 11 is a \(G\)-crossed categorical extension of \(\A\).

We call the categorical extension from Theorem 6 the Connes \(G\)-crossed categorical extension of \(\A\). The main reason to introduce \(G\)-crossed categorical extensions is the following uniqueness theorem, analogous to [13]. Let us denote by \(L^{\boxtimes}\) and \(R^\boxtimes\) the operators in the Connes \(G\)-crossed categorical extension of \(\A\).

4 The category of representations of the fixed-points conformal net↩︎

Given a conformal net \(\A\), acted on by a finite group \(G\), the category \(G-\Loc(\A)\) was introduced in [2] to study the categories of localized endomorphisms of the fixed-points conformal net \(\A^G\), constructed as follows. The category \(\Rep^G(\A)\) serves the same purpose in this article.

Let \((\A,H_0, U, \Omega)\) be a conformal net and let \(G\to \Aut(\A)\) be an injective group homomorphism defining a faithful action of a finite group \(G\) on \(\A\). In order to avoid cluttering the notation, we will continue denoting by \(g\) the image of an element \(g\in G\) in \(\Aut(\A)\). Each element \(g\in G\) comes equipped with a unitary \(V_g\in U(H_0)\) implementing the automorphism \(g\in \Aut(\A)\). Hence, the group \(G\) acts on the vacuum Hilbert space \(H_0\) of \(\A\) via \(g\mapsto V_g\). We denote by \(H_0^G\) the subspace of \(H_0\) of \(G\)-invariant vectors. The vacuum vector \(\Omega\) is in \(H_0^G\) by hypothesis. Given an interval \(I\in \Jcal\), we define \(\A^G(I):=\A(I)^G|_{H_0^G}\), which is a von Neumann algebra acting on \(H_0^G\). The projective action \(U\) of \(\Diff^+(S^1)\) on \(H_0\) restricts to a projective action \(U^G\) on \(H_0^G\), see [1] or [6] and [27]. The tuple \((\A^G, H_0^G, U^G, \Omega)\) is a conformal net, see for example [28].

The main goal of this section is to obtain an equivalence of balanced \(\mathrm{W}^*\)-tensor categories \[\Rep(\A^G)\cong (\Rep^G(\A))^G.\] We outline the proof strategy here. The main part of the proof is producing an equivalence of braided \(\mathrm{W}^*\)-tensor categories \(\Rep(\A^G)\cong (\Rep^G(\A))^G,\) which is done in Theorem 8. Checking that the equivalence is compatible with the balance is done in Theorem 9. In order to produce a braided \(\mathrm{W}^*\)-tensor equivalence \(\Rep(\A^G)\cong (\Rep^G(\A))^G\) we introduce an auxiliary category \(\M\), as follows. The vacuum representation \(H_0\in\Rep(\A)\) can be restricted to an honest representation \(r(H_0)\in \Rep(\A^G)\) of \(\A^G\) under the inclusion \(\A^G\subset \A\). Then, \(r(H_0)\) has a canonical structure of a commutative separable \(\mathrm{C}^*\)-Frobenius algebra object in \(\Rep(\A^G)\) in the sense of [20]. We denote this algebra object by \(\Gamma\) and write \(\M\) for its \(\mathrm{W}^*\)-tensor category of unitary modules, with tensor structure \(\boxtimes_\Gamma\). We define on \(\M\) a \(G\)-grading, a \(G\)-action \(\mathrm{T}\) and a natural family of unitary \(G\)-crossed braid isomorphisms \(\mathbb{B}^\Gamma\), without arguing the hexagon conditions yet, see Proposition [Prop:32ModGammaIsGGraded] and Lemma 2. The category \(\M\) is the de-equivariantization of \(\Rep(\A^G)\), but we will not argue this statement, see Remark [rk:32CommentDeequivariantization]. Instead, we obtain in Proposition [prop:32MEquivalenceOfGCats] an equivalence \(\mathfrak{M}:\M\to \Rep^G(\A)\) of \(G\)-graded \(\mathrm{W}^*\)-categories with a \(G\)-action. Identifying \(\M\) and \(\Rep^G(\A)\) via \(\mathfrak{M}\), we extend \((\boxtimes_\Gamma, \mathbb{B}^\Gamma)\) to a \(G\)-crossed categorical extension of \(\A\) in the sense of Definition 6. Then, by uniqueness of categorical extensions (see Theorem 7) we find that \((\M, \mathrm{T}, \boxtimes_\Gamma, \mathbb{B}^\Gamma)\) is a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category and that \(\mathfrak{M}\) can be upgraded to an equivalence of \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories \(\mathfrak{M}: \M\to \Rep^G(\A)\), see Proposition [prop:32MisEquivalenceOFG-X-BraidedCats]. Taking \(G\)-equivariantizations, we obtain an equivalence of braided \(\mathrm{W}^*\)-tensor categories \(\mathfrak{M}^G: \M^G\to (\Rep^G(\A))^G\). The proof is concluded by defining a functor of braided \(\mathrm{W}^*\)-tensor categories \(\mathfrak{D}: \Rep(\A^G)\to \M^G\) and finding an inverse functor \(\mathfrak{R}\) to the composition \[\Rep(\A^G)\xrightarrow{\mathfrak{D}}\M^G\xrightarrow{\mathfrak{M}^G}(\Rep^G(\A))^G.\] The functor \(\mathfrak{R}: (\Rep^G(\A))^G\to \Rep(\A^G)\) restricts a \(G\)-equivariant, \(G\)-twisted representation of \(\A\) on a Hilbert space \(H\) to the honest representation of \(\A^G\) on the \(G\)-fixed points of \(H\), see above Remark [rk:32NonHomogeneous]. In order to upgrade the equivalence \[\begin{tikzcd} {\Rep(\A^G)} && {(\Rep^G(\A))^G} \arrow["{\mathfrak{M}^G\circ\mathfrak{D}}", shift left, from=1-1, to=1-3] \arrow["{\mathfrak{R}}", shift left, from=1-3, to=1-1] \end{tikzcd}\] of braided \(\mathrm{W}^*\)-tensor categories to an equivalence of balanced \(\mathrm{W}^*\)-tensor categories, we argue in Theorem 9 that \(\mathfrak{R}\) preserves the balance.

4.1 The equivalence of braided \(\mathrm{W}^*\)-tensor categories↩︎

In this section, we follow [20] to produce an equivalence of braided \(\mathrm{W}^*\)-tensor categories \[\Rep(\A^G)\cong (\Rep^G(\A))^G.\] In order to differentiate the Connes fusions of representations in \(\Rep(\A^G)\) and \(\Rep^G(\A)\), we denote the former by \(-\boxtimes ^G-\), and keep the notation \(-\boxtimes-\) for the Connes fusion of twisted representations in \(\Rep^G(\A)\). We also write \(\beta\) for the braiding on \(\Rep(\A^G)\), and keep the notation \(\B\) for the \(G\)-crossed braiding on \(\Rep^G(\A)\) and \(\B^G\) for the braiding on its equivariantization \((\Rep^G(\A))^G\). We write \(L^G\) and \(R^G\) for the families of operators of the (non-crossed) Connes categorical extension of \(\A^G\).

Let us recall the notion of a \(\mathrm{C}^*\)-Frobenius algebra object in \(\Rep(\A^G)\), following [20].

Let \(r: \Rep^G(\A)\to \Rep(\A^G)\) be the functor that restricts a \(G\)-twisted representation of \(\A\) to an honest representation of \(\A^G\) under the inclusion \(\A^G\subset \A\). We will produce a commutative Frobenius \(\mathrm{C}^*\)-algebra structure on the object \(r(H_0)\in\Rep(\A^G)\). Let \(\iota: H_0^G\hookrightarrow r(H_0)\) be the inclusion of \(\A^G\)-representations. Given \(\tilde{I}\in\Jcal_\R\), we define the linear map \[\begin{array}{cccc} \mu_{\tilde{I}}: &r(H_0)\boxtimes^G r(H_0) & \to &r(H_0) \\ & L^G(\pi_0(x)\Omega,\tilde{I})\xi&\mapsto & \pi_0(x)\xi \end{array}\] for \(x\in\A(I)\) and \(\xi\in H_0\). Since \(G\) is finite, the inclusion \(\A^G(I)\subset \A(I)\) is of finite index. Therefore, the same arguments as step 1 of the proof of the implication \((3)\implies (4)\) in [20] show that \(\mu_{\tilde{I}}\) is bounded. If \(y\in\A^G(I^c)\), it holds that \[\begin{align} \pi_0(y)\mu_{\tilde{I}} (L^G(\pi_0(x)\Omega, \tilde{I})\xi) &= \pi_0(y)\pi_0(x)\xi = \pi_0(x)\pi_0(y)\xi \\&= \mu_{\tilde{I}}(L^G(\pi_0(x)\Omega, \tilde{I})\pi_0(y)\xi) = \mu_{\tilde{I}}(\pi^{r(H_0)\boxtimes^G r(H_0)}_{I^c}(y)L^G(\pi_0(x)\Omega, \tilde{I})\xi), \end{align}\] showing that \(\mu_{\tilde{I}}\) intertwines the actions of \(\A(I^c)\). If \(\tilde{J}\in\Jcal_\R\) is an interval with \(\tilde{J}\subset\tilde{I}\) and \(x\in \A(J)\), it then holds that \(\mu_{\tilde{J}}(L^G(\pi_0(x)\Omega, \tilde{J})\xi) = \mu_{\tilde{I}}(L^G(\pi_0(x)\Omega, \tilde{I})\xi)\) for all \(\xi\in H_0\). Hence, \(\mu_{\tilde{I}}\) and \(\mu_{\tilde{J}}\) agree. Therefore, \(\mu_{\tilde{I}} = \mu_{\tilde{J}}\) for any intervals \(\tilde{I}, \tilde{J}\in\Jcal_\R\), and we may write \(\mu:=\mu_{\tilde{I}}\), which is a morphism of \(\A^G\)-representations.

When referring to \(r(H_0)\) as a \(\mathrm{C}^*\)-Frobenius algebra object in \(\Rep(\A^G)\), we will denote it by \(\Gamma\). Recall that we write \(\Delta = \mu^*\circ (\mu\mu^*)^{-1}: \Gamma\to \Gamma\boxtimes^G\Gamma\) for the splitting of \(\mu\) as a \((\Gamma-\Gamma)\)-bimodule witnessing the fact that \(\Gamma\) is separable. It is straightforward to check that the diagram \[\label{eq:32DeltaAdjointCommutativity}\begin{tikzcd} \Gamma && {\Gamma\boxtimes^G\Gamma} \\ {\Gamma\boxtimes^G\Gamma} && {\Gamma\boxtimes^G\Gamma\boxtimes^G\Gamma} \arrow["\Delta", from=1-1, to=1-3] \arrow["\Delta"', from=1-1, to=2-1] \arrow["{\id\boxtimes^G\Delta}", from=1-3, to=2-3] \arrow["{\Delta\boxtimes^G \id}"', from=2-1, to=2-3] \end{tikzcd}\tag{15}\] commutes adjointly. We next introduce the category of unitary modules over \(\Gamma\).

We may endow \(\M\) with a tensor structure as follows. Given \((H,\mu_H),(K,\mu_K)\in \M\), we define their tensor product to be \[\label{eq:32TensorModules} H\boxtimes_\Gamma K:= \Gamma\boxtimes^G_{\Gamma\boxtimes^G \Gamma}(H\boxtimes ^G K),\tag{17}\] where the right-hand side denotes the tensor product of the left \((\Gamma\boxtimes ^G\Gamma)\)-module \(H\boxtimes ^G K\) with the right \((\Gamma\boxtimes ^G\Gamma)\)-module \(\Gamma\) in \(\Rep(\A^G).\) Note that, since \(\Gamma\) is separable, the right-hand side of Equation 17 makes sense, even though \(\Rep(\A^G)\) is not abelian. With this tensor product, the category \(\M\) is a \(\mathrm{W}^*\)-tensor category with unit \(\Gamma\), see [26]. By definition, there is a canonical morphism \(H\boxtimes^G K\to H\boxtimes _\Gamma K\), which we denote \(\mu_{H,K}\). The tensor product \(H\boxtimes_\Gamma K\) can also be defined by regarding \(H\) as a \(\Gamma\)-bimodule via the right-action \(H\boxtimes^G \Gamma\xrightarrow{\beta^{-1}_{\Gamma, H}} \Gamma\boxtimes^G H \xrightarrow{\mu_H} H\), see [26]. This means that \(H\boxtimes_\Gamma K\) is the splitting of the idempotent \[\begin{align} \label{eq:32Idempotent} p_{H, K}:H\boxtimes^G K\cong H\boxtimes^G H_0^G\boxtimes^G K&\xrightarrow{\id\boxtimes^G (\Delta\circ \iota)\boxtimes^G \id} H\boxtimes^G \Gamma\boxtimes^G\Gamma\boxtimes^G K\\&\xrightarrow{\beta^{-1}_{\Gamma, H}\boxtimes^G\id}\Gamma\boxtimes^G H\boxtimes^G \Gamma\boxtimes^G K\xrightarrow{\mu_H\boxtimes^G \mu_K} H\boxtimes^G K.\nonumber \end{align}\tag{18}\] Since \(\Rep(\A^G)\) is idempotent complete, the object \(H\boxtimes_\Gamma K\in \Rep(\A^G)\) exists. In addition, \(H\boxtimes_\Gamma K\) is also the coequalizer of \(H\boxtimes^G\Gamma\boxtimes^G K\xrightarrow{\id\boxtimes^G \mu_K} H\boxtimes^G K\) and \(H\boxtimes^G\Gamma\boxtimes^G K\xrightarrow{\beta^{-1}_{\Gamma, H}\boxtimes^G \id} \Gamma\boxtimes^G H\boxtimes^G K\xrightarrow{\mu_H\boxtimes^G\id} H\boxtimes^G K\), see for example the proof of [29]. We stress again that \(\Rep(\A^G)\) is not abelian, but since \(\Gamma\) is separable, the coequalizer exists, equals the splitting of 18 and the morphism \(\mu_{H,K}\) is a projection onto the object splitting \(p_{H, K}\). The object \(H\boxtimes_\Gamma K\) gets equipped with the structure of a unitary left \(\Gamma\)-module by the morphism \(\Gamma\boxtimes^G H\boxtimes ^G K\xrightarrow{\mu_H\boxtimes^G\id} H\boxtimes^G K\). The left unitor on a module \((H, \mu_H)\in\M\) is induced by the map \(\mu_H: \Gamma\boxtimes^G H\to H\), and the right unitor is induced by \(H\boxtimes^G\Gamma\xrightarrow{\beta_{\Gamma, H}^{-1}}\Gamma\boxtimes^G H\xrightarrow{\mu_H}H\). We will need the following properties of \(\mu_{H, K}\).

Given \(g\in G\), we continue denoting by \(V_g: \Gamma\to \Gamma\) the algebra automorphism induced by \(V_g: H_0\to H_0\). For an element \(g\in G\) and a module \((H,\mu_H)\in \M\), we say that \((H,\mu_H)\) is a \(g\)-graded module if the diagram \[\begin{tikzcd} {\Gamma\boxtimes^GH} && {\Gamma\boxtimes^GH} \\ {\Gamma\boxtimes^GH} && H \arrow["{\beta_{H,\Gamma}\circ\beta_{\Gamma, H}}", from=1-1, to=1-3] \arrow["{V_{g^{-1}}\boxtimes^G \id}"', from=1-1, to=2-1] \arrow["{\mu_H}", from=1-3, to=2-3] \arrow["{\mu_H}"', from=2-1, to=2-3] \end{tikzcd}\] commutes. We write \(\M_g\) for the full subcategory of \(\M\) on the \(g\)-graded modules.

We can also define an action \(\mathrm{T}\) of \(G\) on \(\M\). Given \(g\in G\) and \((H,\mu_H)\in \M\), we define \(\mathrm{T}_g(H, \mu_H) = (H, g^{-1}\mu_H)\), where \[g^{-1}\mu_H: \Gamma\boxtimes^G H\xrightarrow{V_{g^{-1}}\boxtimes^G\id} \Gamma\boxtimes ^G H\xrightarrow{\mu_H} H.\] The action of \(G\) on morphisms is trivial, and it is easy to upgrade the functors \(\mathrm{T}_g\) to an action of \(G\) on \(\M\) compatible with the \(G\)-grading, see [26] or [21]. Finally, given \((H,\mu_H),(K,\mu_K)\in \M\) modules such that \((H,\mu_H)\) is homogeneous of degree \(g\), the crossed braiding \(\mathbb{B}^\Gamma_{(H,\mu_H), (K,\mu_K)}; H\boxtimes_\Gamma K\xrightarrow{\cong} T_g(K)\boxtimes_\Gamma H\) is the unique morphism making the following diagram commute, \[\label{eq:32braidingModGamma}\begin{tikzcd} {H\boxtimes^G K} &&& {K\boxtimes^GH} \\ {H\boxtimes_\Gamma K} &&& {\mathrm{T}_g(K)\boxtimes_\Gamma H}. \arrow["{\beta_{H,K}}", from=1-1, to=1-4] \arrow["{\mu_{H,K}}"', from=1-1, to=2-1] \arrow["{\mu_{\mathrm{T}_g(K), H}}", from=1-4, to=2-4] \arrow["{\mathbb{B}^\Gamma_{(H,\mu_H), (K,\mu_K)}}"', from=2-1, to=2-4] \end{tikzcd}\tag{21}\]

We will show that, with the tensor \(\boxtimes^\Gamma\), the grading \(\M = \bigoplus_{g\in G}\M_g\), the action \(\mathrm{T}\) and the unitaries \(\B^\Gamma\), the \(\mathrm{W}^*\)-category \(\M\) is a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category equivalent to \(\Rep^G(\A)\). We construct first the underlying functor \(\mathfrak{M}: \M\to \Rep^G(\A)\), inspired by [20]. We need the following lemma, which is analogous to [20] for our choice of braiding.

By Proposition [Prop:32ExtensionFunctor], we can define a \(\mathrm{W}^*\)-functor \[\mathfrak{M}: \M\to \Rep^G(\A)\] by linearly extending the assignment obtained by sending a \(g\)-graded module \((H,\mu_H)\in \M_g\) to \(\mathfrak{M}(H,\mu_H):=(H,\pi^{\mathfrak{M}(H)})\in \Rep^g(\A)\) and a morphism \(F\) in \(\M\) to the same map of Hilbert spaces. Indeed, fix \(g\in G\) and let \(F:(H, \mu_H)\to (K, \mu_K)\) be a morphism in \(\M_g\). We show that \(F\) intertwines the \(g\)-twisted actions \(\pi^{\mathfrak{M}(H)}\) and \(\pi^{\mathfrak{M}(K)}\) on \(H\) and \(K\) respectively. Let \(\tilde{I}\in\Jcal_\R\), \(x\in \A_\R(\tilde{I})\) and \(\xi\in H\). Then, we can compute \[\begin{align} \pi_{\tilde{I}}^{\mathfrak{M}(K)}(x)\circ F(\xi) &= \mu_K\circ L^G(\pi_0(x)\Omega, \tilde{I}) F(\xi)\\ & = \mu_K\circ (\id\boxtimes^G F)\circ L^G(\pi_0(x)\Omega, \tilde{I})\xi \\& = F\circ \mu_H\circ L^G(\pi_0(x)\Omega, \tilde{I})\\ & = F\circ\pi_{\tilde{I}}^{\mathfrak{M}(H)}(x)(\xi). \end{align}\]

We will next show that \(\M\) is naturally a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category equivalent to \(\Rep^G(\A)\) via (an upgrade to a tensor functor of) the functor \(\mathfrak{M}\). To do so, we first need the following lemma, which is analogous to [20]. Recall that we write \(\iota: H_0^G\hookrightarrow \Gamma\) for the inclusion of \(\A^G\)-representations.

We may now introduce the following notation. Let \(H^g\in \Rep^g(\A)\) and \(K^h\in \Rep^h(\A)\) be representations twisted by \(g,h\in G\) respectively. Let \(\mu_H: \Gamma\boxtimes^Gr(H)\to r(H)\) be the morphism \(L^G(\pi_0(x)\Omega, \tilde{I})\xi\mapsto \pi_{\tilde{I}}^H(x)(\xi)\) defined in the proof of Proposition [prop:32MEquivalenceOfGCats] (see Equation 24 ). We similarly define \(\mu_K\). Then, \((H, \mu_H),(K, \mu_K)\in \M\) and \(\mathfrak{M}(H, \mu_H) = H\) and \(\mathfrak{M}(K, \mu_K) = K\). Given \(\tilde{I}\in \Jcal_\R\) and \(\xi\in H^g_+(\tilde{I})\), we define the maps of Hilbert spaces \[\begin{align} L^\Gamma(\xi, \tilde{I}):&\;K\xrightarrow{L^G(\xi, \tilde{I})} H\boxtimes ^G K \xrightarrow{\mu_{H, K}} H\boxtimes_\Gamma K\\ R^\Gamma(\xi, \tilde{I}):&\;K\xrightarrow{R^G(\xi, \tilde{I})} K\boxtimes ^G H \xrightarrow{\mu_{K, H}} K\boxtimes_\Gamma H. \end{align}\]

Let \(\M^G\) denote the \(G\)-equivariantization of \(\M\), and let us write \[\mathfrak{D}: \Rep(\A^G)\to \M^G\] for the functor sending \(H\in \Rep(\A^G)\) to the unitary \(\Gamma\)-module \((\Gamma\boxtimes^G H, \mu\boxtimes^G\id_H)\in \M\) together with the family of isomorphisms \(u_g = V_{g}\boxtimes^G\id_H\) for all \(g\in G\). For a morphism \(F\in \Hom_{\Rep(\A^G)}(H, K)\), we write \(\mathfrak{D}(F):=\id_\Gamma\boxtimes^G F\in \Hom_{\M}(\mathfrak{D}(H), \mathfrak{D}(K))\), which is trivially compatible with the \(G\)-equivariant data. We also write \(\mathfrak{M}^G: \M^G\to (\Rep^G(\A))^G\) for the equivalence of braided \(\mathrm{W}^*\)-tensor categories induced by \(\mathfrak{M}.\) We therefore obtain a \(\mathrm{W}^*\)-functor \[\label{eq:32InductionFunctor} \Rep(\A^G)\xrightarrow{\mathfrak{D}}\M^G\xrightarrow{\mathfrak{M}^G} (\Rep^G(\A))^G,\tag{25}\] which we will show produces an equivalence of braided \(\mathrm{W}^*\)-tensor categories. Given \(((H, \pi_H), u)\in (\Rep^G(\A))^G\), we write \(H^G : = \{\xi\in H\,|\, u_g\xi = \xi\;\text{ for all g \in G}\}\) for the fixed-points Hilbert space of \(H\) under \(u: G\to U(H)\). Note that \(\pi^H\) restricts to an honest action of \(\A^G\) on \(H^G\), which we continue denoting by \(\pi_H\). In addition, if \(F: (H, u)\to (K, v)\) is a morphism in \((\Rep^G(\A))^G\), it restricts to a morphism \(F|_{H^G}:H^G\to K^G\) in \(\Rep(\A^G)\). We write \[\mathfrak{R}: (\Rep^G(\A))^G\to \Rep(\A^G)\] for the \(\mathrm{W}^*\)-functor sending \(((H, \pi^H), u)\in (\Rep^G(\A))^G\) to \((H^G, \pi^H)\) and \(F: (H, u)\to (K, v)\) to \(F|_{H^G}:H^G\to K^G\).

We claim that \(\mathfrak{R}\) produces an inverse functor to the functor \(\mathfrak{M}^G\circ\mathfrak{D}\) in Equation 25 . Before proving this, we need some preliminaries.

Recall that, given \(I_0\in \Jcal\), by Theorem 5, there is an equivalence of \(G\)-crossed braided \(\mathrm{W}^*\)-tensor categories \(\mathfrak{E}: G-\Loc_{I_0}(\A)\to \Rep^G(\A)\), which induces an equivalence between the equivariantizations \(\mathfrak{E}^G: (G-\Loc_{I_0}(\A))^G\to (\Rep^G(\A))^G\). Let us denote by \((G-\Loc_{I_0}(\A))^G_0\) the full subcategory of \((G-\Loc_{I_0}(\A))^G\) on objects whose \(G\)-equivariance data is the identity. The inclusion \((G-\Loc_{I_0}(\A))^G_0\hookrightarrow (G-\Loc_{I_0}(\A))^G\) is an equivalence, by [2]. We shall continue writing \(\mathfrak{E}^G\) for the equivalence \[\mathfrak{E}^G: (G-\Loc_{I_0}(\A))^G_0\hookrightarrow (G-\Loc_{I_0}(\A))^G\xrightarrow{\mathfrak{E}^G}(\Rep^G(\A))^G.\] Non-zero objects of \((G-\Loc_{I_0}(\A))^G_0\) can be constructed as follows. Fix \(g\in G\) and denote by \(Z_G(g)\) the centralizer of \(g\) in \(G\) and by \(C_g\) the conjugacy class of \(g\). For every \(h\in C_g\), we fix \(h_g\in G\) an element conjugating \(g\) to \(h\). Fix \(\rho_g\in (g-\Loc_{I_0}(\A))^{Z_G(g)}_0\), that is, \(\rho_g\) is a \(g\)-localized-in-\(I_0\) transportable endomorphism of \(\A\) which is fixed by the action of \(Z_G(g)\) on the nose. We write \(\rho_h:=\gamma_{h_g}(\rho_g)\in (h-\Loc_{I_0}(\A))^{Z_G(h)}_0\) for \(h\in C_g\). Note that, since \(\rho_g\) is \(Z_G(g)\)-invariant, \(\rho_h\) is independent of the chosen \(h_g\). Then, we can construct \(\rho: = \bigoplus\limits_{h\in C_g}\rho_h\in (G-\Loc_{I_0}(\A))^G_0\), and all objects of \((G-\Loc_{I_0}(\A))^G_0\) are direct sums of objects constructed as such. The image of \(\rho\) under \(\mathfrak{E}^G\) has underlying \(G\)-twisted representation \(\bigoplus\limits_{h\in C_g}(H_0, \pi_0\circ \rho_h)\) and the following \(G\)-equivariant data \(u\). Given \(k,h\in G\) and \(\xi\in (H_0, \pi_0\circ \rho_h)\), then \(u_k\xi : = V_k\xi\in (H_0, \pi_0\circ \rho_{khk^{-1}}).\)

We also need the following relation between \(\A\) and \(\A^G\). Let \(I\in \Jcal\) and fix \(x\in \A(I)\). Then, the vector \(\pi_0(x)\Omega\in r(H_0)\) is trivially \(I\)-bounded as witnessed by the \(\A^G(I^c)\)-equivariant map \(Z^G(\pi_0(x)\Omega,\tilde{I}) = \pi_0(x): H^G_0\to H_0\). Given another \(y\in\A(I)\), we can consider the composition \[H_0^G\xrightarrow{Z^G(\pi_0(x)\Omega, \tilde{I})}H_0\xrightarrow{Z^G(\pi_0(y)\Omega, \tilde{I})^*}H_0^G,\] which is \(\A^G(I^c)\)-equivariant, hence an element of \(\A^G(I)\) by Haag duality.

We can now prove the main result of this section.

4.2 Compatibility with the balance↩︎

We finish this paper by upgrading the equivalence of braided \(\mathrm{W}^*\)-tensor categories in Theorem 8 to an equivalence of balanced \(\mathrm{W}^*\)-tensor categories \(\Rep(\A^G)\cong (\Rep^G(\A))^G\). We denote by \(\vartheta\) the canonical balance on \(\Rep(\A^G)\), so that we reserve the notation \(\theta\) for the \(G\)-crossed balance on \(\Rep^G(\A)\) and \(\theta^G\) for the balance on \((\Rep^G(\A))^G\). Recall that, on a representation \(H\in\Rep(\A^G)\), the unitary \(\vartheta_H:H\to H\) is given by the action \(e^{-2\pi i L_0}\) of \(\widetilde{\exp}(-2\pi i L_0)\in\widetilde{\Mob}\).

5 Bi-involutive structures↩︎

In [1], we constructed an involutive \(\mathrm{W}^*\)-tensor structure on the \(\mathrm{W}^*\)-tensor category \(\Rep^{\Aut(\A)}(\A)\). By restricting, one also obtains an involutive \(\mathrm{W}^*\)-tensor structure on \(\Rep^G(\A)\), and on \(\Rep(\A^G)\). The category \((\Rep^G(\A))^G\) also comes equipped with an involutive \(\mathrm{W}^*\)-tensor structure by the fact that it is an equivariantization of an involutive \(\mathrm{W}^*\)-tensor category and \(G\) acts by \(\mathrm{W}^*\)-tensor functors, see below. We note that, in [14], what we refer to as involutive \(\mathrm{W}^*\)-tensor categories are called bi-involutive \(\mathrm{W}^*\)-tensor categories.

An involutive structure on a \(\mathrm{W}^*\)-tensor category \(\mathcal{C}\) consists of an anti-linear \(\mathrm{W}^*\)-functor \(\overline{\,\cdot\,}:\mathcal{C}\to \mathcal{C}\), equipped with a unitary anti-tensor structure \(\nu_{X, Y}: \overline{X}\otimes \overline{Y}\to \overline{X\otimes Y}\) natural in \(X,Y\in\mathcal{C}\) and compatible with the associator, a unitary isomorphism \(r: \1\to \overline{\1}\) and a natural family of unitary isomorphisms \(\varphi_X: X\to \overline{\overline{X}}\). The obvious compatibilities between these pieces of data are spelled out, for example, in [34] or [1]. An involutive \(\mathrm{W}^*\)-functor between two involutive \(\mathrm{W}^*\)-tensor categories consists of a \(\mathrm{W}^*\)-tensor functor \(F\) together with a unitary natural transformation \(\chi: \overline{F(-)}\cong F(\overline{\,-\,})\). The family \(\chi\) satisfies the obvious compatibility conditions with the tensor structure data of \(F\), see [1]. Natural transformations between involutive \(\mathrm{W}^*\)-tensor functors are also defined in the obvious way. We write \(\underline{\Aut_{\otimes}^{\overline{\,\cdot\,}}}(\mathcal{C})\) for the tensor category of involutive \(\mathrm{W}^*\)-tensor automorphisms of \(\mathcal{C}\) and unitary involutive natural transformations.

Let \(\mathcal{C}\) be an involutive \(\mathrm{W}^*\)-tensor category and \(G\) be a discrete group acting on the underlying \(\mathrm{W}^*\)-tensor category of \(\mathcal{C}\) by a tensor functor \(T:\underline{G}\to \underline{\Aut_\otimes(\mathcal{C})}\). Recall that \(T\) consists of tensor functors \((T_g, s_g): \mathcal{C}\to \mathcal{C}\) for every \(g\in G\) together with unitary natural transformations \(n_{g,h}: T_gT_h\cong T_{gh}\) for all \(g, h\in G\). A lift of \(T\) to an action by involutive \(\mathrm{W}^*\)-tensor functors consists of a unitary family of natural isomorphisms \(\chi_g: \overline{T_g(-)}\cong T_g(\overline{\,-\,})\) upgrading \((T_g,s_g)\) to an involutive \(\mathrm{W}^*\)-tensor functor for every \(g\in G\) and making the following diagram commute \[\begin{tikzcd}\label{eq:32CompatibilityNChi} {\overline{T_gT_h(X)}} && {\overline{T_{gh}(X)}} \\ {T_g(\overline{T_h(X)})} && {T_{gh}(\overline{X})} \\ & {T_gT_h(\overline{X})} \arrow["{\overline{n_{g,h}(X)}}", from=1-1, to=1-3] \arrow["{\chi_g(T_h(X))}"', from=1-1, to=2-1] \arrow["{\chi_{gh}(X)}", from=1-3, to=2-3] \arrow["{T_g(\chi_h(X))}"', from=2-1, to=3-2] \arrow["{n_{g,h}(\overline{X})}"', from=3-2, to=2-3] \end{tikzcd}\tag{26}\] for all \(g,h\in G\) and \(X\in \mathcal{C}.\) In particular, the families \(\chi_g\) upgrade \(T\) to a tensor functor \(T: \underline{G}\to \underline{\Aut_{\otimes}^{\overline{\,\cdot\,}}}(\mathcal{C})\).

A \(G\)-crossed braided involutive \(\mathrm{W}^*\)-tensor category consists of a \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category \(\mathcal{C}\), together with a lift of the \(G\)-action to an action by involutive \(\mathrm{W}^*\)-tensor functors and such that \(\overline{\,\cdot\,}:\mathcal{C}\to \mathcal{C}\) restricts to equivalences \(\mathcal{C}_g\to \mathcal{C}_{g^{-1}}\) for all \(g\in G\). An involutive \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor category is an involutive \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category \(\mathcal{C}\) together with a \(G\)-crossed balance \(\theta\) on its underlying \(G\)-crossed braided \(\mathrm{W}^*\)-tensor category such that the diagram \[\begin{tikzcd}\label{eq:32RibbonCondition} {\overline{T_g(X)}} && {\overline{X}} \\ && {T_e(\overline{X})} \\ {T_g(\overline{X})} && {T_{g^{-1}}\circ T_g(\overline{X})} \arrow["{\overline{\theta_X^*}}", from=1-1, to=1-3] \arrow["{\chi_g(X)}"', from=1-1, to=3-1] \arrow["\cong"', from=2-3, to=1-3] \arrow["{\theta_{T_g(\overline{X})}}"', from=3-1, to=3-3] \arrow["{n_{g^{-1}, g}(\overline{X})}"', from=3-3, to=2-3] \end{tikzcd}\tag{27}\] commutes for all \(g\in G\) and \(X\in\mathcal{C}_g\). The commutativity of Diagram 27 is a generalization to the context of involutive \(\mathrm{W}^*\)-tensor categories of the usual ribbon condition in rigid semisimple \(G\)-crossed balanced tensor categories, see [1] and [35]. Functors between involutive \(G\)-crossed braided and \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor functors are defined in the obvious way.

In Section 2.2, we have argued how the equivariantization of a \(G\)-crossed balanced \(\mathrm{W}^*\)-tensor category is canonically balanced. We generalize this statement here to the context of involutive \(\mathrm{W}^*\)-tensor categories.

Let \(\A\) be a conformal net. We now recall from [1] how to endow the \(\mathrm{W}^*\)-tensor category \(\Rep^{\Aut(\A)}(\A)\) with the structure of an involutive \(\mathrm{W}^*\)-balanced tensor category. If \(G\) is a discrete group acting on \(\A\) by a group homomorphism \(G\to \Aut(\A)\), we can pull back the balanced involutive \(\mathrm{W}^*\)-tensor structure on \(\Rep^{\Aut(\A)}(\A)\) under this homomorphism to obtain a \(G\)-crossed balanced involutive \(\mathrm{W}^*\)-tensor structure on \(\Rep^G(\A)\).

Recall that we fix the point \(\mathrm{p}=1\in S^1\), and that we define \(\widetilde{S^1_+} := (0,\frac{1}{2})\in \Jcal_\R\), and \(S^1_+:=q(\widetilde{S^1_+})\), the upper semi-circle. The von Neumann algebra \(\A(S^1_+)\) acts on \(H_0\) by definition, and the vacuum vector \(\Omega\in H_0\) is cyclic for the action of \(\A(S^1_+)\), by the Reeh-Schlieder Theorem. One can therefore define an anti-linear unbounded operator \(S: \A(S^1_+)\Omega\to \A(S^1_+)\Omega\) such that, for every \(x\in \A(S^1_+)\), \[S\pi_{0, S_+^1}(x)(\Omega) = \pi_{0, S_+^1}(x^*)(\Omega),\] following Tomita-Takesaki theory. The operator \(S\) is preclosed and we continue denoting its closure by \(S\). Let \(S = \mathrm{J}\circ \Delta^{1/2}\) be the polar decomposition of \(S\). Then, the anti-unitary map \(\mathrm{J}: H_0\to H_0\) is called the modular conjugation and satisfies \(\mathrm{J}\Omega = \Omega\) and \(\mathrm{J}^2 = \id_{H_0}\). By the Bisognano-Wichmann Theorem [5], it also holds that, for every \(I\in\Jcal\) and \(x\in \A(I)\), \[\mathrm{J}\circ \pi_{0, I}(x)\circ \mathrm{J}\in \Hom_{\A(\overline{I})^c}(H_0,H_0)\cong \A(\overline{I}),\] where \(\overline{I} := \{\overline{z}\,|\, z\in I\}\in \Jcal\) is the image of \(I\) under complex conjugation in \(S^1\). Hence, we obtain a von Neumann algebra isomorphism \[\begin{array}{cccc} j: &\A(I) &\to & \A(\overline{I})^{\text{op}} \\ & x&\mapsto & j(x), \end{array}\] where \(j(x)\) is the unique element in \(\A(\overline{I})\) such that \(\pi_{0, \overline{I}}(j(x)) = \mathrm{J}\circ \pi_{0, I}(x^*)\circ \mathrm{J}\). The isomorphisms \(j\) are compatible with inclusions of intervals and commute with automorphisms of \(\A\).

Fix an automorphism \(\varphi\in\Aut(\A)\) and let \((H^\varphi, \pi^H)\in \Rep^\varphi(\A)\) be a \(\varphi\)-twisted representation of \(\A\). We define a new representation \(\overline{(H^\varphi, \pi^H)}:=(\overline{H^\varphi}, \overline{\pi^H})\in \Rep^{\varphi^{-1}}(\A)\). Here, \(\overline{H^\varphi}\) denotes the complex conjugate Hilbert space of \(H^\varphi\) and \(\overline{\pi^H}\) is defined as follows. Let \(\tilde{I}\in \Jcal_{\R}\) and \(x\in \A_\R(\tilde{I})\). Then, we write \[\label{eq:32defInvolutionRepGA} \overline{\pi^H}_{\tilde{I}}(x)(\overline{\xi}) = \overline{\pi^H_{-\tilde{I}}(j(x)^*)(\xi)}.\tag{29}\] for all \(\overline{\xi}\in \overline{H^\varphi}\). It is easy to see that Equation 29 indeed defines a \(\varphi^{-1}\)-twisted action of \(\A\) on \(\overline{H^\varphi}\). It is also clear that, if \(K^\varphi\in \Rep^\varphi(\A)\) is a twisted representation and \(F: H^\varphi\to K^\varphi\) is a morphism of \(\varphi\)-twisted representations, the complex conjugate map \(\overline{F}: \overline{H^\varphi}\to \overline{K^\varphi}\) intertwines the \(\varphi^{-1}\)-twisted actions \(\overline{\pi^H}\) and \(\overline{\pi^K}\) of \(\A\). We write \(\overline{\,\cdot\,}: \Rep^{\Aut(\A)}(\A)\to \Rep^{\Aut(\A)}(\A)\) for the anti-linear functor that sends a twisted representation \((H^\varphi, \pi^H)\in \Rep^\varphi(\A)\) to \((\overline{H^\varphi}, \overline{\pi^H})\), and a morphism of twisted representations \(F: H^\varphi\to K^\varphi\) to the complex conjugate morphism \(\overline{F}: \overline{H^\varphi}\to \overline{K^\varphi}\).

The functor \(\overline{\,\cdot\,}\) is upgraded to involutive \(\mathrm{W}^*\)-tensor structure on \(\Rep^{\Aut(\A)}(\A)\) as follows. The compatibility with the unit is given by the unitary \(i:(H_0, \pi_0)\xrightarrow{\cong} \overline{(H_0,\pi_0)}\), \(\xi\mapsto \overline{\mathrm{J}\xi}\), and the trivialization of the square of \(\overline{\,\cdot\,}\) is defined to be the family \(\phi_{H}: (H^\varphi,\pi^H)\xrightarrow{\cong}\overline{\overline{(H^\varphi, \pi^H)}}\), \(\xi\mapsto \overline{\overline{\xi}}\), for \(H\in\Rep^{\Aut(\A)}(\A)\).

Let us next introduce the anti-tensor structure \(\nu\). Given an interval \(\tilde{I}\in \Jcal_\R\) and a twisted representation \(H^\varphi\in \Rep^\varphi(\A)\), there are canonical isomorphisms \(H_-^\varphi(\tilde{I})\cong \overline{H^\varphi}_+(-\tilde{I})\) and \(H_+^\varphi(\tilde{I})\cong \overline{H^\varphi}_-(-\tilde{I})\) given by \(\xi\mapsto \overline{\xi}\). In addition, for \(\xi\in H_{\pm}^\varphi(\tilde{I})\), we have \[Z^{\mp}(\overline{\xi}, -\tilde{I}) = \overline{Z^{\pm}(\xi, \tilde{I})}\circ i.\] Therefore, given another twisted representation \(K^\mu\in \Rep^\mu(\A)\), the map \(\overline{H^\varphi}_+(\tilde{I})\otimes \overline{K^\mu}\to \overline{K^\mu\otimes H^\varphi_-(-\tilde{I})}\) given by \(\overline{\xi}\otimes\overline{\eta}\mapsto \overline{\eta\otimes\xi}\) defines a unitary isomorphism \[\nu^{\tilde{I}}_{H, K}: \overline{H^\varphi}_+(\tilde{I})\boxtimes \overline{K^\mu}\to \overline{K^\mu\boxtimes H^\varphi_-(-\tilde{I})},\] see [1]. The anti-tensor structure \(\nu\) of \(\overline{\,\cdot\,}\) is then defined, on the twisted representations \(H^\varphi\) and \(K^\varphi\), as \[\nu_{H, K}: \overline{H^\varphi}\boxtimes \overline{K^\mu} = \overline{H^\varphi}_+(\widetilde{S^1_-})\boxtimes \overline{K^\mu}\xrightarrow{\nu_{H, K}^{\widetilde{S^1_-}}}\overline{K^\mu\boxtimes H_-^\varphi(-\widetilde{S_-^1})} = \overline{K^\mu\boxtimes H_-^\varphi(\widetilde{S_+^1})} = \overline{K^\mu\boxtimes H^\varphi}.\] The lift of the action \(T\) of \(\Aut(\A)\) on \(\Rep^{\Aut(\A)}(\A)\) to an action by involutive \(\mathrm{W}^*\)-tensor automorphisms is given trivially by \[T_\mu(\overline{\,\cdot\,}) = \overline{T_\mu(-)}\] for all \(\mu\in\Aut(\A)\). The data presented here upgrades \(\Rep^{\Aut(\A)}(\A)\) to an involutive \(\Aut(\A)\)-crossed balanced \(\mathrm{W}^*\)-tensor category.

If \(G\) is a discrete group acting on \(\A\) by a group homomorphism \(\Phi: G\to \Aut(\A)\), we obtain that \(\Rep^G(\A)\) is canonically a \(G\)-crossed balanced involutive \(\mathrm{W}^*\)-tensor category.

We are now ready to show that the equivalence of balanced \(\mathrm{W}^*\)-tensor categories \(\Rep(\A^G)\cong (\Rep^G(\A))^G\) in Theorem 9, for \(\A\) being acted on faithfully by a finite group \(G\), can be upgraded to an equivalence of balanced involutive \(\mathrm{W}^*\)-tensor categories. We will need the following two lemmas.

We write \(\upsilon\) for the anti-tensor structure of the involution on \(\Rep(\A^G)\), so that we can reserve \(\nu\) for the anti-tensor structure of the involution on \(\Rep^G(\A)\) and \(\nu^G\) for the anti-tensor structure on the equivariantization \((\Rep^G(\A))^G\). We can now prove the main result of the Appendix.

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