# Wightman fields for two-dimensional conformal field theories with pointed
representation category

Maria Stella Adamo<EMAIL_ADDRESS>Department of Mathematical
Sciences, The University of Tokyo,
3-8-1 Komaba, Tokyo, 153-8914, JAPAN  Luca Giorgetti<EMAIL_ADDRESS>Dipartimento di Matematica, Università di Roma Tor Vergata,
Via della Ricerca Scientifica 1, I-00133 Roma, Italy  Yoh Tanimoto
<EMAIL_ADDRESS>Dipartimento di Matematica, Università di Roma Tor
Vergata,
Via della Ricerca Scientifica 1, I-00133 Roma, Italy

###### Abstract

Two-dimensional full conformal field theories have been studied in various
mathematical frameworks, from algebraic, operator-algebraic to categorical. In
this work, we focus our attention on theories with chiral components having
pointed braided tensor representation subcategories, namely having
automorphisms whose equivalence classes necessarily form an abelian group. For
such theories, we exhibit the explicit Hilbert space structure and construct
primary fields as Wightman fields for the two-dimensional full theory. Given a
finite collection of chiral components with automorphism categories with
trivial total braiding, we also construct a local extension of their tensor
product as a chiral component. We clarify the relations with the Longo–Rehren
construction, and illustrate these results with concrete examples including
the ${\mathrm{U}}(1)$-current.

Dedicated to Roberto Longo in occasion of his 70th birthday

##  1 Introduction

Two-dimensional conformal field theories (CFTs) have been studied extensively
[DMS97] and have attracted the interest of mathematicians for their algebraic,
analytic and geometric structures. In particular, the conformal symmetry in
two-dimensional Minkowski spacetime is described by the diffeomorphism group
of the lightrays, therefore it is infinite-dimensional. This allows to study
first the theories that depend only on one of the lightray coordinates (the
chiral components) and then their two-dimensional (full) extensions.

From the operator-algebraic point of view (Haag–Kastler axioms), a general
quantum field theory can be formulated as a net of von Neumann algebras
associated with the open spacetime regions [Haa96]. The relations between the
full theory and the chiral components have been obtained, e.g., in [Reh00],
and they resulted in classification schemes for certain classes of two-
dimensional CFTs, see, e.g., [KL04b], [BKL15]. In this course, it was
important that from two copies of a single chiral theory and a family of
charged sectors, one can construct two-dimensional theories as extensions of
the two-dimensional theory obtained just by taking tensor products of the
chiral components, and then extended it by introducing non-chiral (bulk)
fields. Among such extensions, the most studied ones are the “diagonal”
extensions, obtained by letting the chiral theories act simultaneously on a
direct sum of copies of their vacuum representation, and then introducing
“charged” fields that mix the different components.

Since [LR95], in the operator-algebraic setting, the (finite index, i.e.,
“relatively small”) extensions can be equivalently well described by Q-systems
(i.e., $C^{*}$-Frobenius algebra objects) in the unitary braided tensor
representation category of the net that one wants to extend. Locality of the
extension can also be characterized by means of a commutativity condition on
the associated Q-system. In the operator-algebras context, the Q-system
associated with a finite index “diagonal” extension is called a Longo–Rehren
Q-system.

It is worth mentioning that the method of Q-systems applies both to chiral and
to two-dimensional theories (in fact even in four-dimensions), and that
similar ideas (commutative Frobenius algebra objects in tensor categories)
emerged independently in other algebraic approaches to CFT, such as vertex
operator algebras [HKL15], [KO02], both in one and two dimensions, see, e.g.,
[FRS02], [HK07], [Kon07], [RFFS07]. In the unitary VOA [CKLW18], [DL14],
[Gui22] and unitary tensor category context [GLR85], [DR89b], [LR97], these
notions (Frobenius algebras, $C^{*}$-Frobenius algebras, Q-systems, to
describe extensions) have been recently shown to be equivalent [CGGH23].

The relationship between chiral and full two-dimensional CFTs (not restricted
to a single Minkowski spacetime) can also be cast and studied [BGS22] in the
more general categorical/operadic framework of locally covariant AQFT [BFV03],
[BSW19], [BSW21].

Let us note, however, that most of the purely algebraic frameworks, see, e.g.,
[HK07], are designed particularly for conformal field theories, and do not
apply, as they are, to massive theories. On the other hand, it is natural to
expect that a two-dimensional full CFT can be described in a more traditional
framework for quantum field theory, the Wightman axioms [SW00]. Wightman
fields are operator-valued distributions on a Hilbert space, and a reasonable
description of such fields and the Hilbert space would be desirable.

Having explicit Wightman fields is not just interesting on its own, but could
be a starting point for constructing non-conformal field theory, e.g., by
perturbing the dynamics of the CFT by these fields. Such an idea is presented
in [Zam89], where certain massive integrable fields are associated with
charged fields in CFT. Perturbing the dynamics of the free field on the same
Hilbert space has been carried out on the de Sitter space [BJM23]. Therefore,
developing a theory of Wightman fields in two dimensions will be a basis for
rigorously studying the relations between CFT and massive models [JT23].

In this paper, we study full two-dimensional CFTs whose chiral components
admit a (finite or infinite) collection of automorphisms (invertible objects
in the language of tensor categories) among their irreducible representations
(superselection sectors). We will define and construct both two-dimensional
conformal Haag–Kastler nets and two-dimensional conformal Wightman fields
explicitly in terms of the Hilbert space of the chiral components and of their
charged sectors. We introduce charged primary fields as operators between
different charged sectors of the chiral components, and combine them to obtain
local bulk fields.

To be more specific, let ${\mathcal{A}}_{\mathrm{L}}$ and
${\mathcal{A}}_{\mathrm{R}}$ be chiral conformal nets on $S^{1}$ admitting a
family of automorphisms (including the defining vacuum representation) among
their charged representations (in the sense of Doplicher–Haag–Roberts
[DHR69a], [DHR69b], [DHR71] [DHR74], but in one and two dimensions instead of
four) parametrized by the same abelian group $G$. These automorphisms,
respectively denoted by ${\mathrm{L}}(g)$ and ${\mathrm{R}}(g)$, $g\in G$, are
defined on the same Hilbert spaces, but we denote them as
${\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(g)},{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(g)}$
to distinguish the representation. On the Hilbert space $\bigoplus_{g\in
G}{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(g)}\otimes{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(g)}$,
the chiral observables
${\mathcal{A}}_{\mathrm{L}}\otimes{\mathcal{A}}_{\mathrm{R}}$ act diagonally.
We will add charged fields that shift the sectors in the direct sum, under
some condition on their braiding (which is satisfied in many cases, see, e.g.,
the end of Section 6) and obtain full two-dimensional conformal nets and
conformal Wightman fields. Moreover, with a similar technique, we construct
some (presumably new) local conformal nets on $S^{1}$ by taking the tensor
product of a family of conformal nets and then extending it, in such a way
that a certain trivial total braiding condition is fulfilled. This generalizes
the well-known extensions of a single chiral ${\mathrm{U}}(1)$-current net
[BMT88], and it can be seen as a variation of the “gluing” construction due to
[CKM22] for VOAs.

We describe explicitly the fields for the ${\mathrm{U}}(1)$-current algebra.
Charged primary fields of a single chiral component are given as formal series
between charged sectors [TZ12], [TL97]. We expect that our construction works
for loop group nets (for a simply laced, simple, simply connected, compact
group) at level $1$ as well [Was98], [TL97]. With a more involved combination
of left and right chiral components, it should also be possibile to generalize
it to other completely rational nets (when the charged fields are available),
in the presence of irreducible representations with statistical dimension
greater that 1 (i.e., in the non-pointed tensor category case).

This paper is organized as follows. In Section 2, we recall the fact that two-
dimensional conformal field theories extend to the Einstein cylinder, then set
out the operator-algebraic formulation of the chiral components and of the
full two-dimensional CFTs. We also collect some facts about representations
and charged fields of chiral components, in the case of automorphisms. In
Section 3, from a family of chiral components equipped with a collection of
automorphisms and satisfying certain conditions on the braiding, we construct
an extension of their tensor product on $S^{1}$. In Section 4, we take a pair
of left and right chiral components with a collection of automorphisms and
assume that their braidings cancel in a certain sense, and we construct a full
two-dimensional CFT extending the tensor product of the chiral components. In
Section 5, we study the case where the charged primary fields are given as
formal series. Under similar assumptions on the braiding and assuming energy
bounds, we exhibit the Wightman fields responsible for the extensions at the
level of nets constructed in the previous sections. In Section 6, we consider
the explicit example of the ${\mathrm{U}}(1)$-current. We exhibit its
superselection charge structure, the associated braiding and charged fields,
and we show that they fit in the general construction of the previous
sections. In Section 7, we summarize our outlook.

### Notations

In order to keep uniform notations throughout the paper, we label chiral (one-
dimensional) objects, while we use symbols without a label for two-dimensional
objects.

  * •

${\mathcal{A}}_{\kappa},\phi_{\kappa,j},\psi_{\kappa},{\mathcal{A}}_{\mathrm{L}},{\mathcal{A}}_{\mathrm{R}},\psi_{\mathrm{L}},\psi_{\mathrm{R}}$:
chiral net/chiral fields/chiral charged fields.

  * •

${\mathcal{A}}_{K},\psi_{K}$: chiral extension/chiral charged local fields.

  * •

${\mathcal{A}}$: two-dimensional net.

  * •

$\widetilde{\mathcal{A}},\widetilde{\psi}$: two-dimensional net/two-
dimensional charged local fields.

##  2 Preliminaries

###  2.1 Einstein cylinder

As suggested by [LM75], it is natural to study four-dimensional conformally
covariant Wightman fields on the Einstein cylinder111More precisely, [LM75]
showed that, assuming that there are Wightman fields and their Euclidean
$n$-point functions are invariant under Euclidean conformal group, the
representation of the Poincaré group can be extended to $\mathrm{SO}(4,2)$.
See [KQR21, Section 8.3] for a recent review.. Moreover, the conformal group
acts on the Einstein cylinder in a natural way and the fields are covariant
with respect to its action. In two dimensions, an analogous procedure can be
carried out with one more step because the two-dimensional Einstein cylinder
is not simply connected. To state these results, let us first discuss the
conformal geometry. We follow [KL04b], and start the discussion with the
lightrays ${\mathbb{R}}$ in ${\mathbb{R}}^{1+1}$.

The Möbius group
${\mathrm{P}}{\mathrm{S}}{\mathrm{L}}(2,{\mathbb{R}})\cong\mathrm{M\ddot{o}b}$
acts on the one-point compactification $S^{1}$ of ${\mathbb{R}}$, where
${\mathbb{R}}$ is identified with $S^{1}\setminus\\{-1\\}$ through the
stereographic projection. Hence, by lifting such action, its universal
covering group $\overline{\mathrm{M\ddot{o}b}}$ acts on ${\mathbb{R}}$, the
universal covering of $S^{1}$. The original lightray ${\mathbb{R}}$ is
identified with the interval $(-\pi,\pi)$ in ${\mathbb{R}}$ as the universal
covering of $S^{1}$. For an interval $I$ such that
$\overline{I}\subset(-\pi,\pi)$, there is a neighborhood ${\mathcal{U}}$ of
the unit element in $\mathrm{M\ddot{o}b}$ such that if
$\gamma\in{\mathcal{U}}$, then $\gamma\cdot I\subset(-\pi,\pi)$. In this
sense, the group $\mathrm{M\ddot{o}b}$ acts locally on the lightray.

The two-dimensional Minkowski space ${\mathbb{R}}^{1+1}$ has the metric
$(a,b)=a_{0}b_{0}-a_{1}b_{1}$, where $a,b\in{\mathbb{R}}^{1+1}$. With the
lightcone coordinates
$(a_{-},a_{+})=\left(\frac{a_{0}-a_{1}}{\sqrt{2}},\frac{a_{0}+a_{1}}{\sqrt{2}}\right)$,
the Minkowski space is the product of two lightrays
${\mathbb{R}}^{1+1}={\mathbb{R}}\times{\mathbb{R}}$ (in the lightcone
coordinates, not the $(a_{0},a_{1})$-coordinates) and the metric can be
written as $(a,b)=a_{-}b_{+}+a_{+}b_{-}$. A conformal transformation of
${\mathbb{R}}^{1+1}$ is, by definition, a transformation of
${\mathbb{R}}^{1+1}$ that preserves the metric up to a scalar. From the above
expression, it is clear that a product of any pair of orientation-preserving
diffeomorphisms of lightrays is a conformal transformation.

The diamond $D_{0}=\\{(a_{-},a_{+}):-\pi<a_{\pm}<\pi\\}$ can be mapped to the
Minkowski space ${\mathbb{R}}^{1+1}$ by the conformal transformation
$(a_{-},a_{+})\mapsto(\tan(\frac{1}{2}a_{-}),\tan(\frac{1}{2}a_{+}))$. Through
this transformation, ${\mathbb{R}}^{1+1}$ can be identified with $D_{0}$. On
$D_{0}=(-\pi,\pi)\times(-\pi,\pi)$, the group
$\overline{\mathrm{M\ddot{o}b}}\times\overline{\mathrm{M\ddot{o}b}}$ acts
locally in the sense above.

Let $R_{t}$ be the lift of the rotation by $t$ in
$\overline{\mathrm{M\ddot{o}b}}$. In a two-dimensional conformal field theory,
the correlation functions are invariant under the local action of
$\overline{\mathrm{M\ddot{o}b}}\times\overline{\mathrm{M\ddot{o}b}}$, and
moreover, the spacelike $2\pi$-rotations $\mathfrak{R}:=\\{R_{2n\pi}\times
R_{-2n\pi}:n\in{\mathbb{Z}}\\}$ are often trivial. If this holds, the field
theory can be extended to the Einstein cylinder222This is topologically
equivalent to $S^{1}\times{\mathbb{R}}$, but the product structure is
different from the lightray decomposition.
${\mathcal{E}}={\mathbb{R}}^{1+1}/\mathfrak{R}$. In the most favorable case,
the theory extends to ${\mathcal{E}}$ and there is also a local action of the
conformal group
${\mathscr{C}}=\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}/\mathfrak{R}$.

$a_{1}$$a_{0}$${\mathbb{R}}^{1+1}$$a_{+}$$a_{-}$ Figure 1: The Minkowski space
${\mathbb{R}}^{1+1}$, depicted as the diamond. When the spacelike rotation
$R_{2\pi}\times R_{-2\pi}$ is trivial, the dotted lines are identified and it
is a subset of the Einstein cylinder ${\mathcal{E}}$.

###  2.2 Chiral components and representation theory

####  2.2.1 Conformal net on $S^{1}$

Among two-dimensional conformal fields, there are those that do not depend on
one of the lightray coordinates. They are called chiral fields, and can be
restricted to the lightray, then extended to the circle $S^{1}$ by locality.
We put the index $\kappa$ to the objects in this section, although we study a
single chiral field theory at a time. Note that we do not use the index
$\kappa$ to distinguish between left and right chiral components. This
convention will be useful later when we combine different chiral fields on
either component of a two-dimensional conformal field theory, while we denote
two-dimensional objects without index.

Let us start with a net on ${\mathbb{R}}$, and see how it extends to $S^{1}$.
Note that if a unitary projective $U_{\kappa}$ representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$ on a Hilbert space
${\mathcal{H}}_{\kappa}$ is restricted to the semisimple subgroup
$\overline{\mathrm{M\ddot{o}b}}$, there is a unique true (non-projective)
representation whose quotient in
${\mathrm{P}}{\mathrm{U}}({\mathcal{H}}_{\kappa})$ coincides with
$U_{\kappa}$. In this sense, we can consider the spectrum of $U_{\kappa}$
restricted to $\overline{\mathrm{M\ddot{o}b}}$ without ambiguity [Bar54,
Theorem 7.1].

We call a triple $({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ a
conformal net on ${\mathbb{R}}$ if ${\mathcal{A}}_{\kappa}$ assigns to each
open non-dense non-empty interval $I\subset{\mathbb{R}}$ a von Neumann algebra
${\mathcal{A}}_{\kappa}(I)$ on a Hilbert space ${\mathcal{H}}_{\kappa}$,
$U_{\kappa}$ is a unitary projective representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$ and
$\Omega_{\kappa}\in{\mathcal{H}}_{\kappa}$ such that

  1. (1dCN1)

Isotony: if $I_{1}\subset I_{2}$, then
${\mathcal{A}}_{\kappa}(I_{1})\subset{\mathcal{A}}_{\kappa}(I_{2})$.

  2. (1dCN2)

Locality: if $I_{1}$ and $I_{2}$ are disjoint, then
${\mathcal{A}}_{\kappa}(I_{1})\subset{\mathcal{A}}_{\kappa}(I_{2})^{\prime}$.

  3. (1dCN3)

Diffeomorphism covariance: For a bounded interval $I\subset{\mathbb{R}}$,
there is a neighborhood ${\mathcal{U}}$ of the unit element of
$\operatorname{Diff_{+}}(S^{1})$ such that if $\gamma\in{\mathcal{U}}$ then
$\gamma\cdot I\subset{\mathbb{R}}$ and

$U_{\kappa}(\gamma){\mathcal{A}}_{\kappa}(I)U_{\kappa}(\gamma)^{*}={\mathcal{A}}_{\kappa}(\gamma\cdot
I).$

Furthermore, if $\operatorname{supp}\gamma$ is disjoint from $I$, then
$\operatorname{Ad}U_{\kappa}(\gamma)(x)=x$ for
$x\in{\mathcal{A}}_{\kappa}(I)$.

  4. (1dCN4)

Positivity of energy: the restriction of $U_{\kappa}$ to the translation
subgroup ${\mathbb{R}}\subset\overline{\mathrm{M\ddot{o}b}}$ has the spectrum
contained in ${\mathbb{R}}_{+}$.

  5. (1dCN5)

Vacuum and the Reeh-Schlieder property: there exists a unique (up to a phase)
vector $\Omega_{\kappa}\in{\mathcal{H}}_{\kappa}$ such that
$U_{\kappa}(g)\Omega_{\kappa}=\Omega_{\kappa}$ for
$g\in\overline{\mathrm{M\ddot{o}b}}$ and
$\overline{{\mathcal{A}}_{\kappa}(I)\Omega_{\kappa}}={\mathcal{H}}_{\kappa}$.

If we assume only the covariance with respect to
$\overline{\mathrm{M\ddot{o}b}}$, we call it a Möbius-covariant net.

If $U_{\kappa}$ factors through $\operatorname{Diff_{+}}(S^{1})$ as a
projective representation, that is, if
$U_{\kappa}(R_{2n\pi}),n\in{\mathbb{Z}}$ is a scalar, then
${\mathcal{A}}_{\kappa}$ can be extended to a net defined on the set
${\mathcal{I}}$ of non-dense, non-trivial open intervals on $S^{1}$ and it
satisfies the usual axioms of conformal net on $S^{1}$, that is a triple
$({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ satisfying isotony,
locality, $\operatorname{Diff_{+}}(S^{1})$-covariance, positivity of energy
and the vacuum properties. See, e.g., [KL04a, Section 2.1], [CKLW18, Chapter
3].

We also recall that $U_{\kappa}$ can be made into a unitary multiplier
representation (rather than projective) of
$\overline{\operatorname{Diff_{+}}(S^{1})}$, that is,
$U_{\kappa}(\gamma_{1})U_{\kappa}(\gamma_{2})=c(\gamma_{1},\gamma_{2})U_{\kappa}(\gamma_{1}\gamma_{2})$
for some $c(\gamma_{1},\gamma_{2})\in{\mathbb{C}}$ [Car04, Theorem A.2],
[FH05, Proposition 5.1]. Such $c$ is called the cocycle of the multiplier
representation.

######  Proposition 2.1.

Let $({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ be a conformal net
on ${\mathbb{R}}$. Assume furthermore that $U_{\kappa}(R_{2\pi})=\mathbbm{1}$
(in ${\mathrm{P}}{\mathrm{U}}({\mathcal{H}}_{\kappa})$), that is, the lift of
$2\pi$-rotation is trivial. Then
$({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ extends to $S^{1}$.

###### Proof.

If $U_{\kappa}(R_{2\pi})=\mathbbm{1}$, then its adjoint action is trivial, and
the natural extension of ${\mathcal{A}}_{\kappa}$ on $(-\pi,\pi)$ to
${\mathbb{R}}$ (the universal covering of $S^{1}$) is periodic, hence we can
regard it as a net on $S^{1}$. It is known that the positivity of energy
restricted to the rotation subgroup and that to the translation subgroup are
equivalent (see, e.g., [Wei06, Lemma 3.1]). The remaining axioms of [KL04a,
Section 2.1] follow easily. ∎

The assumption $U_{\kappa}(R_{2\pi})=\mathbbm{1}$ is necessary, but we do not
know whether this follows from more general assumptions (cf. the
Bisognano–Wichmann property is necessary and sufficient [GLW98, Theorem 1.4],
but we are not aware whether it is automatic for conformal nets on
${\mathbb{R}}$).

There is an example of a net on ${\mathbb{R}}$ with a weaker covariance and
not extending to $S^{1}$ ($U_{\kappa}$ is only a projective representation of
the group generated by translations, dilations and diffeomorphisms of
${\mathbb{R}}$ with compact support): the ${\mathrm{U}}(1)$-current net with
the perturbed stress-energy tensor [BSM90], see also the discussion in [MT19,
Section 5.2].

####  2.2.2 Representations of chiral conformal nets

A representation of a conformal net ${\mathcal{A}}_{\kappa}$ (or more
precisely of $({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$) on $S^{1}$
is a family of representations $\rho=\\{\rho_{I}\\}$ of
$\\{{\mathcal{A}}_{\kappa}(I)\\}_{I\in{\mathcal{I}}}$ on a single Hilbert
space ${\mathcal{H}}_{\kappa}^{\rho}$, which is compatible in the sense that
if $I_{1}\subset I_{2}$, then
$\rho_{I_{2}}|_{{\mathcal{A}}_{\kappa}(I_{1})}=\rho_{I_{1}}$. If
${\mathcal{H}}_{\kappa}^{\rho}$ is equal to ${\mathcal{H}}_{\kappa}$ and
$\rho_{I}({\mathcal{A}}_{\kappa}(I))={\mathcal{A}}_{\kappa}(I)$, equivalently
if the $C^{*}$-tensor categorical or statistical dimension $d_{\rho}$ equals
$1$, we say that $\rho$ is an automorphism of ${\mathcal{A}}_{\kappa}$.

Let us summarize the general theory of representations in the case of
automorphisms (cf. [MTW18, Section 6]):

  * •

Any automorphism is irreducible, hence diffeomorphism covariant in the
following sense (cf. [DFK04, Theorem 6] for projective representations,
[CDVIT21, Section 3.2] for local multiplier representations): there is a local
unitary multiplier representation $U_{\kappa}^{\rho}$ of a neighborhood
${\mathcal{U}}$ of the unit element of
$\overline{\operatorname{Diff_{+}}(S^{1})}$ with the same cocycle as that of
$U_{\kappa}$ such that $U_{\kappa}^{\rho}(\gamma)=\rho(U_{\kappa}(\gamma))$ if
$\operatorname{supp}\gamma\subset I$ for some $I$ and
$\operatorname{Ad}U_{\kappa}^{\rho}(\gamma)(\rho(x))=\rho(\operatorname{Ad}U_{\kappa}(\gamma)(x))$
for $x\in{\mathcal{A}}_{\kappa}(I)$ for some $I$, $\gamma\in{\mathcal{U}}$
(the last restriction is why we call it a local representation).
$U_{\kappa}^{\rho}$ extends to $\overline{\operatorname{Diff_{+}}(S^{1})}$ as
a projective representation because
$\overline{\operatorname{Diff_{+}}(S^{1})}$ is simply connected. Moreover, its
restriction to $\overline{\mathrm{M\ddot{o}b}}$ is a true representation and
it has automatically positive energy [Wei06, Theorem 3.8]. Therefore, the
lowest eigenvalue of the generator of lift rotations $L_{0}^{\rho}$ is
uniquely determined, and we denote it by $D_{\kappa,\rho}$. This is called the
conformal dimension. As the $2\pi$-rotation is trivial, the spectrum of
$L_{0}^{\rho}$ is contained in $D_{\kappa,\rho}+({\mathbb{N}}\cup\\{0\\})$. By
the first part of the proof of [FH05, Theorem 5.1], the representation
$U_{\kappa}^{\rho}$ extends to a unitary multiplier representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$.

  * •

If $\rho,\rho^{\prime}$ are two representations of ${\mathcal{A}}_{\kappa}$
and there is a unitary $V\in{\mathrm{U}}({\mathcal{H}}_{\kappa})$ such that
$V\rho_{I}(x)=\rho_{I}^{\prime}(x)V$, for every
$x\in{\mathcal{A}}_{\kappa}(I)$ and for all $I\in{\mathcal{I}}$ we say that
they are unitarily equivalent. We also say that $V$ is a unitary intertwiner
between $\rho$ and $\rho^{\prime}$.

  * •

An automorphism is said to be localized in $I$ if
$\rho_{I^{\prime}}=\operatorname{id}$, where $I^{\prime}$ is the interior of
$S^{1}\setminus I$. In this case, $\rho_{I}$ maps ${\mathcal{A}}_{\kappa}(I)$
to itself, i.e., it is an endomorphism (in this case an automorphism) of
${\mathcal{A}}_{\kappa}(I)$, by Haag duality on $S^{1}$, [GF93, Theorem 2.19]
(i.e., ${\mathcal{A}}_{\kappa}(I^{\prime})={\mathcal{A}}_{\kappa}(I)^{\prime}$
for every $I\in{\mathcal{I}}$, cf. [BSM90]). Given $\rho$ localized in
$I_{1}$, one can always find a unitarily equivalent automorphism localized in
another interval $I_{2}$. (More generally, every representation of a conformal
net on a separable Hilbert space can be localized in any given interval,
[GF93, Lemma 4.6]). If $I_{1}\cup I_{2}$ is not dense in $S^{1}$, one can take
a proper interval $I$ that contains $I_{1}\cup I_{2}$, and by Haag duality on
$S^{1}$, the unitary operator $V$ implementing the equivalence belongs to
${\mathcal{A}}_{\kappa}(I)$. Such a $V$ is called a charge transporter.

  * •

The operator
$z_{\kappa,\rho}(\gamma):=U_{\kappa}(\gamma)U_{\kappa}^{\rho}(\gamma)^{*}$ for
$\gamma\in\overline{\operatorname{Diff_{+}}(S^{1})}$ is a charge transporter
between $\rho$ and $\rho^{\gamma}$, where
$\rho^{\gamma}:=\operatorname{Ad}U_{\kappa}(\gamma)\circ\rho\circ\operatorname{Ad}U_{\kappa}(\gamma^{-1})$
is localized in $\gamma\cdot I$ if $\rho$ is localized in $I$. We call
$z_{\kappa,\rho}(\gamma)$ the covariance cocycle, or just cocycle, of $\rho$.

  * •

The DHR tensor (or monoidal) product of representations $\rho_{1}$ and
$\rho_{2}$ is defined by the composition of the associated endomorphisms
(automorphisms in this case) localized in each $I$. Note that the tensor
product is not given by considering the tensor product Hilbert space, which
would not give a well-defined family of representations of the local algebras.
In symbols,
$(\rho_{1})_{I}\otimes(\rho_{2})_{I}:=(\rho_{1})_{I}\circ(\rho_{2})_{I}$, or
just $(\rho_{1})_{I}(\rho_{2})_{I}$, for short. The resulting representation
is denoted by $\rho_{1}\otimes\rho_{2}$, or just by $\rho_{1}\rho_{2}$. See,
e.g., [GF93, Section IV.2]. For strongly additive conformal nets
${\mathcal{A}}_{\kappa}$ on $S^{1}$ (i.e.,
${\mathcal{A}}_{\kappa}(I_{1})\vee{\mathcal{A}}_{\kappa}(I_{2})={\mathcal{A}}_{\kappa}(I)$
for every $I\in{\mathcal{I}}$ and $I_{1},I_{2}\in{\mathcal{I}}$ arising as
connected components of $I\smallsetminus\\{p\\}$, $p\in I$), or equivalently
for conformal nets on ${\mathbb{R}}$ that satisfy Haag duality on
${\mathbb{R}}$, the tensor product can be defined globally, see, [KLM01,
Appendix B], [GR18, Section 2], which is closer in spirit to the original
definition of Doplicher–Haag–Roberts [DHR69a], [DHR69b].

  * •

Let $\rho_{1},\rho_{2}$ be two automorphisms localized in $I$. In order to
define the DHR braiding [DHR71], [FRS89] between $\rho_{1}$ and $\rho_{2}$,
let us choose a point on $S^{1}$, identified with the point at infinity by
means of the corresponding stereographic projection onto ${\mathbb{R}}$. Take
$\widetilde{\rho}_{1},\widetilde{\rho}_{2}$ localized in
$\widetilde{I}_{1},\widetilde{I}_{2}$, respectively, such that
$\widetilde{I}_{1}\cap\widetilde{I}_{2}=\emptyset$,
$\widetilde{I}_{1},\widetilde{I}_{2}$ are away from infinity, and take charge
transporters $V_{1},V_{2}$ between $\rho_{1}$ and $\widetilde{\rho}_{1}$,
$\rho_{2}$ and $\widetilde{\rho}_{2}$, respectively. The operator
$\epsilon^{\pm}_{\rho_{1},\rho_{2}}:=\rho_{2}(V_{1}^{*})V_{2}^{*}V_{1}\rho_{1}(V_{2})$,
where $\pm$ depends on whether $I_{1}$ is on the left or right of $I_{2}$, is
a unitary intertwiner between $\rho_{1}\rho_{2}$ and $\rho_{2}\rho_{1}$ with
all the properties of a unitary braiding in a unitary tensor category. It does
not depend on the choice of $\widetilde{\rho}_{1},\widetilde{\rho}_{2}$ or
$V_{1},V_{2}$ under the same configuration of
$\widetilde{I}_{1},\widetilde{I}_{2}$, or on the chosen point at infinity,
see, e.g., [GF93, Section IV.4].

  * •

Let $\rho$ be an automorphism localized in $I$. The conjugate automorphism
$\overline{\rho}$ of $\rho$ is the automorphism localized in $I^{\prime}$
given by $j\circ\rho\circ j$, where $j=\operatorname{Ad}J_{I}$ and $J_{I}$ is
the modular conjugation of ${\mathcal{A}}_{\kappa}(I)$ with respect to the
vacuum. It follows that $\rho\circ\overline{\rho}\cong\operatorname{id}$, see
[GL92, Theorem 8.3], [GL96, Theorem 2.11].

Let $({\mathcal{B}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ be a conformal net
on $S^{1}$ and let $\\{{\mathcal{A}}_{\kappa}\\}_{I\in{\mathcal{I}}}$ be a
family of von Neumann subalgebras
${\mathcal{A}}_{\kappa}(I)\subset{\mathcal{B}}_{\kappa}(I)$ satisfying
covariance with respect to $U_{\kappa}$. Then, on the subspace
${\mathcal{H}}_{{\mathcal{A}}_{\kappa}}=\overline{\bigcup_{I\in{\mathcal{I}}}{\mathcal{A}}_{\kappa}(I)\Omega_{\kappa}}$,
$({\mathcal{A}}_{\kappa}|_{{\mathcal{H}}_{{\mathcal{A}}_{\kappa}}},U_{\kappa}|_{{\mathcal{H}}_{{\mathcal{A}}_{\kappa}}},\Omega_{\kappa})$
is a Möbius-covariant net with respect to $U_{\kappa}$ because
$U_{\kappa}(\gamma)x\Omega_{\kappa}=\operatorname{Ad}U_{\kappa}(\gamma)(x)\Omega_{\kappa}\in{\mathcal{A}}(\gamma\cdot
I)\Omega$ for $x\in{\mathcal{A}}_{\kappa}(I)$, thus $U_{\kappa}(\gamma)$
preserves ${\mathcal{H}}_{{\mathcal{A}}_{\kappa}}$. The restriction of
$U_{\kappa}|_{{\mathcal{H}}_{{\mathcal{A}}_{\kappa}}}$ to
$\mathrm{M\ddot{o}b}$ often extends to $\operatorname{Diff_{+}}(S^{1})$ and
${\mathcal{A}}_{\kappa}|_{{\mathcal{H}}_{{\mathcal{A}}_{\kappa}}}$ is
covariant with respect to it, in which case we say that
${\mathcal{A}}_{\kappa}$ is a (conformal) subnet of ${\mathcal{B}}_{\kappa}$
and write ${\mathcal{A}}_{\kappa}\subset{\mathcal{B}}_{\kappa}$ for
simplicity. In this case, we say that ${\mathcal{B}}_{\kappa}$ is an extension
of ${\mathcal{A}}_{\kappa}$.

####  2.2.3 Charged fields associated with automorphisms

Let $\\{({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})\\}$ be a conformal
net on $S^{1}$. In this section, we assume that among its irreducible
representations there are non-trivial automorphisms. In the language of tensor
categories, the representations of ${\mathcal{A}}_{\kappa}$ contain a
(braided) pointed tensor subcategory.

The construction of charged fields and extensions performed in this section is
inspired by [DHR69b] and [DR89a], [DR90] in the four-dimensional context, cf.
[Bau95, Chapter 3], where the unitary tensor categories in question are in
addition symmetrically braided (hence dual to a compact group [DR90]). In
[DHR69b] the pointed case is investigated, as in our case. See also [Müg01]
for comparison with the one and two-dimensional context.

Fix an interval $I\in{\mathcal{I}}$. Let $\Delta_{\kappa}$ be a choice of
mutually inequivalent automorphisms of ${\mathcal{A}}_{\kappa}$ localized in
$I$, one for each unitary equivalence class, including the trivial
automorphism $\operatorname{id}$ (the defining vacuum representation) of
${\mathcal{A}}_{\kappa}$. Under the present assumption, the equivalence
classes of automorphisms form a _discrete (finite or infinite) abelian group_
$G$ under class multiplication $[\rho][\sigma]=[\rho\sigma]$ (where
$\rho\sigma$ is the composition of automorphisms) and inversion
$[\rho]^{-1}=[\rho^{-1}]$. Let us assume that $G$ is _finitely generated_ ,
whose elements we denote by $g,h,g^{-1},\ldots$, and we denote by $\iota$ the
identity element. Let $\kappa(g)\in\Delta_{\kappa}$, for every $g\in G$, be
the previously made choice of automorphisms of ${\mathcal{A}}_{\kappa}$. We
assume that $\kappa(\iota)=\operatorname{id}$. Without loss of generality, by
suitably changing the localization of the automorphisms inside $I$, we may
assume that $\kappa(g)\kappa(h)=\kappa(h)\kappa(g)$ for every $g,h\in G$. Note
that we are _not_ assuming $\kappa(g)\kappa(h)=\kappa(gh)$, as
$\Delta_{\kappa}$ need _not_ be closed under composition and inverses. For
every $g,h\in G$, let $V^{g,h}$ be a unitary intertwiner in
${\mathcal{A}}_{\kappa}(I)$ between $\kappa(gh)$ and $\kappa(g)\kappa(h)$
realizing the equivalence $[\kappa(gh)]=[\kappa(g)][\kappa(h)]$. Namely,
$V^{g,h}\kappa(gh)(x)=\kappa(g)(\kappa(h)(x))V^{g,h}$ for every
$x\in{\mathcal{A}}_{\kappa}(I)$. We may assume that $V^{g,h}=V^{h,g}$ and that
$V^{g,\iota}=V^{\iota,h}=1$.

For each $g\in G$, we define the operator $\psi_{\kappa}^{g}$ acting on the
Hilbert space $\hat{\mathcal{H}}_{\kappa}:=\bigoplus_{g\in
G}{\mathcal{H}}_{\kappa}^{\kappa(g)}$, where
${\mathcal{H}}_{\kappa}^{\kappa(g)}:={\mathcal{H}}_{\kappa}$ the vacuum
Hilbert space of ${\mathcal{A}}_{\kappa}$, by setting

$\displaystyle(\psi_{\kappa}^{g}\Psi)_{h}:=V^{g,h}(\Psi)_{gh},$

for every $\Psi\in\hat{\mathcal{H}}_{\kappa}$. We call the $\psi_{\kappa}^{g}$
charged field operators. They satisfy the following:

  * •

Each $\psi_{\kappa}^{g}$ is unitary on $\hat{\mathcal{H}}_{\kappa}$ and it
maps ${\mathcal{H}}_{\kappa}^{\kappa(h)}$ to
${\mathcal{H}}_{\kappa}^{\kappa(gh)}$ for every $h\in G$.

  * •

Consider the representation $\hat{\kappa}(x):=\bigoplus_{h\in G}\kappa(h)(x)$
of $x\in{\mathcal{A}}_{\kappa}(I)$ on $\hat{\mathcal{H}}_{\kappa}$. Then

$\displaystyle(\psi_{\kappa}^{g}\hat{\kappa}(x)\Psi)_{h}$
$\displaystyle=V^{g,h}\left(\hat{\kappa}(x)\Psi\right)_{gh}$
$\displaystyle=V^{g,h}\kappa(gh)(x)\left(\Psi\right)_{gh}$
$\displaystyle=\kappa(g)(\kappa(h)(x))V^{g,h}\left(\Psi\right)_{gh}$
$\displaystyle=\kappa(h)(\kappa(g)(x))V^{g,h}\left(\Psi\right)_{gh}$
$\displaystyle=\kappa(h)(\kappa(g)(x))\left(\psi_{\kappa}^{g}\Psi\right)_{h}$
$\displaystyle=\left(\hat{\kappa}(\kappa(g)(x))\psi_{\kappa}^{g}\Psi\right)_{h},$

for every $\Psi\in\hat{\mathcal{H}}_{\kappa}$, where we used that $\kappa(h)$
and $\kappa(g)$ commute in the 4th equality.

Therefore, we have the “charged field intertwiner” property (in the sense of
Doplicher–Roberts [DR72]) together with its conjugate (by substituting $x$
with $\kappa(g)^{-1}(x^{*})$):

$\displaystyle\psi_{\kappa}^{g}\hat{\kappa}(x)$
$\displaystyle=\hat{\kappa}(\kappa(g)(x))\psi_{\kappa}^{g},$ (2.1)
$\displaystyle(\psi_{\kappa}^{g})^{*}\hat{\kappa}(x)$
$\displaystyle=\hat{\kappa}(\kappa(g)^{-1}(x))(\psi_{\kappa}^{g})^{*},$

for every $x\in{\mathcal{A}}_{\kappa}(I)$, as operators on
$\hat{\mathcal{H}}_{\kappa}$.

  * •

The covariance cocycles $z_{\kappa(g)}(\gamma)$ for
$\gamma\in\overline{\operatorname{Diff_{+}}(S^{1})}$ are unitary charged
transporters between $\kappa(g)$ and $\kappa(g)^{\gamma}$, and they fulfill
the following tensoriality property.

_Tensoriality of cocycles_ :
$z_{\kappa(g)}(\gamma)\kappa(g)(z_{\kappa(h)}(\gamma))=z_{\kappa(g)\kappa(h)}(\gamma)$,
or equivalently written as $z_{\kappa(g)}(\gamma)\otimes
z_{\kappa(h)}(\gamma)=z_{\kappa(g)\otimes\kappa(h)}(\gamma)$.

See [Lon97, Appendix A], [DG18, Section 7], [MTW18, Proposition 6.1].

  * •

Let $\hat{U}_{\kappa}(\gamma):=\bigoplus_{h\in
G}U_{\kappa}^{\kappa(h)}(\gamma)$ be a representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$. As in [DG18, Theorem 7.7],
[MTW18, Section 6], we get

$\displaystyle(\hat{U}_{\kappa}(\gamma)\psi_{\kappa}^{g}\hat{U}_{\kappa}(\gamma)^{*}\Psi)_{h}$
$\displaystyle=U_{\kappa}^{\kappa(h)}(\gamma)(\psi_{\kappa}^{g}\hat{U}_{\kappa}(\gamma)^{*}\Psi)_{h}$
$\displaystyle=U_{\kappa}^{\kappa(h)}(\gamma)V^{g,h}(\hat{U}_{\kappa}(\gamma)^{*}\Psi)_{gh}$
$\displaystyle=U_{\kappa}^{\kappa(h)}(\gamma)V^{g,h}U_{\kappa}^{\kappa(gh)}(\gamma)^{*}(\Psi)_{gh}$
$\displaystyle=U_{\kappa}^{\kappa(h)}(\gamma)U_{\kappa}^{\kappa(g)\kappa(h)}(\gamma)^{*}V^{g,h}(\Psi)_{gh}$
$\displaystyle=U_{\kappa}^{\kappa(h)}(\gamma)U_{\kappa}^{\kappa(\iota)}(\gamma)^{*}U_{\kappa}^{\kappa(\iota)}(\gamma)U_{\kappa}^{\kappa(g)\kappa(h)}(\gamma)^{*}V^{g,h}(\Psi)_{gh}$
$\displaystyle=z_{\kappa(h)}(\gamma)^{*}z_{\kappa(h)\kappa(g)}(\gamma)V^{g,h}(\Psi)_{gh}$
$\displaystyle=\kappa(h)(z_{\kappa(g)}(\gamma))V^{g,h}(\Psi)_{gh}$
$\displaystyle=(\hat{\kappa}(z_{\kappa(g)}(\gamma))\psi_{\kappa}^{g}\Psi)_{h},$

for every $\Psi\in\hat{\mathcal{H}}_{\kappa}$. We used that
$U_{\kappa}^{\kappa(g)}(\gamma)=\kappa(g)(U_{\kappa}(\gamma))$ if
$\operatorname{supp}\gamma\subset I$ in the 4th equality (see Section 2.2.2),
the definition of $z_{\kappa(g)}(\gamma)$ in the 6th equality, and the
tensoriality of the cocycles in the 7th equality.

Therefore, we have the following “covariance property” of the charged fields:
for every $\gamma\in\overline{\operatorname{Diff_{+}}(S^{1})}$, as operators
on $\hat{\mathcal{H}}_{\kappa}$,

$\displaystyle\operatorname{Ad}\hat{U}_{\kappa}(\gamma)(\psi_{\kappa}^{g})$
$\displaystyle=\hat{\kappa}(z_{\kappa(g)}(\gamma))\psi_{\kappa}^{g},$ (2.2)
$\displaystyle\operatorname{Ad}\hat{U}_{\kappa}(\gamma)((\psi_{\kappa}^{g})^{*})$
$\displaystyle=\hat{\kappa}(z_{\kappa(g)^{-1}}(\gamma))(\psi_{\kappa}^{g})^{*},$

where we used that
$(\psi_{\kappa}^{g})^{*}\hat{\kappa}(z_{\kappa(g)}(\gamma)^{*})=\hat{\kappa}(\kappa(g)^{-1}(z_{\kappa(g)}(\gamma)^{*}))(\psi_{\kappa}^{g})^{*}=\hat{\kappa}(z_{\kappa(g)^{-1}}(\gamma))(\psi_{\kappa}^{g})^{*}$
by the charged field intertwiner property of $(\psi_{\kappa}^{g})^{*}$ and the
tensoriality of cocycle.

######  Remark 2.2.

Note that the charged fields constructed above are highly non-canonical, as
they depend, e.g., on the choice of $V^{g,h}$ for every $g,h\in G$ (each
$V^{g,h}$ is unique up to unitary equivalence). Note also that we are
demanding neither $\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{gh}$, nor
${\psi_{\kappa}^{g}}^{*}=\psi_{\kappa}^{g^{-1}}$ for $g,h\in G$, i.e., the
charged field operators need not form a group, cf. [Reh90a, Section 3].
Nevertheless, by the intertwiner and covariance properties, for every choice
of $V^{g,h}$, they give rise to _relatively local_ (but possibly neither
irreducible, nor factorial) extensions of ${\mathcal{A}}_{\kappa}$.

Later we shall need the condition
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}\psi_{\kappa}^{g}$
(established in Proposition 2.3 below) on the charged fields just defined, in
order to produce local extensions of tensor products by “gluing” conformal
nets. Note that
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}\psi_{\kappa}^{g}$,
alone, does not mean locality of the extension. Note also that the group
multiplication condition
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{gh}$ would imply the
commutativity condition
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}\psi_{\kappa}^{g}$, as
$G$ is abelian.

######  Proposition 2.3.

In the notation of this section, suppose that $G={\mathbb{Z}}_{n}$ whose
elements we label by $\\{0,1,\ldots,n-1\\}$ modulo $n$, and let
$\alpha\in\Delta_{\kappa}$ be an automorphism localized in $I$ such that
$\alpha^{k}\in\Delta_{\kappa}$ for all $k=0,1,\ldots,n-1$ and
$[\alpha^{n}]=[\operatorname{id}]$. Choose a unitary
$V\in{\mathcal{A}}_{\kappa}(I)$ intertwining $\operatorname{id}$ with
$\alpha^{n}$, i.e., $Vx=\alpha^{n}(x)V$ for every
$x\in{\mathcal{A}}_{\kappa}(I)$. Let the $V^{g,h}$, $g,h\in G$, be defined by
$V^{g,h}:=V$ if $g+h\geq n$, where $g,h\in\\{0,1,\cdots,n-1\\}$ are thought of
as representatives, and by $V^{g,h}:=\mathbbm{1}$ if $g+h<n$. Then
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}\psi_{\kappa}^{g}$ and
$(\psi_{\kappa}^{g})^{*}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}(\psi_{\kappa}^{g})^{*}$
for every $g,h\in G$.

###### Proof.

Only in this proof, we use the additive notation $g+h$ for the group
operation.

We first note that with our choice
$V^{g,h}\kappa(g+h)=\kappa(g)\kappa(h)V^{g,h}$. By taking the representative
$g,h\in\\{0,1,\ldots,n-1\\}$, this can be checked by cases depending on
whether $g+h\geq n$ or $g+h<n$ and using that $\kappa(g)=\kappa(1)^{g}$.

From the definition, it is straightforward that $\psi^{1}_{\kappa}$ (where the
upper index $1$ is the group element, not the exponent) is the product of a
shift on $\bigoplus_{g\in G}{\mathcal{H}}_{\kappa}^{\kappa(g)}=\bigoplus_{g\in
G}{\mathcal{H}}_{\kappa}$ as a Hilbert space followed by $1$ or $V$, and that
$\psi_{\kappa}^{g}=(\psi_{\kappa}^{1})^{g}$, again with
$g\in\\{0,1,\ldots,n-1\\}$. From this it is clear that
$\psi_{\kappa}^{g},(\psi_{\kappa}^{g})^{*}$ and $\psi_{\kappa}^{h}$ commute. ∎

Now let $G$ be a finitely generated abelian group. Then it is isomorphic to a
finite product of cyclic groups ${\mathbb{Z}}_{n}$ or ${\mathbb{Z}}$. Let us
assume that $\Delta_{\kappa}$ is a choice of mutually inequivalent
automorphisms of ${\mathcal{A}}_{\kappa}$ localized in $I$ and their
equivalence classes form $G$ as a group. For each of the finite cyclic groups
${\mathbb{Z}}_{n}$ in $G$, we can choose a generator in $\Delta_{\kappa}$
localized in a smaller interval $\underline{I}\subset I$. For each of these
generators, we choose $V^{h,g}\in{\mathcal{A}}(\underline{I})$ as in
Proposition 2.3, constructing the operator $\psi^{h}_{\kappa}$ for such
$h,g\in{\mathbb{Z}}_{n}$. We can make this choice for each copy of
${\mathbb{Z}}_{n}$ by choosing finitely many mutually disjoint smaller
intervals inside $I$, obtaining commuting operators
$\psi^{h}_{\kappa},\psi^{g}_{\kappa}$ where $g,h$ are in different finite
cyclic groups in $G$. For each of the copies of the infinite cyclic group
${\mathbb{Z}}$, we can take a generator $1$ and we may assume that
$\kappa(g)=\kappa(1)^{g}$ for all $g\in{\mathbb{Z}}$, thus we can set
$V^{g,h}:=\mathbbm{1}$ and
$\psi^{g}_{\kappa}\psi^{h}_{\kappa}=\psi^{gh}_{\kappa}$ for every
$g,h\in{\mathbb{Z}}$. Altogether, we obtain charged fields $\psi^{h}_{\kappa}$
for every $h\in G$, satisfying
$\psi^{h}_{\kappa}\psi^{g}_{\kappa}=\psi^{g}_{\kappa}\psi^{h}_{\kappa}$.

######  Remark 2.4.

Even if $G$ is not finitely generated, if there is a choice of
$\Delta_{\kappa}$ such that $\kappa(g)\kappa(h)=\kappa(gh)$, then one can
construct $\psi^{h}_{\kappa}$ satisfying
$\psi^{g}_{\kappa}\psi^{h}_{\kappa}=\psi^{gh}_{\kappa}=\psi^{h}_{\kappa}\psi^{g}_{\kappa}$
as in [MTW18, Section 6], thus $(\psi^{g})^{*}=\psi^{g^{-1}}$ and the rest of
our construction works as well. This happens with the
${\mathrm{U}}(1)$-current net with $G={\mathbb{Q}}$ or ${\mathbb{R}}$.

###  2.3 Two-dimensional conformal nets

In [KL04b], it was shown that any Haag–Kastler net on ${\mathbb{R}}^{1+1}$
that is locally conformally covariant extends to the Einstein cylinder, using
a slight modification of the conformal spin-statistics theorem [GL96], [MT19,
Theorem A.5]. To state this result precisely, let us start with a net on
${\mathbb{R}}^{1+1}$. We call a triple $({\mathcal{A}},U,\Omega)$ a conformal
net on ${\mathbb{R}}^{1+1}$ if ${\mathcal{A}}$ assigns to each open region
$O\subset{\mathbb{R}}^{1+1}$ a von Neumann algebra ${\mathcal{A}}(O)$ on a
Hilbert space ${\mathcal{H}}$, $U$ is a unitary projective representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$
and $\Omega\in{\mathcal{H}}$ satisfying

  1. (2dCN1)

Isotony: if $O_{1}\subset O_{2}$, then
${\mathcal{A}}(O_{1})\subset{\mathcal{A}}(O_{2})$.

  2. (2dCN2)

Locality: if $O_{1}$ and $O_{2}$ are spacelike separated, then
${\mathcal{A}}(O_{1})\subset{\mathcal{A}}(O_{2})^{\prime}$.

  3. (2dCN3)

Diffeomorphism covariance: For a bounded region $O\subset{\mathbb{R}}^{1+1}$,
there is a neighborhood ${\mathcal{U}}$ of the unit element of
$\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$
such that if $\gamma\in{\mathcal{U}}$ then $\gamma\cdot
O\subset{\mathbb{R}}^{1+1}$ and

$U(\gamma){\mathcal{A}}(O)U(\gamma)^{*}={\mathcal{A}}(\gamma\cdot O).$

Furthermore, if $\operatorname{supp}\gamma$ is disjoint from $O$, then
$\operatorname{Ad}U(\gamma)(x)=x$ for ${\mathcal{A}}(O)$.

  4. (2dCN4)

Positivity of energy: the restriction of $U$ to the translation subgroup
${\mathbb{R}}^{2}\subset\overline{\mathrm{M\ddot{o}b}}\times\overline{\mathrm{M\ddot{o}b}}$
has the joint spectrum contained in the closed forward light cone
$\overline{V_{+}}=\\{(a_{0},a_{1})\in{\mathbb{R}}^{1+1}:a_{0}^{2}-a_{1}^{2}\geq
0,a_{0}\geq 0\\}$.

  5. (2dCN5)

Vacuum and the Reeh-Schlieder property: there exists a unique (up to a phase)
vector $\Omega\in{\mathcal{H}}$ such that $U(g)\Omega=\Omega$ for
$g\in\overline{\mathrm{M\ddot{o}b}}$ and is cyclic for any local algebra,
namely $\overline{{\mathcal{A}}(O)\Omega}={\mathcal{H}}$.

If there is a conformal net on ${\mathbb{R}}^{1+1}$ as above, we can consider
it as a net on the diamond $D_{0}=(-\pi,\pi)\times(-\pi,\pi)$ as in Section
2.1. With this identification, the group
$\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$
acts on ${\mathbb{R}}\times{\mathbb{R}}$, and we can extend the net
${\mathcal{A}}$ by covariance. However, this extension is not very natural.
Indeed, it often happens that the conformal net extends to ${\mathcal{E}}$
rather than to ${\mathbb{R}}\times{\mathbb{R}}$. We say that
$({\mathcal{A}},U,\Omega)$ is a conformal net on ${\mathcal{E}}$ if
${\mathcal{A}}$ is defined for regions in ${\mathcal{E}}$, covariant with
respect to $U$ along the natural action of
$\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$
and local in the sense that ${\mathcal{A}}(O_{1})$ and ${\mathcal{A}}(O_{2})$
commute whenever there is a translation $\gamma$ of the cylinder such that
$D_{0}$ contains $\gamma\cdot O_{1},\gamma\cdot O_{2}$ and they are spacelike
there. A sufficient condition is given in [MT19, Theorem A.5] (the
Bisognano–Wichmann property for wedges), however, it is not immediate to check
it in the examples we construct. Instead, the following is easier to check and
gives immediately the desired extension.

######  Proposition 2.5.

Let $({\mathcal{A}},U,\Omega)$ be a conformal net on ${\mathbb{R}}^{1+1}$.
Assume furthermore that $U(R_{2\pi}\times R_{-2\pi})=\mathbbm{1}$ (in
${\mathrm{P}}{\mathrm{U}}({\mathcal{H}})$), that is, the spacelike
$2\pi$-rotation is trivial. Then $({\mathcal{A}},U,\Omega)$ extends to a
conformal net on ${\mathcal{E}}$.

###### Proof.

This is parallel to Proposition 2.1.

As $U(R_{2\pi}\times R_{-2\pi})=\mathbbm{1}$, the representation $U$ factors
through the group ${\mathscr{C}}$ (see Section 2.1). Furthermore, as
$U(R_{2\pi}\times R_{-2\pi})=\mathbbm{1}$, for any region $O$,
${\mathcal{A}}(O)$ and ${\mathcal{A}}(R_{2\pi}\times R_{-2\pi}\cdot O)$
coincide. Therefore, we can identify any point $x$ on
${\mathbb{R}}\times{\mathbb{R}}$ and $R_{2\pi}\times R_{-2\pi}\cdot x$, and
obtain the Einstein cylinder ${\mathcal{E}}$. Covariance follows by
definition, and locality in the sense above follows from covariance and
locality in $D_{0}$. ∎

In the situation of Proposition 2.5, our net $({\mathcal{A}},U,\Omega)$ is
equivalent to a local conformal net in the sense of [KL04b, Section 2].

Rehren introduced the maximal chiral nets of a two-dimensional conformal net
$({\mathcal{A}},U,\Omega)$ that extends to ${\mathcal{E}}$ [Reh00]: let
$I_{\mathrm{L}}\times I_{\mathrm{R}}$ be a diamond in ${\mathbb{R}}^{1+1}$.
Define
${\mathcal{A}}^{\text{max}}_{\mathrm{L}}(I_{\mathrm{L}})={\mathcal{A}}(I_{\mathrm{L}}\times
I_{\mathrm{R}})\cap
U(\iota\times\overline{\operatorname{Diff_{+}}(S^{1})})^{\prime}$, where
$\iota$ is the unit element of $\overline{\operatorname{Diff_{+}}(S^{1})}$.
Then ${\mathcal{A}}^{\text{max}}_{\mathrm{L}}$ is a priori a conformal net on
${\mathbb{R}}$ defined on the Hilbert subspace
${\mathcal{H}}_{\mathrm{L}}=\overline{{\mathcal{A}}^{\text{max}}_{\mathrm{L}}\Omega}$,
and satisfies the condition of Proposition 2.1, therefore, it extends to
$S^{1}$. Similarly, one can define ${\mathcal{A}}^{\text{max}}_{\mathrm{R}}$
on ${\mathcal{H}}_{\mathrm{R}}$. The original full net ${\mathcal{A}}$
contains the tensor product
${\mathcal{A}}^{\text{max}}_{\mathrm{L}}\otimes{\mathcal{A}}^{\text{max}}_{\mathrm{R}}$
on the subspace
${\mathcal{H}}_{\mathrm{L}}\otimes{\mathcal{H}}_{\mathrm{R}}\subset{\mathcal{H}}$
(more precisely, there is a natural injective homomorphism from
${\mathcal{A}}^{\text{max}}_{\mathrm{L}}\otimes{\mathcal{A}}^{\text{max}}_{\mathrm{R}}$
into ${\mathcal{A}}$). In this sense, a generic two-dimensional conformal net
${\mathcal{A}}$ is an extension of the tensor product net
${\mathcal{A}}^{\text{max}}_{\mathrm{L}}\otimes{\mathcal{A}}^{\text{max}}_{\mathrm{R}}$.

##  3 One-dimensional gluing from trivial total braiding

In this Section, from a family of conformal nets on $S^{1}$ satisfying certain
conditions, we construct _local_ extensions of their tensor products. Let us
start with a _finite_ collection conformal nets
$\\{({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})\\}_{\kappa\in K}$ on
$S^{1}$ labelled by $\kappa\in K$, $|K|<\infty$. We fix an interval
$I\in{\mathcal{I}}$. We assume that each ${\mathcal{A}}_{\kappa}$ admits a
(not necessarily finite) collection $\Delta_{\kappa}$ of mutually inequivalent
and commuting automorphisms localized in $I$, containing the trivial
automorphism $\operatorname{id}_{\kappa}$, and whose fusion rules (up to
unitary equivalence) are isomorphic to the _same_ abelian group $G$, as in
Section 2.2.3. This group $G$ should be either a finitely generated abelian
group, or we assume that we can choose elements in $\Delta_{\kappa}$ and
charged fields $\psi_{\kappa}^{h}$ as in Section 2.2.3, fulfilling the
conclusions of Proposition 2.3 (for each $\kappa\in K$). Denote also by
$\kappa:G\to\Delta_{\kappa}$ a bijection (not necessarily an isomorphism, as
each $\Delta_{\kappa}$ need not be closed under composition and inverses),
such that $\kappa(\iota)=\operatorname{id}_{\kappa}$, for each $\kappa\in K$.
Here $\kappa$ is used as an index as well as a map, with a slight abuse of
notations, hence $\kappa(g)$ is an automorphism of ${\mathcal{A}}_{\kappa}$.
Furthermore, we assume that, for all $g_{1},g_{2},g\in G$,

  * •

$\prod_{\kappa}\epsilon^{\pm}_{\kappa(g_{1}),\kappa(g_{2})}=1$,

  * •

$\prod_{\kappa}\epsilon^{\pm}_{\kappa(g_{1})^{-1},\kappa(g_{2})}=1$,

  * •

$\sum_{\kappa}D_{\kappa,\kappa(g)}\in{\mathbb{Z}}$,

where the choice of $\pm$ above is common for all $\kappa$, and we denoted by
the same symbol $\epsilon^{\pm}_{\kappa(g_{1}),\kappa(g_{2})}$ the phase
multiple of $\mathbbm{1}$ (the trivial intertwiner between
$\kappa(g_{1})\kappa(g_{2})$ and itself) associated with the braiding of
$\kappa(g_{1})$ and $\kappa(g_{2})$. This makes sense by the commutativity
assumption on $\Delta_{\kappa}$, namely
$\kappa(g_{1})\kappa(g_{2})=\kappa(g_{2})\kappa(g_{1})$.

Now we construct a conformal net ${\mathcal{A}}_{K}$ on $S^{1}$ as follows.

  * •

The Hilbert space of our net is ${\mathcal{H}}_{K}:=\bigoplus_{g\in
G}\bigotimes_{\kappa\in K}{\mathcal{H}}_{\kappa}^{\kappa(g)}$. On this space,
we let any operator of the form
$\bigotimes_{\kappa}x_{\kappa}\in\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}(I)$
act333To clarify the notation once more, $\kappa$ in $x_{\kappa}$ and
$\Delta_{\kappa}$ works as an index to distinguish different tensor
components, while $\kappa$ in $\kappa(g)$ indicates the choice of automorphism
in $\Delta_{\kappa}$ labelled by $g\in G$. Therefore, $\kappa(g)(x_{\kappa})$
is a bounded operator on ${\mathcal{H}}_{\kappa}^{g}$. In general, $\kappa$
works as an index except for $\kappa(g)$.  as $\bigoplus_{g\in
G}\bigotimes_{\kappa}\kappa(g)(x_{\kappa})$. Denote this representation of the
tensor product net $\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}$ by $K$.

  * •

We also consider an auxiliary tensor product space
$\hat{\mathcal{H}}_{K}:=\bigotimes_{\kappa\in K}\bigoplus_{g\in
G}{\mathcal{H}}_{\kappa}^{\kappa(g)}\supset{\mathcal{H}}_{K}$. The
representation of the element $\bigotimes_{\kappa}x_{\kappa}$ on this space is
denoted by $\hat{K}(\bigotimes_{\kappa}x_{\kappa})$. We have
$\hat{K}(\bigotimes_{\kappa}x_{\kappa})|_{{\mathcal{H}}_{K}}=K(\bigotimes_{\kappa}x_{\kappa})$.

  * •

The vacuum vector of the net ${\mathcal{A}}_{K}$ will be
$\Omega_{K}:=\bigotimes_{\kappa}\Omega_{\kappa}\in\bigotimes_{\kappa}{\mathcal{H}}_{\kappa}^{\kappa(\iota)}$,
belonging to the $\iota$-th component of ${\mathcal{H}}_{K}$.

  * •

The covariance is given by $U_{K}(\gamma):=\bigoplus_{g\in
G}\bigotimes_{\kappa}U_{\kappa}^{\kappa(g)}(\gamma)$, which is a unitary
multiplier representation of $\overline{\operatorname{Diff_{+}}(S^{1})}$. By
the condition that $\sum_{\kappa}D_{\kappa,\kappa(g)}\in{\mathbb{Z}}$, $U_{K}$
satisfies the assumptions of Proposition 2.1 and is a unitary multiplier
representation of $\overline{\operatorname{Diff_{+}}(S^{1})}$. Thus by
ignoring the phase, it is a unitary projective representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$. This also extends naturally to
$\hat{U}_{K}$ on $\hat{\mathcal{H}}_{K}$.

  * •

For each element $g\in G$, we introduce the charged field operator
$\psi_{K}^{g}$ as follows. Our Hilbert space
${\mathcal{H}}_{K}=\bigoplus_{g\in
G}\bigotimes_{\kappa}{\mathcal{H}}_{\kappa}^{\kappa(g)}$ is a “diagonal”
subspace of the auxiliary Hilbert space
$\hat{\mathcal{H}}_{K}=\bigotimes_{\kappa}\hat{\mathcal{H}}_{\kappa}=\bigotimes_{\kappa}\bigoplus_{g\in
G}{\mathcal{H}}_{\kappa}^{\kappa(g)}$ in a natural way, where recall that
$\hat{\mathcal{H}}_{\kappa}$ was defined and used in Section 2.2.3. Let

$\displaystyle\psi_{K}^{g}:=\bigotimes_{\kappa}\psi_{\kappa}^{g},$

where $\psi_{\kappa}^{g}$ are charged fields acting as in Section 2.2.3 on
$\hat{\mathcal{H}}_{\kappa}$, hence $\psi_{K}^{g}$ acts on
$\bigotimes_{\kappa}\hat{\mathcal{H}}_{\kappa}$ but preserves
${\mathcal{H}}_{K}$. Now, for the $\psi_{\kappa}^{g}$ we make the choice of
the $V^{g,h}$ leading to the conclusions of Proposition 2.3 and comments
thereafter. Namely, we choose charged fields such that
$\psi_{\kappa}^{g}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}\psi_{\kappa}^{g}$ and
$(\psi_{\kappa}^{g})^{*}\psi_{\kappa}^{h}=\psi_{\kappa}^{h}(\psi_{\kappa}^{g})^{*}$
for every $g,h\in G$.

  * •

From the charged intertwiner property (2.1), it follows that

$\displaystyle\psi_{K}^{g}\hat{K}(\bigotimes_{\kappa}x_{\kappa})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}\kappa(g)(x_{\kappa})\right)\psi_{K}^{g},$
(3.1) $\displaystyle(\psi_{K}^{g})^{*}\hat{K}(\bigotimes_{\kappa}x_{\kappa})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}\kappa(g)^{-1}(x_{\kappa})\right)(\psi_{K}^{g})^{*}.$
(3.2)

  * •

The local algebras are given as follows. For the interval $I$ fixed above, we
set

${\mathcal{A}}_{K}(I):=\hat{K}\left(\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}(I)\right)\vee\\{\psi_{K}^{g},(\psi_{K}^{g})^{*}\\}_{g\in
G}.$

For any other interval $\gamma\cdot I\subset{\mathbb{R}}$ given by the action
of a diffeomorphism $\gamma$, we set ${\mathcal{A}}_{K}(\gamma\cdot
I):=\operatorname{Ad}U_{K}(\gamma)({\mathcal{A}}_{K}(I))$. We will show below
that this net is well-defined, diffeomorphism covariant, local, and it extends
to $S^{1}$.

######  Theorem 3.1.

Let $\\{({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})\\}_{\kappa\in K}$
as above and assume that $\\{\Delta_{\kappa}\\}_{\kappa\in K}$, their braiding
and $D_{\kappa,\kappa(g)}$ satisfy the conditions stated at the beginning of
this section. Then the net $({\mathcal{A}}_{K},U_{K},\Omega_{K})$ satisfies
(1dCN1)–(1dCN5) and the condition of Proposition 2.1, hence provides a
conformal net on $S^{1}$, extending the chiral tensor product net
$\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}$.

###### Proof.

Although the calculations will be similar to those of [MTW18, Section 6], the
setting is different because we construct a chiral net on $S^{1}$, instead of
two-dimensional on ${\mathcal{E}}$. Let us present the proofs to exhibit why
this works for tensor products of $|K|$ factors.

Let us first compute on the auxiliary Hilbert space
$\bigotimes_{\kappa}\hat{\mathcal{H}}_{\kappa}$ with
$\hat{U}_{K}(\gamma):=\bigotimes_{\kappa}\hat{U}_{\kappa}(\gamma)$, where
$\hat{U}_{\kappa}(\gamma)=\bigoplus_{h\in G}U_{\kappa}^{\kappa(h)}(\gamma)$
was defined and used in Section 2.2.3. Using the covariance property of
charged fields (2.2) (i.e., the formula for the adjoint action of
$\hat{U}_{\kappa}(\gamma)$ on $\psi_{\kappa}^{g}$ acting on each
$\hat{\mathcal{H}}_{\kappa}$), we get

$\displaystyle\operatorname{Ad}\hat{U}_{K}(\gamma)(\psi_{K}^{g})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g)}(\gamma)\right)\psi_{K}^{g},$
$\displaystyle\operatorname{Ad}\hat{U}_{K}(\gamma)((\psi_{K}^{g})^{*})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g)^{-1}}(\gamma)\right)(\psi_{K}^{g})^{*}.$

Furthermore, we also have $\psi_{K}^{g}\psi_{K}^{h}=\psi_{K}^{h}\psi_{K}^{g}$,
$(\psi_{K}^{g})^{*}\psi_{K}^{h}=\psi_{K}^{h}(\psi_{K}^{g})^{*}$ by the
commutation condition we imposed on the charged fields $\psi_{\kappa}^{g}$,
$\psi_{\kappa}^{h}$, and $(\psi_{\kappa}^{g})^{*}$.

We have to make sure that ${\mathcal{A}}_{K}(\gamma\cdot I)$ is well-defined,
that is, the definition does not depend on the choice of $\gamma$. This
question reduces to whether
$\operatorname{Ad}U_{K}(\gamma)({\mathcal{A}}_{K}(I))={\mathcal{A}}_{K}(I)$ if
$\gamma\cdot I=I$. This is true because we can check the inclusion for
$\gamma$ with compact support and generating elements of
${\mathcal{A}}_{K}(I)$:

$\displaystyle\operatorname{Ad}U_{K}(\gamma)\left(K\left(\bigotimes_{\kappa}x_{\kappa}\right)\right)$
$\displaystyle=K\left(\bigotimes_{\kappa}\operatorname{Ad}U_{\kappa}(\gamma)(x)\right)\in{\mathcal{A}}_{K}(I),$
$\displaystyle\operatorname{Ad}\hat{U}_{K}(\gamma)(\psi_{K}^{g})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g)}(\gamma)\right)\psi_{K}^{g}\in{\mathcal{A}}_{K}(I),$
$\displaystyle\operatorname{Ad}\hat{U}_{K}(\gamma)((\psi_{K}^{g})^{*})$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g)^{-1}}(\gamma)\right)(\psi_{K}^{g})^{*}\in{\mathcal{A}}_{K}(I).$

because $z_{\kappa(g)}(\gamma)\in{\mathcal{A}}_{\kappa}(I)$. As a general
$\gamma$ can be written as a product of diffeomorphisms with compact supports,
this gives the inclusion
$\operatorname{Ad}U_{K}(\gamma)({\mathcal{A}}_{K}(I))\subset{\mathcal{A}}_{K}(I)$,
and the converse inclusion is obtained by applying this to $\gamma^{-1}$.

With this well-definedness, the first part of covariance (1dCN3) follows by
definition. Concerning the second part, if $\operatorname{supp}\gamma$ is
disjoint from $I$, then $U_{K}(\gamma)$ commutes both with
$\hat{K}\left(\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}(I)\right)$ by
covariance of ${\mathcal{A}}_{\kappa}$. Furthermore, $U_{K}(\gamma)$ commutes
with $\psi_{K}^{g}$ because componentwise $\psi_{K}^{g}$ is a product of the
shift and an element in $\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}(I)$, while
$U_{K}(\gamma)\in\hat{K}\left(\bigotimes_{\kappa}{\mathcal{A}}_{\kappa}(I^{\prime})\right)$
and this commutes with the shift because the $\kappa(g)$ are localized in $I$.

Positivity of energy (1dCN4) follows because $U_{K}$ is a direct sum of
positive-energy representations. The vacuum $\Omega_{K}$ is invariant under
$U_{K}|_{\mathrm{M\ddot{o}b}}$ because it is the tensor product of the vacuum
vectors of ${\mathcal{A}}_{\kappa}$. It is cyclic for
${\mathcal{A}}_{K}(I_{1})$, where $I_{1}\subset I^{\prime}$, because
${\mathcal{A}}_{K}(I_{1})\Omega_{K}$ spans
$\bigotimes_{\kappa}{\mathcal{H}}_{\kappa}$ due to the cyclicity of the vacua
of ${\mathcal{A}}_{\kappa}$, and the whole ${\mathcal{H}}_{K}$, since
${\mathcal{A}}_{K}(I_{1})$ contains the shifts
$\hat{K}(z_{g}(\gamma))\psi_{K}^{g}$ (up to a unitary on the left). From the
assumption $\sum_{\kappa}D_{\kappa,\kappa(g)}\in{\mathbb{Z}}$ for each $g\in
G$, we can apply Proposition 2.1.

As for locality, we take $\gamma$ such that $\gamma\cdot I$ is disjoint from
$I$. We have to show that generating elements
$\hat{K}(\bigotimes_{\kappa}x_{\kappa})$, $\psi^{g_{1}}_{K}$ of
${\mathcal{A}}_{K}(I)$ and
$\operatorname{Ad}U_{K}(\gamma)\left(\hat{K}(\bigotimes_{\kappa}y_{\kappa})\right)$,
$\operatorname{Ad}U_{K}(\gamma)\left(\psi^{g_{2}}_{K}\right)$ of
${\mathcal{A}}_{K}(\gamma\cdot I)$ commute. This is easy except the one
involving $\psi^{g_{1}}_{K}$ and
$\operatorname{Ad}U_{K}(\gamma)(\psi^{g_{2}}_{K})$. As we have seen,
$\operatorname{Ad}U_{K}(\gamma)(\psi_{K}^{g})=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g)}(\gamma)\right)\psi_{K}^{g}$.
Therefore, to compute the commutator,

$\displaystyle\psi_{K}^{g_{1}}\operatorname{Ad}U_{K}(\gamma)(\psi_{K}^{g_{2}})$
$\displaystyle=\psi_{K}^{g_{1}}\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g_{2})}(\gamma)\right)\psi_{K}^{g_{2}}$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}\kappa(g_{1})(z_{\kappa(g_{2})}(\gamma))\right)\psi_{K}^{g_{1}}\psi_{K}^{g_{2}},$

and

$\displaystyle\operatorname{Ad}U_{K}(\gamma)(\psi_{K}^{g_{2}})\psi_{K}^{g_{1}}$
$\displaystyle=\hat{K}\left(\bigotimes_{\kappa}z_{\kappa(g_{2})}(\gamma)\right)\psi_{K}^{g_{2}}\psi_{K}^{g_{1}}.$

Now observe that
$z_{\kappa(g_{2})}(\gamma)^{*}\kappa(g_{1})(z_{\kappa}(g_{2})(\gamma))=\epsilon^{\pm}_{\kappa(g_{1}),\kappa(g_{2})}$
(the DHR braiding, a scalar in the case of automorphisms $\kappa(g_{1})$ and
$\kappa(g_{2})$) where the $\pm$ sign depends only on whether $\gamma$ moves
$I$ to the left or to the right, and the choice of $\pm$ is common for all
$\kappa$). Hence the two left hand sides above are equal if and only if
$\prod_{\kappa}\epsilon^{\pm}_{\kappa(g_{1}),\kappa(g_{2})}=1$ and
$\psi_{K}^{g_{1}}\psi_{K}^{g_{2}}=\psi_{K}^{g_{2}}\psi_{K}^{g_{1}}$ for every
$g_{1},g_{2}\in G$. The commutation between $(\psi_{K}^{g_{1}})^{*}$ and
$\operatorname{Ad}U_{K}(\gamma)(\psi_{K}^{g_{2}})$ follows too by replacing
$\kappa(g_{1})$ by $\kappa(g_{1})^{-1}$ and using the condition
$\prod_{\kappa}\epsilon^{\pm}_{\kappa(g_{1})^{-1},\kappa(g_{2})}=1$ and
$(\psi_{K}^{g_{1}})^{*}\psi_{K}^{g_{2}}=\psi_{K}^{g_{2}}(\psi_{K}^{g_{1}})^{*}$.
Locality for general intervals follows from the previous paragraph and
covariance. ∎

If all ${\mathcal{A}}_{\kappa}$ are strongly additive, that is,
${\mathcal{A}}_{\kappa}(I_{3})={\mathcal{A}}_{\kappa}(I_{1})\vee{\mathcal{A}}_{\kappa}(I_{2})$
where $I_{1},I_{2}$ is obtained by removing one point from $I_{3}$, then
${\mathcal{A}}_{K}$ is strongly additive, too. This can be seen by taking
$I_{1}=I$, then ${\mathcal{A}}_{K}(I_{3})$ is generated by the same
$\psi_{K}^{g}\in{\mathcal{A}}_{K}(I)$ and $K(\bigotimes_{\kappa}x_{\kappa})$,
$x_{\kappa}\in{\mathcal{A}}_{\kappa}(I_{3})={\mathcal{A}}_{\kappa}(I)\vee{\mathcal{A}}_{\kappa}(I_{2})$.

This construction can be carried out even if there is only one index $\kappa$,
if the condition $\epsilon^{\pm}_{\kappa(g_{1}),\kappa(g_{2})}=1$ is
satisfied. When applied to the ${\mathrm{U}}(1)$-current net, this gives the
extensions considered in [BMT88], cf. Section 6, where $G={\mathbb{Z}}$, the
map $\kappa$ is a group isomorphism, and $\kappa(g_{1})$, $\kappa(g_{2})$ are
powers of the same fixed automorphism $\kappa(h)$ having
$\epsilon^{\pm}_{\kappa(h),\kappa(h)}=1$ (bosonic automorphism).

##  4 Two-dimensional conformal field theory arising from braiding-cancelling
map

With an idea similar to that of Section 3, we construct conformal nets on
${\mathbb{R}}^{1+1}$. Here we start with left and right chiral components,
take their tensor product as a two-dimensional conformal net, then find
extensions of it. The problem of constructing extensions on
${\mathbb{R}}^{1+1}$ by adding “charged fields” associated with suitable
subgroups of automorphisms on top of a tensor product theory has also been
tackled in [Reh90b, Section 5], see in particular [Reh90b, Proposition 5.5],
within the framework of the reduced field bundle [FRS89], [FRS92].

###  4.1 Extensions with pointed tensor category

Let us start with a pair of conformal nets
${\mathcal{A}}_{\mathrm{L}},{\mathcal{A}}_{\mathrm{R}}$ on $S^{1}$. We fix an
interval $I\subset S^{1}$. As in Section 3, we assume that
${\mathcal{A}}_{\mathrm{L}}$ and ${\mathcal{A}}_{\mathrm{R}}$ admit,
respectively, a family $\Delta_{\mathrm{L}},\Delta_{\mathrm{R}}$ (finite or
infinite) of mutually inequivalent and commuting automorphisms localized in
$I$, containing the trivial automorphism
$\operatorname{id}_{\mathrm{L}},\operatorname{id}_{\mathrm{R}}$, and whose
fusion rules (up to unitary equivalence) are isomorphic to the same abelian
group $G$, cf. Section 2.2.3. This group $G$ should be either finitely
generated, or we assume that we can choose elements in
$\Delta_{\mathrm{L}},\Delta_{\mathrm{R}}$ and charged fields
$\psi_{\mathrm{L}}^{g},\psi_{\mathrm{R}}^{g}$ with the commuting property as
in the conclusions of Proposition 2.3 (for $\kappa={\mathrm{L}},{\mathrm{R}}$)
and comments thereafter. In this section, we denote by
${\mathrm{L}},{\mathrm{R}}$ (instead of $\kappa$) the bijections from $G$ to
$\Delta_{\mathrm{L}},\Delta_{\mathrm{R}}$, respectively, corresponding to the
choice of automorphisms, and by $D_{{\mathrm{L}}(g)}$, $D_{{\mathrm{R}}(g)}$,
where $g\in G$, the conformal dimensions of the associated automorphisms.

We assume that

  * •

$\Delta_{\mathrm{L}}$ and $\Delta_{\mathrm{R}}$ contain the trivial
automorphisms $\operatorname{id}_{\mathrm{L}},\operatorname{id}_{\mathrm{R}}$
of ${\mathcal{A}}_{\mathrm{L}}$ and ${\mathcal{A}}_{\mathrm{R}}$,
corresponding to the identity element $\iota$ of $G$ via
${\mathrm{L}}(\iota)=\operatorname{id}_{\mathrm{L}}$,
${\mathrm{R}}(\iota)=\operatorname{id}_{\mathrm{R}}$.

  * •

For every $g_{1},g_{2}\in G$, it holds that
$\overline{\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})}}=\epsilon^{\mp}_{{\mathrm{R}}(g_{1}),{\mathrm{R}}(g_{2})}$
and
$\overline{\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})^{-1}}}=\epsilon^{\mp}_{{\mathrm{R}}(g_{1}),{\mathrm{R}}(g_{2})^{-1}}$
(braiding cancellation), and
$D_{{\mathrm{L}}(g)}-D_{{\mathrm{R}}(g)}\in\mathbb{Z}$ for every $g\in G$.

We denote by the same symbol
$\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})}$ the phase multiple
of $\mathbbm{1}$ (the trivial intertwiner between
${\mathrm{L}}(g_{1}){\mathrm{L}}(g_{2})={\mathrm{L}}(g_{2}){\mathrm{L}}(g_{1})$
and itself) associated with the braiding of ${\mathrm{L}}(g_{1})$ and
${\mathrm{L}}(g_{2})$. Similarly for ${\mathrm{R}}(g_{1})$ and
${\mathrm{R}}(g_{2})$.

######  Remark 4.1.

Let ${\mathcal{A}}_{\mathrm{L}}={\mathcal{A}}_{\mathrm{R}}$. If $G$ is cyclic,
then the braiding cancellation is automatic if we take
$\Delta_{\mathrm{L}}=\Delta_{\mathrm{R}}$ and
${\mathrm{L}}(g)={\mathrm{R}}(g)^{-1}$ for every $g\in G$. Indeed, in general,
for commuting automorphisms, it holds that
$\overline{\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})}}=\epsilon^{\mp}_{{\mathrm{L}}(g_{2})^{-1},{\mathrm{L}}(g_{1})^{-1}}$.
Now, for automorphisms ${\mathrm{L}}(g_{1})$, ${\mathrm{L}}(g_{2})$ that are
powers of the same ${\mathrm{L}}(h)$, namely
${\mathrm{L}}(g_{1})={\mathrm{L}}(h)^{n}$,
${\mathrm{L}}(g_{2})={\mathrm{L}}(h)^{m}$ for $n,m\in{\mathbb{Z}}$, it holds
$\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})}=(\epsilon^{\pm}_{{\mathrm{L}}(h),{\mathrm{L}}(h)})^{nm}=\epsilon^{\pm}_{{\mathrm{L}}(g_{2}),{\mathrm{L}}(g_{1})}$
(namely, the $nm$-th power of the statistical phase of ${\mathrm{L}}(h)$).
Hence $g_{1}$ and $g_{2}$ can be exchanged and we have braiding cancellation.

With this input, we construct a conformal net $\widetilde{\mathcal{A}}$ on
${\mathbb{R}}^{1+1}$ as follows.

  * •

The full Hilbert space is $\widetilde{\mathcal{H}}:=\bigoplus_{g\in
G}{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(g)}\otimes{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(g)}$.
On $\widetilde{\mathcal{H}}$, any operator of the form $x_{\mathrm{L}}\otimes
x_{\mathrm{R}}\in{\mathcal{A}}_{\mathrm{L}}(I_{\mathrm{L}})\otimes{\mathcal{A}}_{\mathrm{R}}(I_{\mathrm{R}})$
acts as $\bigoplus_{g\in
G}{\mathrm{L}}(g)(x_{\mathrm{L}})\otimes{\mathrm{R}}(g)(x_{\mathrm{R}})$ (see
the footnote 3 for this notation). Denote this representation by
$\widetilde{\tau}$.

  * •

The vacuum vector of $\widetilde{\mathcal{A}}$ is
$\Omega_{\mathrm{L}}\otimes\Omega_{\mathrm{R}}\in{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(\iota)}\otimes{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(\iota)}$.

  * •

The covariance is given by
$\widetilde{U}(\gamma_{\mathrm{L}}\times\gamma_{\mathrm{R}}):=\bigoplus_{g\in
G}U_{\mathrm{L}}^{{\mathrm{L}}(g)}(\gamma_{\mathrm{L}})\otimes
U_{\mathrm{R}}^{{\mathrm{R}}(g)}(\gamma_{\mathrm{R}})$, which is a unitary
projective representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$.
By the assumption that $D_{{\mathrm{L}}(g)}-D_{{\mathrm{R}}(g)}\in\mathbb{Z}$,
$\widetilde{U}$ satisfies the condition of Proposition 2.5.

  * •

Note that $\widetilde{\mathcal{H}}$ can be seen as a subspace of
$\hat{\mathcal{H}}_{\mathrm{L}}\otimes\hat{\mathcal{H}}_{\mathrm{R}}:=\left(\bigoplus_{g\in
G}{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(g)}\right)\otimes\left(\bigoplus_{g\in
G}{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(g)}\right)$ in a natural way. We
denote $\hat{\mathrm{L}}:=\bigoplus_{g\in
G}{\mathrm{L}}(g),\hat{\mathrm{R}}:=\bigoplus_{g\in G}{\mathrm{R}}(g)$.

  * •

Let $\psi_{\mathrm{L}}^{g},\psi_{\mathrm{R}}^{g}$ be as in Section 2.2.3, with
the $V^{g,h}$ chosen as in Proposition 2.3. For each $g\in G$, we introduce
the charge operator $\widetilde{\psi}^{g}$ as follows

$\displaystyle\widetilde{\psi}^{g}:=\psi_{\mathrm{L}}^{g}\otimes\psi_{\mathrm{R}}^{g}\in{\mathcal{B}}(\hat{\mathcal{H}}_{\mathrm{L}})\otimes{\mathcal{B}}(\hat{\mathcal{H}}_{\mathrm{R}}).$
(4.1)

It is then clear that $\widetilde{\psi}^{g}$ preserves
$\widetilde{\mathcal{H}}$.

From this definition and by (2.1), it follows that

$\displaystyle\widetilde{\psi}^{g}\hat{\mathrm{L}}(x_{\mathrm{L}})\otimes\hat{\mathrm{R}}(x_{\mathrm{R}})=\hat{\mathrm{L}}({\mathrm{L}}(g)(x_{\mathrm{L}}))\otimes\hat{\mathrm{R}}({\mathrm{R}}(g)(x_{\mathrm{R}}))\widetilde{\psi}^{g},$
(4.2)

just as in (3.1).

  * •

The local algebras of $\widetilde{\mathcal{A}}$ are given as follows: For the
diamond $I\times I$, we set $\widetilde{\mathcal{A}}(I\times
I):=\widetilde{\tau}({\mathcal{A}}_{\mathrm{L}}(I)\otimes{\mathcal{A}}_{\mathrm{R}}(I))\vee\\{\widetilde{\psi}^{g},(\widetilde{\psi}^{g})^{*}\\}_{g\in
G}$. For any other diamond $\gamma_{\mathrm{L}}\cdot
I\times\gamma_{\mathrm{R}}\cdot I$, we set
$\widetilde{\mathcal{A}}(\gamma_{\mathrm{L}}\cdot
I\times\gamma_{\mathrm{R}}\cdot
I):=\operatorname{Ad}\widetilde{U}(\gamma_{\mathrm{L}}\times\gamma_{\mathrm{R}})(\widetilde{\mathcal{A}}(I\times
I))$.

######  Theorem 4.2.

The net $(\widetilde{\mathcal{A}},\widetilde{U},\widetilde{\Omega})$ satisfies
(2dCN1)–(2dCN5) and the condition of Proposition 2.5, hence provides a
conformal net on ${\mathcal{E}}$, extending the two-dimensional conformal net
${\mathcal{A}}_{\mathrm{L}}\otimes{\mathcal{A}}_{\mathrm{R}}$.

###### Proof.

Most of the properties, well-definedness of $\widetilde{\mathcal{A}}$,
(2dCN1)–(2dCN5) except for (2dCN2) and the condition of Proposition 2.5 can be
verified as in [MTW18, Section 6], using the assumption
$D_{{\mathrm{L}}(g)}-D_{{\mathrm{R}}(g)}\in\mathbb{Z}$. As for locality, in
the proof of [MTW18, Section 6], it is only important that
$\overline{\epsilon^{\pm}_{{\mathrm{L}}(g_{1}),{\mathrm{L}}(g_{2})}}=\epsilon^{\mp}_{{\mathrm{R}}(g_{1}),{\mathrm{R}}(g_{2})}$
as scalars, which we assume. The commutation between
$(\widetilde{\psi}^{g_{1}})^{*}$ and
$\operatorname{Ad}U(\gamma_{\mathrm{L}}\times\gamma_{\mathrm{R}})(\widetilde{\psi}^{g_{2}})$
follows from
$\overline{\epsilon^{\pm}_{{\mathrm{L}}(g_{1})^{-1},{\mathrm{L}}(g_{2})}}=\epsilon^{\mp}_{{\mathrm{R}}(g_{1})^{-1},{\mathrm{R}}(g_{2})}$.
Therefore, the proof of locality (2dCN2) works as well. ∎

###  4.2 Comparison with Longo–Rehren extensions

In this section, we review the so-called Longo–Rehren extensions introduced in
[LR95, Proposition 4.10], see also [Mas00], [KLM01, Appendix A], [BKLR15,
Section 3.3] and references therein, and we interpret them as “generalized
shift constructions”. Later, we shall relate them to the extensions presented
in the previous sections.

Let ${\mathcal{N}}$ be an infinite factor (e.g.,
${\mathcal{N}}={\mathcal{A}}_{\kappa}(I)$ a local algebra of a conformal net
on $S^{1}$). Let ${\mathcal{C}}$ be a unitary fusion category (not necessarily
braided, for the moment) realized as a full subcategory of
$\mathrm{End}({\mathcal{N}})$, the set of normal injective unital
*-endomorphisms of ${\mathcal{N}}$. For the preliminaries on unitary fusion
categories (realized, without loss of generality, as endomorphisms of von
Neumann algebras) we refer, e.g., to [BKLR15], [EGNO15]. We use the notation
$\operatorname{Hom}_{\mathcal{C}}(\rho,\sigma)$ for the spaces of intertwiners
$V\in{\mathcal{N}}$ between $\rho$ and $\sigma$ objects in ${\mathcal{C}}$,
namely $V\rho(x)=\sigma(x)V$ for every $x\in{\mathcal{N}}$, the arrows in our
subcategory of endomorphism.

For every unitary equivalence class of irreducible objects in ${\mathcal{C}}$,
choose one representative $\rho_{i}$, $i=0,\ldots,n$, with
$\rho_{0}=\operatorname{id}$, and denote
$\Delta:=\\{\rho_{0},\ldots,\rho_{n}\\}$. Note that, if
${\mathcal{C}},{\mathcal{D}}\subset\mathrm{End}({\mathcal{N}})$ are two
unitary fusion categories, then the Deligne tensor product
${\mathcal{C}}\boxtimes{\mathcal{D}}$ [BK01, Definition 1.1.15] can be
realized in $\mathrm{End}({\mathcal{N}}\bar{\otimes}{\mathcal{N}})$, where
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}$ is the spatial tensor product444In
this section, we denote by $\bar{\otimes}$ the spatial tensor product of
operators, in order to distinguish it from the categorical tensor product
functor $\otimes$, which we shall employ below on intertwiners. von Neumann
algebra. Next, let ${\mathcal{J}}:{\mathcal{N}}\to{\mathcal{N}}^{\mathcal{J}}$
be an antilinear isomorphism of von Neumann algebras, where
${\mathcal{N}}^{\mathcal{J}}:={\mathcal{J}}({\mathcal{N}})$, e.g., the natural
involution ${\mathcal{J}}:x\mapsto x^{*}$, where
${\mathcal{N}}^{\mathcal{J}}\cong{\mathcal{N}}^{\textrm{opp}}$. Consider the
unitary fusion category
${\mathcal{C}}^{\mathcal{J}}\subset\mathrm{End}({\mathcal{N}}^{\mathcal{J}})$
with objects
$\rho^{\mathcal{J}}:={\mathcal{J}}\circ\rho\circ{\mathcal{J}}^{-1}\in\mathrm{End}({\mathcal{N}}^{\mathcal{J}})$,
where $\rho$ is an object in ${\mathcal{C}}$, and with arrows
$t^{\mathcal{J}}:={\mathcal{J}}(t)$, where $t$ is an arrow in ${\mathcal{C}}$.

The Longo–Rehren extension of the spatial tensor product von Neumann algebra
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$ (an irreducible finite
index unital inclusion of factors
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\subset{\mathcal{M}}$)
is specified by the irreducible Q-system
$(\Theta_{\operatorname{LR}},W_{\operatorname{LR}},X_{\operatorname{LR}})$ in
${\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}\subset\textrm{End}({\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}})$
defined as follows. Let

$\Theta_{\operatorname{LR}}:=\bigoplus_{i=0,\ldots,n}\rho_{i}\bar{\otimes}\rho_{i}^{\mathcal{J}}\in\mathrm{End}({\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}).$

The direct sum is defined by choosing a family of $n+1$ isometries
$T_{i,i}\in{\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$,
$i=0,\ldots,n$, “mutually orthogonal” in the sense that
$T_{i,i}^{*}T_{i^{\prime},i^{\prime}}=\delta_{i,i^{\prime}}\mathbbm{1}$, and
“complete” in the sense that
$\sum_{i=0,\ldots,n}T_{i,i}T_{i,i}^{*}=\mathbbm{1}$, and letting
$\Theta_{\operatorname{LR}}(x):=\sum_{i=0,\ldots,n}T_{i,i}\rho_{i}\bar{\otimes}\rho_{i}^{\mathcal{J}}(x)T_{i,i}^{*}$
for every $x\in{\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$. Hence,
by definition,
$T_{i,i}\in\operatorname{Hom}_{{\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}}(\rho_{i}\bar{\otimes}\rho_{i}^{\mathcal{J}},\Theta_{\operatorname{LR}})$
for every $i=0,\ldots,n$. Note that the endomorphism
$\Theta_{\operatorname{LR}}$ is not a “simple tensor” in
${\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}$, unless $n=0$.

Let
$W_{\operatorname{LR}}:=T_{0,0}\in\operatorname{Hom}_{{\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}}(\operatorname{id}\bar{\otimes}\operatorname{id},\Theta_{\operatorname{LR}})$
be the unit of the Q-system and let

$X_{\operatorname{LR}}:=\sum_{i,j,k=0,\ldots,n}\sqrt{\frac{d(\rho_{i})d(\rho_{j})}{d(\rho_{k})}}(T_{i,i}\otimes
T_{j,j})(\sum_{V}V\bar{\otimes}V^{\mathcal{J}})T_{k,k}^{*}\in\operatorname{Hom}_{{\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}}(\Theta_{\operatorname{LR}},\Theta_{\operatorname{LR}}^{2})$

be the comultiplication of the Q-system, where $V\in{\mathcal{N}}$ runs in a
chosen orthonormal basis of $N^{k}_{i,j}(=\text{the multiplicity of $\rho_{k}$
in $\rho_{i}\rho_{j}$}$, possibly $0$) isometries of
$\operatorname{Hom}_{\mathcal{C}}(\rho_{k},\rho_{i}\rho_{j})$ (whose dimension
indeed equals $N^{k}_{i,j}$) for every fixed $i,j,k=0,\ldots,n$, with respect
to the inner product $V^{*}V^{\prime}=\delta_{V,V^{\prime}}1$. The definition
of $X_{\operatorname{LR}}$ is _independent_ of this choice because
${\mathcal{J}}$ is antilinear. Note that $T_{i,i}\otimes T_{j,j}$ denotes the
tensor product of arrows in the category of endomorphisms, and that both
$X_{\operatorname{LR}}$ and $W_{\operatorname{LR}}$ belong to
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$. The normalizations
read $W_{\operatorname{LR}}^{*}W_{\operatorname{LR}}=\mathbbm{1}$ and
$X_{\operatorname{LR}}^{*}X_{\operatorname{LR}}=(\sum_{i}d(\rho_{i})^{2})\mathbbm{1}$.
Being
$(\Theta_{\operatorname{LR}},W_{\operatorname{LR}},X_{\operatorname{LR}})$ a
Q-system, see [LR95, Proposition 4.10] for the proof and cf. [BKLR15,
Definition 3.8], [BKLR15, Proposition 3.19], then by [Lon94]
$\Theta_{\operatorname{LR}}\in\mathrm{End}({\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}})$
is a dual canonical endomorphism of an irreducible finite index unital
extension
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\subset{\mathcal{M}}$
realized on the Hilbert space
${\mathcal{H}}\bar{\otimes}{\mathcal{H}}^{\mathcal{J}}$ if ${\mathcal{N}}$ and
${\mathcal{N}}^{\mathcal{J}}$ are respectively realized on ${\mathcal{H}}$ and
${\mathcal{H}}^{\mathcal{J}}$. The beginning of the Jones/canonical tunnel,
see [LR95, Section 2.5], reads

${\mathcal{M}}_{-2}\subset{\mathcal{M}}_{-1}\subset{\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\subset{\mathcal{M}},$

where
${\mathcal{M}}_{-2}:=\Theta_{\operatorname{LR}}({\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}})$
is the image of $\Theta_{\operatorname{LR}}$, and
${\mathcal{M}}_{-1}:=\langle\Theta_{\operatorname{LR}}({\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}),X_{\operatorname{LR}}\rangle$
is the von Neumann algebra generated together with $X_{\operatorname{LR}}$.
The two subfactors ${\mathcal{M}}_{-2}\subset{\mathcal{M}}_{-1}$ and
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\subset{\mathcal{M}}$
are spatially isomorphic. The Q-system
$(\Theta_{\operatorname{LR}},W_{\operatorname{LR}},X_{\operatorname{LR}})$
also specifies a (unique, by irreducibility, normal faithful) conditional
expectation $E$ from ${\mathcal{M}}$ onto
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$ with Jones projection
$e\in{\mathcal{M}}_{1}$ (the Jones extension of ${\mathcal{M}}$ with respect
to $E$), or equivalently
$E_{-2}(\cdot):=\Theta_{\operatorname{LR}}(W_{\operatorname{LR}}^{*}\cdot
W_{\operatorname{LR}})$ from ${\mathcal{M}}_{-1}$ onto ${\mathcal{M}}_{-2}$
with Jones projection
$e_{-2}:=W_{\operatorname{LR}}W_{\operatorname{LR}}^{*}$.

######  Remark 4.3.

Choosing a state $\omega$ on
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}$ (e.g., the vacuum
state in the QFT context) and working in the GNS Hilbert space of
$({\mathcal{M}},\omega\circ E)$ yields the more common description in QFT of
the extension
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\subset{\mathcal{M}}$
with a vacuum vector $\Omega$ (cyclic and separating for ${\mathcal{M}}$ and
inducing an $E$-invariant state) such that
$e=[{\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}\Omega]$, cf. the
proof of [LR95, Theorem 4.9].

Let
$\psi_{j,j}:=\Theta_{\operatorname{LR}}(T_{j,j}^{*})X_{\operatorname{LR}}\in{\mathcal{M}}_{-1}$,
for every $j=0,\ldots,n$. Each $\psi_{j,j}$ has the “charged field
intertwiner” property on ${\mathcal{M}}_{-2}\subset{\mathcal{M}}_{-1}$ in the
sense of Doplicher–Roberts [DR72], i.e.,

$\psi_{j,j}\Theta_{\operatorname{LR}}(x)=\Theta_{\operatorname{LR}}(\rho_{j}\bar{\otimes}\rho_{j}^{\mathcal{J}}(x))\psi_{j,j},\quad
x\in{\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}.$

We refer to the $\psi_{j,j}$ as the charged fields of the Longo–Rehren
Q-system
$(\Theta_{\operatorname{LR}},W_{\operatorname{LR}},X_{\operatorname{LR}})$.
Such charged fields can be defined for every other Q-system of endomorphisms.

As
$X_{\operatorname{LR}}=\sum_{j=0,\ldots,n}\Theta_{\operatorname{LR}}(T_{j,j})\psi_{j,j}$,
because $\sum_{j=0,\ldots,n}T_{j,j}T_{j,j}^{*}=\mathbbm{1}$, it holds

${\mathcal{M}}_{-1}=\langle{\mathcal{M}}_{-2},\psi_{0,0},\ldots,\psi_{n,n}\rangle.$

Note that
$\psi_{0,0}=\Theta_{\operatorname{LR}}(T_{0,0}^{*})X_{\operatorname{LR}}=\Theta_{\operatorname{LR}}(W_{\operatorname{LR}}^{*})X_{\operatorname{LR}}=\mathbbm{1}$.

Setting
${\mathcal{H}}_{j,j}:=T_{j,j}T_{j,j}^{*}{\mathcal{H}}\bar{\otimes}{\mathcal{H}}^{\mathcal{J}}$,
for every $j=0,\ldots,n$, yields an orthogonal decomposition

$\displaystyle{\mathcal{H}}\bar{\otimes}{\mathcal{H}}^{\mathcal{J}}=\bigoplus_{j=0,\ldots,n}{\mathcal{H}}_{j,j}.$
(4.3)

Note that ${\mathcal{H}}_{0,0}$ is the range of the Jones projection
$e_{-2}=T_{0,0}T_{0,0}^{*}=W_{\operatorname{LR}}W_{\operatorname{LR}}^{*}$.

Moreover, by definition of $X_{\operatorname{LR}}$ and $T_{i,i}\otimes
T_{j,j}=\Theta_{\operatorname{LR}}(T_{j,j})T_{i,i}$, it follows

$\psi_{j,j}=\sum_{i,k=0,\ldots,n}\sqrt{\frac{d(\rho_{i})d(\rho_{j})}{d(\rho_{k})}}T_{i,i}(\sum_{V}V\bar{\otimes}V^{\mathcal{J}})T_{k,k}^{*},$

where the first sum runs over all $i,k$ such that
$\operatorname{Hom}(\rho_{k},\rho_{i}\rho_{j})\neq\langle 0\rangle$ for fixed
$j$, and the second sum runs over $V$ in the chosen orthonormal basis of
isometries in $\operatorname{Hom}_{\mathcal{C}}(\rho_{k},\rho_{i}\rho_{j})$.
Note that, while ${\mathcal{M}}_{-2}$ acts diagonally on
${\mathcal{H}}\bar{\otimes}{\mathcal{H}}^{\mathcal{J}}$ by the very definition
of $\Theta_{\operatorname{LR}}$, the charged fields $\psi_{j,j}$’s mix
different components.

######  Remark 4.4.

In the special case of pointed unitary fusion categories ${\mathcal{C}}$ (in
particular, $d(\rho_{i})=d(\rho_{j})=d(\rho_{k})=1$ and the fusion ring is a
finite group), then all vector spaces
$\operatorname{Hom}_{\mathcal{C}}(\rho_{k},\rho_{i}\rho_{j})$ have either
dimension 1 or 0. In this case each $\psi_{j,j}$ operates as a “right $j^{-1}$
shift operator” on the grading of
${\mathcal{H}}\bar{\otimes}{\mathcal{H}}^{\mathcal{J}}$, mapping each subspace
${\mathcal{H}}_{k,k}$, $k=0,\ldots,n$, to the subspace ${\mathcal{H}}_{i,i}$
such that $[\rho_{k}]=[\rho_{i}\rho_{j}]$, i.e.,
$[\rho_{k}\rho_{j}^{-1}]=[\rho_{i}]$. Moreover, for every
$j,j^{\prime}=0,\ldots,n$,

$\displaystyle\psi_{j,j}\psi_{j^{\prime},j^{\prime}}=\sum_{i,k=0,\ldots,n}T_{i,i}(V\bar{\otimes}V^{\mathcal{J}})T_{k,k}^{*}\sum_{i^{\prime},k^{\prime}=0,\ldots,n}T_{i^{\prime},i^{\prime}}(V^{\prime}\bar{\otimes}V^{\prime{\mathcal{J}}})T_{k^{\prime},k^{\prime}}^{*}$

where $i,k,i^{\prime},k^{\prime}$ are such that
$\dim(\operatorname{Hom}_{\mathcal{C}}(\rho_{k},\rho_{i}\rho_{j}))=1$ and
$\dim(\operatorname{Hom}_{\mathcal{C}}(\rho_{k^{\prime}},\rho_{i^{\prime}}\rho_{j^{\prime}}))=1$.
Note that, as $\Delta$ is fixed, $V,V^{\prime}$ are both unique up to a phase
factor, and ${\mathcal{J}}$ is antilinear, hence the irrelevance of their
choice is immediately evident in the pointed case. If we could choose $\Delta$
to be a group (i.e., closed under multiplication, as
$\rho_{0}=\operatorname{id}$), then using
$T_{k,k}^{*}T_{i^{\prime},i^{\prime}}=\delta_{k,i^{\prime}}\mathbbm{1}$ and
$VV^{\prime}\in\operatorname{Hom}(\rho_{k^{\prime}},\rho_{i}\rho_{j}\rho_{j^{\prime}})=\operatorname{Hom}(\rho_{k^{\prime}},\rho_{i}\rho_{h})$
for some $\rho_{h}\in\Delta$ such that $\rho_{j}\rho_{j^{\prime}}=\rho_{h}$,
then it would follow that $\psi_{j,j}\psi_{j^{\prime},j^{\prime}}=\psi_{h,h}$.
If we chose the inverse (which is a conjugate in the case of automorphisms) of
some $\rho_{j}$ to be some $\rho_{l}\in\Delta$, together with
$\rho_{0}=\operatorname{id}$, then $\psi_{j,j}^{*}=\psi_{l,l}$.

If ${\mathcal{C}}$ is in addition braided, not necessarily pointed, and
${\mathcal{C}}^{\mathcal{J}}$ is endowed with the braiding induced by
${\mathcal{J}}$, then the Longo–Rehren Q-system is _commutative_. See the
discussion following the proof of [LR95, Proposition 4.10] and cf. [BKLR15,
Proposition 4.21]. Let $\epsilon^{\pm}_{\rho_{i},\rho_{j}}$ be the braiding in
${\mathcal{C}}$ and its opposite, then the commutativity condition reads
$\epsilon^{\pm}_{\Theta_{\operatorname{LR}},\Theta_{\operatorname{LR}}}X_{\operatorname{LR}}=X_{\operatorname{LR}}$,
i.e., for every $i,j=0,\ldots,n$,

$\displaystyle\epsilon^{\pm}_{\rho_{i},\rho_{j}}\bar{\otimes}\,{\epsilon^{\pm,{\mathcal{J}}}_{\rho_{i},\rho_{j}}}\,\psi_{i,i}\psi_{j,j}=\psi_{j,j}\psi_{i,i},$
(4.4)

where
$\epsilon^{\pm}_{\rho_{i},\rho_{j}}\bar{\otimes}\,{\epsilon^{\pm,{\mathcal{J}}}_{\rho_{i},\rho_{j}}}$
is the braiding between $\rho_{i}\bar{\otimes}\rho_{i}^{\mathcal{J}}$ and
$\rho_{j}\bar{\otimes}\rho_{j}^{\mathcal{J}}$ in
${\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}$, the irreducible summands
of $\Theta_{\operatorname{LR}}$.

######  Remark 4.5.

If ${\mathcal{C}}$ is braided and pointed, then each braiding
$\epsilon^{\pm}_{\rho_{i},\rho_{j}}$ is a phase multiple of any fixed unitary
intertwiner
$U\in\operatorname{Hom}_{\mathcal{C}}(\rho_{i}\rho_{j},\rho_{j}\rho_{i})$
(both $\rho_{i}\rho_{j}$ and $\rho_{j}\rho_{i}$ are automorphisms hence
irreducible). Thus
$\epsilon^{\pm}_{\rho_{i},\rho_{i}}\bar{\otimes}\,{\epsilon^{\pm,{\mathcal{J}}}_{\rho_{i},\rho_{i}}}=\mathbbm{1}\bar{\otimes}\mathbbm{1}$
for $i=j$, by taking $U=\mathbbm{1}$ and by the antilinearity of
${\mathcal{J}}$. Equivalently, the statistical phase of each
$\rho_{i}\bar{\otimes}\rho_{i}^{\mathcal{J}}$ is $+1$ (all bosonic
automorphisms in ${\mathcal{C}}\boxtimes{\mathcal{C}}^{\mathcal{J}}$) being
equal to the (phase associated with the) self-braiding in the case of
automorphisms. If, in addition, $\rho_{i}\rho_{j}=\rho_{j}\rho_{i}$ for every
$i\neq j$ as automorphisms of ${\mathcal{N}}$ (e.g., by spacelike localization
in the case of DHR automorphisms and
${\mathcal{N}}={\mathcal{A}}_{\kappa}(I)$, or if $\rho_{i}$ and $\rho_{j}$ are
different powers of the same automorphism as for cyclic groups), then one can
take $U=\mathbbm{1}$ as well, and
$\epsilon^{\pm}_{\rho_{i},\rho_{j}}\bar{\otimes}\,{\epsilon^{\pm,{\mathcal{J}}}_{\rho_{i},\rho_{j}}}=\mathbbm{1}\bar{\otimes}\mathbbm{1}$
for every $i,j=0,\ldots,n$. In the latter case of commuting automorphisms, the
commutativity of the Longo–Rehren Q-system for braided pointed fusion
categories reads

$\psi_{i,i}\psi_{j,j}=\psi_{j,j}\psi_{i,i}.$

If we take ${\mathcal{C}}$ to be the representation category of a (completely)
rational conformal net on ${\mathbb{R}}$ or $S^{1}$, then ${\mathcal{C}}$ is
necessarily unitary fusion (finitely many sectors) and braided (also modular,
see, e.g., [BK01]), equipped with the DHR braiding, by a result of [KLM01].
Then the Longo–Rehren Q-system provides an irreducible finite index _local_
(by the commutativity property (4.4)) and “diagonal” extension
$\tilde{\mathcal{A}}_{\operatorname{LR}}$ of the tensor product conformal net
${\mathcal{A}}_{\kappa}\bar{\otimes}{\mathcal{A}}_{\kappa}^{\mathcal{J}}$ on
${\mathbb{R}}^{1+1}$ (with equal left and right chiral components), by
choosing
${\mathcal{N}}\bar{\otimes}{\mathcal{N}}^{\mathcal{J}}={\mathcal{A}}_{\kappa}(I)\bar{\otimes}{\mathcal{A}}_{\kappa}(I)^{\mathcal{J}}$,
where $I$ is an interval on ${\mathbb{R}}$ symmetric around the origin, and
${\mathcal{J}}$ the modular conjugation with respect to
${\mathcal{A}}_{\kappa}({\mathbb{R}}_{+}),\Omega_{\kappa}$ (with this choice,
$\rho_{i}^{\mathcal{J}}\cong\bar{\rho}_{i}$, hence in particular
$\rho_{i}^{\mathcal{J}}\cong\rho_{i}^{-1}$ in the pointed case). See [LR95,
Theorem 4.9], cf. [DG18, Theorem 6.8], for the general construction of
extensions from arbitrary Q-systems in the representation category of a local
net, and the comments after the proof of [LR95, Proposition 4.10] for the
Longo–Rehren Q-system.

##  5 Conformal Wightman fields from charged primary fields

###  5.1 Conformal Wightman axioms

Here we show that to some of the extensions we discussed in Sections 3 and 4
we can associate conformal Wightman fields, as we define below. It is natural
to expect that such Wightman fields on ${\mathbb{R}}^{1+1}$ also extend to
${\mathcal{E}}$ as conformal nets do, and hence we formulate Wightman fields
on ${\mathcal{E}}$.

For simplicity, let us start with chiral fields on $S^{1}$. Let us see
$S^{1}\subset{\mathbb{C}}$ and the counterclockwise direction as the lightlike
direction when $S^{1}$ is seen as the one-point compactification of
${\mathbb{R}}$ as in Section 2.1.

A conformal Wightman field on $S^{1}$ with conformal dimension $D$ is a
quantum field that transforms as the tensor field $f\frac{d}{d\theta}^{\otimes
D-1}$ under diffeomorphisms as (1dW2) below, and such a field is called a
primary field: For $D\in{\mathbb{N}},\gamma\in\operatorname{Diff_{+}}(S^{1})$
and $f\in C^{\infty}(S^{1})$, we set (cf. [RTT22])

$\displaystyle X_{\gamma}(e^{i\theta})$
$\displaystyle:=-i\frac{d}{d\theta}\log(\gamma(e^{i\theta})),$
$\displaystyle(\beta_{D}(\gamma)f)(z)$
$\displaystyle:=(X_{\gamma}(\gamma^{-1}(z)))^{D-1}f(\gamma^{-1}(z)).$

Note that $\gamma$ is orientation-preserving, hence $X$ is strictly positive
and $(X_{\gamma}(\gamma^{-1}(z)))^{D-1}$ is bounded below by some positive
number for a fixed $\gamma$.

A conformal Wightman field theory on $S^{1}$ is a family of operator-valued
distributions $\\{\phi_{\kappa,j}\\}$ ($\kappa$ is a fixed label, while $j$
indexes the family) on $S^{1}$, closed under the conjugate
($\phi_{\kappa,j}^{\dagger}$ is also in the family), with corresponding
conformal dimensions $\\{D_{\kappa,j}\\}$ defined on a common invariant dense
domain ${\mathscr{D}}\subset{\mathcal{H}}_{\kappa}$, a unitary projective
representation $U_{\kappa}$ of $\operatorname{Diff_{+}}(S^{1})$ and a vector
$\Omega_{\kappa}\in{\mathcal{H}}_{\kappa}$ such that

  1. (1dW1)

Locality: for $f,g\in C^{\infty}(S^{1})$ with disjoint supports,
$[\phi_{\kappa,j_{1}}(f),\phi_{\kappa,j_{2}}(g)]=0$ on ${\mathscr{D}}$.

  2. (1dW2)

Diffeomorphism covariance: $U_{\kappa}(\gamma)$ preserves ${\mathscr{D}}$ and
it holds that

$\operatorname{Ad}U_{\kappa}(\gamma)(\phi_{\kappa,j}(f))=\phi_{\kappa,j}(\beta_{D_{\kappa,j}}(\gamma)f),\qquad\text{
for }\gamma\in\operatorname{Diff_{+}}(S^{1}).$

  3. (1dW3)

Positivity of energy: the spectrum of rotations of $U_{\kappa}$ is contained
in ${\mathbb{N}}\cup\\{0\\}$.

  4. (1dW4)

Vacuum and the Reeh-Schlieder property: there exists a unique (up to a phase)
vector $\Omega_{\kappa}\in{\mathcal{H}}_{\kappa}$ such that
$U_{\kappa}(\gamma)\Omega_{\kappa}=\Omega_{\kappa}$ for
$\gamma\in\mathrm{M\ddot{o}b}$ and vectors of the form
$\phi_{\kappa,j_{1}}(f_{1})\cdots\phi_{\kappa,j_{\ell}}(f_{\ell})\Omega_{\kappa}$,
where $f_{1},\cdots,f_{\ell}\in C^{\infty}(S^{1})$, are total in
${\mathcal{H}}_{\kappa}$.

As fields $\phi_{\kappa,j}$ are operator-valued distributions, they can be
smeared with functions $\mathfrak{e}_{n}(\theta)=e^{in\theta}$ to give their
Fourier components $\phi_{\kappa,j,n}=\phi_{\kappa,j}(\mathfrak{e}_{n})$. Let
$L_{\kappa,0}$ be the generator of $U_{\kappa}(R_{t})$, where
$R_{t}\in\mathrm{M\ddot{o}b}$ are rotations. We assume that
${\mathscr{D}}=C^{\infty}(L_{\kappa,0}):=\bigcap_{\ell\in{\mathbb{N}}}\operatorname{Dom}(L_{\kappa,0}^{\ell})$.
Furthermore,

  1. (1dW5)

Polynomial energy bounds: there are $r_{j},p_{j},C>0$ such that, for $\Psi\in
C^{\infty}(L_{\kappa,0})$,

$\|\phi_{\kappa,j,n}\Psi\|\leq
C(1+|n|)^{r_{j}}\|(L_{\kappa,0}+\mathbbm{1})^{p_{j}}\Psi\|$

According to [CKLW18, Section 6], polynomial energy bounds allow one to define
the smeared fields. For $f\in C^{\infty}(S^{1})$ whose Fourier components are
$f_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(e^{-in\theta})d\theta$, we define

$\displaystyle\phi_{\kappa,j}(f):=\sum_{n\in{\mathbb{Z}}}f_{n}\phi_{\kappa,j,n}.$

This is convergent on any vector $\Psi\in C^{\infty}(L_{\kappa,0})$ and
$\phi_{\kappa,j}(f)$ preserves the domain $C^{\infty}(L_{\kappa,0})$.

We know from (the proofs of) [CKLW18, Proposition 6.4, Theorem 8.3] the
following:

######  Lemma 5.1.

Let $\\{\phi_{\kappa,j}\\}$ satisfy polynomial energy bounds with $p_{j}\leq
1$. Then it holds that
$\operatorname{Ad}U(\gamma)(\phi_{\kappa,j}(f))=\phi_{\kappa,j}(\beta_{D_{\kappa,j}}(\gamma)(f))$
and $\phi_{\kappa,j_{1}}(f)$ and $\phi_{\kappa,j_{2}}(g)$ commute strongly for
$f,g$ with disjoint supports.

A general two-dimensional conformal field is not chiral and depends on both
lightlike variables. There are fields called primary fields and they are
distinguished by the conformal dimensions $(D_{\mathrm{L}},D_{\mathrm{R}})$,
where $D_{\mathrm{L}},D_{\mathrm{R}}>0$ and can be non-integer. We set
$\beta_{D_{\mathrm{L}},D_{\mathrm{R}}}(\gamma)=\beta_{D_{\mathrm{L}}}(\gamma_{\mathrm{L}})\beta_{D_{\mathrm{R}}}(\gamma_{\mathrm{R}})$
for
$\gamma=\gamma_{\mathrm{L}}\times\gamma_{\mathrm{R}}\in\overline{\operatorname{Diff_{+}}(S^{1})}\times\overline{\operatorname{Diff_{+}}(S^{1})}$.
This can be extended to ${\mathscr{C}}$ if
$D_{\mathrm{L}}-D_{\mathrm{R}}\in{\mathbb{Z}}$, which we always assume (cf.
Section 4.1). Let $H$ be the generator of $\\{R_{t}\times
R_{t}:t\in{\mathbb{R}}\\}$ in $U$, which plays a similar role to that of
$L_{0}$ in chiral fields.

A conformal Wightman field theory on ${\mathcal{E}}$ is a family of operator-
valued distributions $\\{\widetilde{\psi}_{j}\\}$ on ${\mathcal{E}}$, closed
under conjugation, with corresponding conformal dimensions
$\\{(D_{{\mathrm{L}},j},D_{{\mathrm{R}},j})\\}$ defined on a common invariant
domain ${\mathscr{D}}\subset{\mathcal{H}}$ consisting of vectors of the form
$\widetilde{\psi}_{j_{1}}(f_{1})\cdots\widetilde{\psi}_{j_{n}}(f_{n})\Omega$,
a unitary projective representation $U$ of ${\mathscr{C}}$ and a vector
$\Omega\in{\mathcal{H}}$ such that

  1. (2dW1)

Locality: for $f,g\in C^{\infty}({\mathcal{E}})$ with spacelike separated
supports, $[\widetilde{\psi}_{j_{1}}(f),\widetilde{\psi}_{j_{2}}(g)]=0$.

  2. (2dW2)

Diffeomorphism covariance: $U(\gamma)$ preserves ${\mathscr{D}}$ and it holds
that

$\operatorname{Ad}U(\gamma)(\widetilde{\psi}_{j}(f))=\widetilde{\psi}_{j}(\beta_{D_{{\mathrm{L}},j},D_{{\mathrm{R}},j}}(\gamma)f),\qquad\text{
for }\gamma\in{\mathscr{C}}$

  3. (2dW3)

Positivity of energy: the joint spectrum of the translation subgroup of
${\mathbb{R}}^{1+1}$ in $U$ is contained in the closed forward light cone
$\overline{V_{+}}=\\{(a_{0},a_{1})\in{\mathbb{R}}^{1+1}:a_{0}^{2}-a_{1}^{2}\geq
0,a_{0}\geq 0\\}$.

  4. (2dW4)

Vacuum and the Reeh-Schlieder property: there exists a unique (up to a phase)
vector $\Omega\in{\mathcal{H}}$ such that $U(\gamma)\Omega=\Omega$ for
$\gamma\in\overline{\mathrm{M\ddot{o}b}}\times\overline{\mathrm{M\ddot{o}b}}/\mathfrak{R}$
and ${\mathcal{H}}$ is spanned by vectors of the form
$\widetilde{\psi}_{j_{1}}(f_{1})\cdots\widetilde{\psi}_{j_{k}}(f_{k})\Omega$.

  5. (2dW5)

Polynomial energy bounds: it holds that ${\mathscr{D}}=C^{\infty}(H)$ and
there are $p_{j}>0$ such that, for $\Psi\in C^{\infty}(H)$,

$\|\psi_{j}(f)\Psi\|\leq C_{f}\|H^{p_{j}}\Psi\|,$

for some $C_{f}>0$ depending only on $f$.

As is well-known, linear energy bounds ($p_{j}=1$) assure that the conformal
Wightman fields commute strongly.

######  Lemma 5.2.

Let $\\{\widetilde{\psi}_{j}\\}$ satisfy the bound
$\|\widetilde{\psi}_{j}(f)\Psi\|\leq C_{f,j}\|H\Psi\|$. Then
$\widetilde{\psi}_{j_{1}}(f)$ and $\widetilde{\psi}_{j_{2}}(g)$ commute
strongly for $f,g$ with spacelike supports.

###### Proof.

As $H$ is the generator of the one-parameter group $R_{t}\times R_{t}$ in
$\overline{\mathrm{M\ddot{o}b}}\times\overline{\mathrm{M\ddot{o}b}}$, we have
$[H,\widetilde{\psi}_{j}(f)]=i\widetilde{\psi}_{j}(f^{\prime})$ as in
[CDVIT21, (3.7)], where $f^{\prime}$ is the derivative of $f$ with respect to
translations $R_{t}\times R_{t}$ of the cylinder ${\mathcal{E}}$. Furthermore,
we have
$\|[H,\widetilde{\psi}_{j}(f)]\Psi\|=\|\widetilde{\psi}_{j}(f^{\prime})\Psi\|\leq
C_{f^{\prime},j}\|H\Psi\|$. By applying commutator with $H$,
$\|\delta^{k}(\widetilde{\psi}_{j}(f))\Psi\|=\|\widetilde{\psi}_{j}(f^{(k)})\Psi\|\leq
C_{f^{(k)},j}\|H\Psi\|$, $k=2,3$, where $\delta(A)=i[H,A]$. Therefore,
$\widetilde{\psi}_{1}(f)$ and $\widetilde{\psi}_{2}(g)$ satisfy the assumption
of Theorem A.2 with $H$ as the reference operator and they strongly commute. ∎

###  5.2 Formal series of operators

Let $A_{s}$ be a family of operators parametrized by $s\in{\mathbb{R}}$. By a
formal series we mean a symbol of the form $\sum_{s}A_{s}z^{s}$, where the
summation actually has no meaning. We refer to $A_{s}$ as Fourier components
of a formal series. One can consider sums and scalar multiples of such formal
series: $\sum_{s}A_{s}z^{s}+\sum_{s}B_{s}z^{s}=\sum_{s}(A_{s}+B_{s})z^{s}$,
$c\sum_{s}A_{s}z^{s}=\sum_{s}cA_{s}z^{s}$. The product of two formal series is
not always defined, but in some cases there is a natural product. We define
the product of $\sum_{s}A_{s}z^{s}$ and $\sum_{s}B_{s}z^{s}$ as the formal
series $\sum_{s}C_{s}z^{s}$ with the coefficient
$C_{s}=\sum_{t\in{\mathbb{R}}}A_{s-t}B_{t}$, whenever these sums make sense.

Similarly, we consider formal series in two or more variables. If we have two
formal series in two different variables $z,w$, the product
$\sum_{s}A_{s}z^{s}\sum_{t}B_{t}w^{t}=\sum_{s,t}A_{s}B_{t}z^{s}w^{t}$ makes
always sense. A product of such formal series in two variables is defined
similarly as above.

A typical case we use in this paper is when the vector spaces are graded as
$V^{j}=\bigoplus_{t\in{\mathbb{N}}+h_{j}}V^{j}_{t}$, where $j=1,2$,
$h_{j}\in{\mathbb{R}}$ and $A_{s}$ are operators $V^{1}_{t}\to V^{2}_{t-s}$
and they are nonzero only for $s\in{\mathbb{Z}}+D$ for some $D\in{\mathbb{R}}$
and $t-s\in{\mathbb{N}}+h_{2}$. We also consider the situation where
$V=\bigoplus_{j}V^{j}$ and $\sum_{s}A_{s}$ is a formal series on $V$, that has
the form for each pair $V^{j_{1}},V^{j_{2}}$ as above (and $D_{j_{1},j_{2}}$
may depend on $j_{1},j_{2}$).

Let $\sum_{s}A_{s}z^{s},\sum_{s}B_{s}z^{s}$ such two formal series with
$[A_{s_{1}},B_{s_{2}}]=0$ for all $s_{1},s_{2}\in{\mathbb{R}}$. In the
situation of the previous paragraph, we can define the product
$\sum_{s}A_{s}z^{s}\sum_{t}B_{s}z^{s}=\sum_{s}z^{s}\sum_{t}A_{s-t}B_{t}$ (this
is a special case of normal product, see [CTW23, Section 2.1]): indeed, for
each vector $\Psi\in V$ the sum over $t$ is finite because either
$A_{s-t}\Psi$ or $B_{t}\Psi$ vanishes for large $|t|$. Therefore, the sum
$\sum_{t}A_{s-t}B_{t}$ defines an operator on $V$ and this is a formal series
on $V$.

###  5.3 Charged primary fields

We continue studying a single chiral component, but we omit $\kappa$ from
certain parameters that do not appear later. We assume that a conformal net
$({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ on $S^{1}$ is generated
by conformal Wightman fields $\\{\phi_{\kappa,j}\\}$ which commute strongly
when smeared with test functions with disjoint supports.

Unitary operators $e^{i\phi_{\kappa,j}(f)}$ with compactly supported $f$ are
represented in a representation $\rho$ as a unitary operator, and the question
arises whether the operator $\rho(\phi_{\kappa,j}(f))$ makes sense and whether
it is the generator of $\rho(e^{i\phi_{\kappa,j}(f)})$. This property is
called strong integrability, and veryfing it in examples is a non-trivial
problem (although it holds in interesting cases, like the U(1)-current [CWX,
Gui20]). Furthermore, we can also consider localized unbounded intertwiners
between representations and there is the problem whether the commutation
relations between them hold strongly (strong intertwining property and strong
braiding) [Gui20], see also [Ten19b, Ten19a]. These properties should be
useful in comparing the extension of nets and Wightman fields, see Remark 5.8.

Let $\kappa(g)$ be an automorphism of the conformal net
${\mathcal{A}}_{\kappa}$ for $g\in G$ as in Section 2.2. We consider the
Fourier components of chiral fields and assume that they have corresponding
operators $\phi_{\kappa,j,n}^{\kappa(g)}$ on the representation spaces
${\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$, where
${\mathcal{V}}^{\text{fin}}$ denotes the linear span of eigenspaces
${\mathcal{V}}_{t}=\ker(L_{\kappa,0}-t)$ of $L_{\kappa,0}$, where
$L_{\kappa,0}$ is represented on ${\mathcal{V}}$ as an unbounded operator. We
denote the algebraic direct sum of the $\phi^{\kappa(g)}_{\kappa,j,n}$ over
$g\in G$ by $\hat{\phi}_{\kappa,j,n}$ defined on the (algebraic) linear span
$\bigoplus_{g\in G}^{\text{alg}}{\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$
in $\bigoplus_{g\in G}{\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$.
Similarly, we denote $\hat{L}_{\kappa,n}=\bigoplus_{g\in
G}L_{\kappa,n}^{\kappa(g)}$, where $L_{\kappa,n}^{\kappa(g)}$ is a
representation of the Virasoro algebra on
${\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$.

Let $({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})$ be a conformal net
on $S^{1}$ generated by fields $\\{\phi_{\kappa,j}\\}$ with conformal
dimensions $\\{D_{j}\\}$ as above and a collection $\Delta_{\kappa}$ of
automorphisms to which there is a bijective map $\kappa$ from a group $G$. A
charged primary field $\\{\psi_{\kappa}^{h}(\xi,s)\\}_{s\in{\mathbb{R}}}$ is a
family of operators (labelled by $h\in G$ and $\xi$ in some index
set555Examples where we need all these labels are the WZW models associated
with a simply laced, simply connected, simple, connected compact group $G$
(different from the group above) at level $1$. The sectors are parametrized by
the center $Z(G)$, which can be identified with $\Lambda_{W}/\Lambda_{R}$, the
weight lattice quotiented by the root lattice [TL97, Lemma I.2.1.1]. The
primary fields are of the form $\Phi_{\mu}(z)$, where $\mu$ is a minimal
weight of $G$ [TL97, Section V.5.3]. In this example, $h$ is
$\mu+\Lambda_{R}\in\Lambda_{W}/\Lambda_{R}$, while $\mu$ is one of minimal
weights, playing the role of $\xi$ in our notations. See [TL97, Theorem
V.5.3.2] for relative locality of these fields. $\Xi$ depending on $\kappa$
and $h$), acting on the domain $\bigoplus_{g\in
G}^{\text{alg}}{\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$ with the
associated formal series
$\psi_{\kappa}^{h}(\xi,z)=\sum_{s\in{\mathbb{R}}}\psi_{\kappa}^{h}(\xi,s)z^{-s-D_{h}}$,
where $D_{h}>0$ is the conformal dimension and $z$ is the formal variable,
such that

  * •

$\psi_{\kappa}^{h}(\xi,s)$ maps
${\mathcal{H}}_{\kappa}^{\kappa(g),\text{fin}}$ to
${\mathcal{H}}_{\kappa}^{\kappa(hg),\text{fin}}$, and there is
$\ell_{h,g}\in{\mathbb{R}}$ such that $\psi_{\kappa}^{h}(\xi,s)\neq 0$ only
for $s\in\ell_{h,g}+{\mathbb{Z}}$.

  * •

(primarity)
$[\hat{L}_{\kappa,m},\psi_{\kappa}^{h}(\xi,s)]=((D_{h}-1)m-s)\psi_{\kappa}^{h}(\xi,m+s)$.

  * •

(relative locality)
$[\hat{\phi}_{\kappa,j,m},\psi_{\kappa}^{h}(\xi,s)]=\sum_{\xi^{\prime}}X_{j,\xi^{\prime}}^{\xi}\psi_{\kappa}^{h}(\xi^{\prime},m+s)$,
where $X_{j,\xi^{\prime}}^{\xi}\in{\mathbb{C}}$.

  * •

As formal series, the braiding relation holds:

$\displaystyle(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}w^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)=(1-\textstyle{\frac{w}{z}})^{\alpha(h_{1},h_{2})}z^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{2}}(\xi_{1},z)\psi_{\kappa}^{h_{1}}(\xi_{2},w)$

where $\alpha(h_{1},h_{2})\in{\mathbb{R}}$. Recall that
$(1-u)^{\beta}=\sum_{n\geq 0}\binom{\beta}{n}(-u)^{n}$ as a formal series,
where $\binom{\beta}{n}=\frac{\beta(\beta-1)\cdots(\beta-n+1)}{n!}$.

  * •

We assume that the family of charged primary fields is closed under
conjugation, in the sense that for given $h,\xi$ there are $\bar{h},\bar{\xi}$
such that $\psi^{h}_{\kappa}(\xi,z)^{*}=\psi^{\bar{h}}_{\kappa}(\bar{\xi},z)$,
with the convention $z^{*}=z^{-1}$ in the formal series expansion.

We further assume polynomial energy bounds:

  * •

for each $h$, there are $r_{\xi},p_{\xi},C_{\xi}\geq 0$ such that
$\|\psi_{\kappa}^{h}(\xi,s)\Psi\|\leq
C_{\xi}(1+|s|)^{r_{\xi}}\|(L_{\kappa,0}+\mathbbm{1})^{p_{\xi}}\Psi\|$ for
$\Psi\in C^{\infty}(L_{\kappa,0})$.

This allows us to define smeared charged primary fields: For $f\in
C^{\infty}(S^{1}\setminus\\{-1\\})$, we put
$f_{s}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(e^{i\theta})e^{-is\theta}d\theta$ and

$\displaystyle\psi_{\kappa}^{h}(\xi,f):=\sum_{s\in{\mathbb{R}}}f_{s}\psi_{\kappa}^{h}(\xi,s),$

and this defines operators on $C^{\infty}(\hat{L}_{\kappa,0})$ (the sum makes
sense because $\psi_{\kappa}^{h}(\xi,s)=0$ except for countable $s$ on each
${\mathcal{H}}_{\kappa,t}^{\kappa(g)}$).

Let666We use the same notation for the braiding of the charged fields and the
braiding of sectors in Section 2.2. We plan to show that they are indeed equal
under assumptions such as strong braiding [Gui20] in a separate publication.
$\epsilon^{+}_{\kappa(h_{1}),\kappa(h_{2})}:=\lim_{\Im\zeta>0,\zeta\to-1}\zeta^{\alpha(h_{1},h_{2})}$,
where $\Im\zeta$ is the imaginary part of $\zeta$,
$w^{\alpha(h_{1},h_{2})}=e^{\alpha(h_{1},h_{2})\log w}$ and we take the branch
of $\log w$ on ${\mathbb{C}}\setminus(-\infty,0]$ such that $\log
w\in{\mathbb{R}}$ on $(0,\infty)$.

######  Lemma 5.3.

Let $f,g\in C^{\infty}(S^{1}\setminus\\{-1\\})$ such that
$\arg\operatorname{supp}f<\arg\operatorname{supp}g$, that is, they are in the
counterclockwise order on $S^{1}\setminus\\{-1\\}$. Then it holds that
$\psi_{\kappa}^{h_{1}}(\xi_{1},f)\psi_{\kappa}^{h_{2}}(\xi_{2},g)=\epsilon^{+}_{\kappa(h_{1}),\kappa(h_{2})}\psi_{\kappa}^{h_{2}}(\xi_{2},g)\psi_{\kappa}^{h_{1}}(\xi_{1},f)$.

###### Proof.

Let $z_{0}\in S^{1},z_{0}\neq-1$ such that $\arg\operatorname{supp}f<\arg
z_{0}<\arg\operatorname{supp}g$, and $I_{+}$ be the interval on $S^{1}$ from
$z_{0}$ to $-1$ (counterclockwise) while $I_{-}$ be the interval on $S^{1}$
from $-1$ to $z_{0}$, hence we have $\operatorname{supp}f\subset
I_{-},\operatorname{supp}g\subset I_{+}$ (see Figure 2), and put
$f_{s}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(e^{i\theta})e^{-is\theta}d\theta$,
$g_{t}=\frac{1}{2\pi}\int_{-\pi}^{\pi}g(e^{i\theta})e^{-it\theta}d\theta$. As
$f,g$ are smooth, these coefficients are rapidly decaying.

The product $\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)$
is a formal series in $w,z$. For finite-energy vectors $\Psi_{1},\Psi_{2}$ in
some
${\mathcal{H}}_{\kappa}^{\kappa(h_{3}),\text{fin}},{\mathcal{H}}_{\kappa}^{\kappa(h_{4}),\text{fin}}$,
the scalar product
$\varphi_{1}(w,z):=\langle\Psi_{1},\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)\Psi_{2}\rangle$
can be considered as a formal series in $w,z$ with coefficients in
${\mathbb{C}}$, and by the first assumption on charged primary fields, it is
just a countable sum. Furthermore, these coefficients of $w^{s}z^{t}$ vanish
for $s$ large and $t$ small, because each
${\mathcal{H}}_{\kappa}^{\kappa(h_{j}),\text{fin}}$ is a positive-energy
representation of the Virasoro algebra. By polynomial energy bounds, it can be
seen as a distribution in $w,z$, which we denote again by $\varphi_{1}(w,z)$.
Then it holds that $\varphi_{1}(f,g)=\sum\varphi_{1}(s,t)f_{s}g_{t}$, where
$\varphi_{1}(s,t)$ is the coefficient of the formal series $\varphi_{1}(w,z)$
of $w^{s}z^{t}$ (with a slight abuse of notation).

Let $a<1$ and $b>1$. We introduce
$\varphi_{1}^{a,b}(z,w):=\langle\Psi_{1},\psi_{\kappa}^{h_{1}}(\xi_{1},bw)\psi_{\kappa}^{h_{2}}(\xi_{2},az)\Psi_{2}\rangle$,
then it has only finitely many terms with negative powers in $z$ and those
with positive powers in $w$, therefore, it is a convergent series for
$|z|=|w|=1$ again by the polynomial energy bounds and the choice $a<1,b>1$.
Now it can be seen as a function on $I_{-}\times I_{+}$, thus defines a
distribution and
$\varphi_{1}^{a,b}(f,g)=\sum\varphi^{a,b}_{1}(s,t)f_{s}g_{t}$, in the sense
above. Furthermore, as $a\to 1,b\to 1$, this converges to $\varphi_{1}(f,g)$.

The formal series $(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}$ has only
negative powers in $w$ and positive powers in $z$, therefore, the product

$\displaystyle\langle\Psi_{1},(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)\Psi_{2}\rangle$

makes sense again as a formal series (see the remark on formal series above).
Again by polynomial energy bounds, these coefficients grow only polynomially
in $s,t$. Therefore, it can be seen as a distribution with two variables $w\in
I_{-},z\in I_{+}$, which we denote by $\varphi_{2}(w,z)$. Define also
$\varphi^{a,b}_{2}(w,z)=\varphi_{2}(bw,az)$. As $a\to 1,b\to 1$, each
coefficients converge, and $\varphi^{a,b}_{2}(f,g)$ converge as distributions
to $\varphi_{2}(f,g)$ by polynomial energy bounds.

Note that the expansion $(1-u)^{\beta}=\sum_{n\geq 0}\binom{\beta}{n}(-u)^{n}$
converges for $|u|<1$, and can be analytically continued to
${\mathbb{C}}\setminus[1,\infty)$. This expansion coincides with the
definition $(1-u)^{\beta}=e^{\beta\log(1-u)}$ as an analytic function in
$u\in{\mathbb{C}}\setminus[1,\infty)$, where $\log w$ is defined on
${\mathbb{C}}\setminus(-\infty,0]$ and $\log w\in{\mathbb{R}}$ when
$w\in(0,\infty)$ (the last condition fixes a branch of $\log w$ uniquely). We
use the same continuation for $\log w$ when $w^{\beta}=e^{\beta\log w}$ is
considered as a function. If we consider
$(1-\frac{w}{z})^{\alpha(h_{1},h_{2})}$ as a function in $w,z$, we can
multiply it to the above distribution $\varphi_{1}$ and obtain a distribution
in $I_{-}\times I_{+}$, which we denote by $\varphi_{3}(w,z)$. We also
introduce $\varphi_{3}^{a,b}(w,z)=\varphi_{3}(bw,az)$. Then
$\varphi_{3}^{a,b}(f,g)$ converges to $\varphi_{3}(f,g)$ as $a\to 1,b\to 1$
because both $\varphi_{1}^{a,b}(w,z)$ and
$(1-\frac{bw}{az})^{\alpha(h_{1},h_{2})}$ converge as distributions and smooth
functions on $I_{-}\times I_{+}$, respectively.

This shows that $\varphi_{2}(f,g)=\varphi_{3}(f,g)$. That is,
$(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)$
can be seen as the operator-valued distribution multiplied by a smooth
function $(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}$ on $I_{-}\times
I_{+}$.

$-1$$z_{0}$$\operatorname{supp}f$$w$$\operatorname{supp}g$$z$ Figure 2: The
circle, the point of infinity ($-1$) and two functions with disjoint supports.

Similarly,
$(1-\textstyle{\frac{w}{z}})^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{2}}(\xi_{1},z)\psi_{\kappa}^{h_{1}}(\xi_{2},w)$
can be seen as an operator-valued distribution. Therefore, by the braiding
relation of the primary fields, we have the equality

$\displaystyle(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}w^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)=(1-\textstyle{\frac{w}{z}})^{\alpha(h_{1},h_{2})}z^{\alpha(h_{1},h_{2})}\psi_{\kappa}^{h_{2}}(\xi_{1},z)\psi_{\kappa}^{h_{1}}(\xi_{2},w)$

in the sense of operator-valued distributions, under the restriction on the
supports of $f$ and $g$. Seen as functions, the quotient of
$(1-\textstyle{\frac{w}{z}})^{\alpha(h_{1},h_{2})}z^{\alpha(h_{1},h_{2})}$ by
$(1-\textstyle{\frac{z}{w}})^{\alpha(h_{1},h_{2})}w^{\alpha(h_{1},h_{2})}$ is
exactly $\lim_{\Im\zeta>0,\zeta\to-1}\zeta^{\alpha(h_{1},h_{2})}$. Therefore,
by dividing the equation by the former factor, we obtain

$\displaystyle\psi_{\kappa}^{h_{1}}(\xi_{1},w)\psi_{\kappa}^{h_{2}}(\xi_{2},z)=\epsilon^{+}_{\kappa(h_{1}),\kappa(h_{2})}\psi_{\kappa}^{h_{2}}(\xi_{1},z)\psi_{\kappa}^{h_{1}}(\xi_{2},w)$

as desired, as operator-valued distributions. ∎

The following is essentially due to [CKL08, Section 6.3], where the case
$D=\frac{1}{2}$ is treated, see also [CKLW18, Proposition 6.4], [TL99].

######  Lemma 5.4.

Assume that a charged primary field $\psi_{\kappa}^{h}$ has conformal
dimension $D$ and satisfies polynomial energy bounds as above (we omit the
dependence on $\xi$). Then it is diffeomorphism covariant, that is, with
$\hat{U}_{\kappa}$ the projective unitary representation of
$\overline{\operatorname{Diff_{+}}(S^{1})}$,
$\operatorname{Ad}\hat{U}_{\kappa}(\gamma)(\psi_{\kappa}^{h}(f))=\psi_{\kappa}^{h}(\beta_{D}(\gamma)(f))$.

###### Proof.

Let us sketch the arguments of [CKL08, Section 6.3]. Note that
$\hat{L}_{\kappa,n}$ is a representation of the Virasoro algebra on the dense
domain $\hat{\mathcal{H}}_{\kappa}^{\text{fin}}$ in the Hilbert space
$\hat{\mathcal{H}}_{\kappa}$. By polynomial energy bounds they extend to
$C^{\infty}(\hat{L}_{\kappa,0})$. Furthermore, on this domain we have the
commutation relations

$\displaystyle
i[\hat{L}_{\kappa}(f_{1}),\psi_{\kappa}^{h}(f_{2})]=\psi_{\kappa}^{h}(Df_{1}^{\prime}f_{2}-f_{1}f_{2}^{\prime}).$

From this, the following relation follows:

$\displaystyle\frac{d}{dt}\psi_{\kappa}^{h}(\beta_{D}(\exp(tf_{1}))f_{2})|_{t=0}=i[\hat{L}_{\kappa}(f_{1}),\psi_{\kappa}^{h}(f_{2})].$

Again by energy bounds for $\hat{L}_{\kappa,n}$, it holds that
$U_{\kappa}(\exp(tf))$ preserves the domain $C^{\infty}(L_{\kappa,0})$.

For $\Psi\in C^{\infty}(\hat{L}_{\kappa,0})$, we can consider two vector-
valued functions of $t\in{\mathbb{R}}$

$\displaystyle\Psi_{1}(t)$
$\displaystyle=\psi_{\kappa}^{h}(\beta_{D}(\exp(tf_{1})f_{2})\hat{U}_{\kappa}(\exp(tf_{1}))\Psi,$
$\displaystyle\Psi_{2}(t)$
$\displaystyle=\hat{U}_{\kappa}(\exp(tf_{1}))\psi_{\kappa}^{h}(f_{2})\Psi,$

and they both satisfy the differential equation
$\Psi^{\prime}(t)=i\hat{L}_{\kappa}(f)\Psi(t)$. As they satisfy the same
initial condition $\Psi_{1}(0)=\Psi_{2}(0)=\psi_{\kappa}^{h}(f_{2})\Psi$, they
must coincide. Therefore, we have
$\operatorname{Ad}\hat{U}_{\kappa}(\exp(tf_{1}))(\psi(f_{2}))=\psi_{\kappa}^{h}(\beta_{D}(\exp(tf_{1}))f_{2})$.

Recall that $\operatorname{Diff_{+}}(S^{1})$ is algebraically simple [Thu74]
while the subgroup generated by one-parameter groups is a normal subgroup of
$\operatorname{Diff_{+}}(S^{1})$. Therefore, they must coincide. This implies
that any element in $\overline{\operatorname{Diff_{+}}(S^{1})}$ is a product
of elements in some one-parameter group and an element in the center
$\\{R_{2\pi n}\\}$.

Now the relation holds for all
$\gamma\in\overline{\operatorname{Diff_{+}}(S^{1})}$ which is a product of
elements in one-parameter groups, and it is straightforward to verify it for
$R_{2\pi}$, therefore, we have the desired covariance. ∎

###  5.4 One-dimensional Wightman fields arising from trivial total braiding

Let us first assume that there are two conformal nets
${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ on $S^{1}$ and charged primary fields
$\psi_{1},\psi_{2}$ of these nets (here we omit the dependence on $h\in G$ and
$\xi\in\Xi$).

######  Lemma 5.5.

Let $\psi_{1}(z)\otimes\mathbbm{1},\mathbbm{1}\otimes\psi_{2}(z)$ be charged
primary fields satisfying polynomial energy bounds with respect to an operator
$\hat{L}_{0}\otimes\mathbbm{1}+\mathbbm{1}\otimes\hat{L}_{0}$. Then
$\psi_{1}(z)\otimes\psi_{2}(z)$ satisfies polynomial energy bounds.

###### Proof.

Let us assume polynomial energy bounds: there are $r_{j},p_{j},C>0$, for
$j=1,2$ such that

$\displaystyle\|\psi_{j}(s)\Psi\|\leq
C(1+|s|)^{r_{j}}\|(\hat{L}_{0}+\mathbbm{1})^{p_{j}}\Psi\|$

for $\Psi\in C^{\infty}(\hat{L}_{0})$. Note that

$\displaystyle(\hat{L}_{0}+\mathbbm{1})\psi_{2}(t-s)$
$\displaystyle=[\hat{L}_{0},\psi_{2}(t-s)]+\psi_{2}(t-s)\hat{L}_{0}+\psi_{2}(t-s)$
$\displaystyle=(s-t+1)\psi_{2}(t-s)+\psi_{2}(t-s)\hat{L}_{0}$
$\displaystyle=\psi_{2}(t-s)(\hat{L}_{0}+(s-t+1)),$

hence
$(\hat{L}_{0}+\mathbbm{1})^{q}\psi_{2}(t-s)=\psi_{2}(t-s)(\hat{L}_{0}+(s-t+1)\mathbbm{1})^{q}$.
Then, for a fixed $s$ and $\Psi$ such that $H\Psi=\ell\Psi$ where
$H=\hat{L}_{0}\otimes\mathbbm{1}+\mathbbm{1}\otimes\hat{L}_{0}$,

$\displaystyle\|\sum_{t}\psi_{1}(s-t)\otimes\psi_{2}(t)\Psi\|$
$\displaystyle\leq\sum_{t}C(1+|s-t|)^{r_{1}}\|(H+\mathbbm{1})^{p_{1}}(\mathbbm{1}\otimes\psi_{2}(t))\Psi\|$
$\displaystyle\leq\sum_{t}C(1+|s-t|)^{r_{1}}\|(H+\mathbbm{1})^{\lceil
p_{1}\rceil}(\mathbbm{1}\otimes\psi_{2}(t))\Psi\|$
$\displaystyle\leq\sum_{t}C(1+|s-t|)^{r_{1}}\|(\mathbbm{1}\otimes\psi_{2}(t))(H+(t+1)\mathbbm{1})^{\lceil
p_{1}\rceil}\Psi\|$
$\displaystyle=\sum_{t}C(1+|s-t|)^{r_{1}}(\ell+(t+1))^{\lceil
p_{1}\rceil}\|(\mathbbm{1}\otimes\psi_{2}(t))\Psi\|$
$\displaystyle\leq\sum_{t}C^{2}(1+|s-t|)^{r_{1}}(1+|t|)^{r_{2}}(\ell+(t+1))^{\lceil
p_{1}\rceil}\|(H+\mathbbm{1})^{p_{2}}\Psi\|$ $\displaystyle\leq
C^{2}(1+|s|)^{2r_{1}+r_{2}+\lceil
p_{1}\rceil}\|(H+\mathbbm{1})^{r_{1}+r_{2}+2\lceil p_{1}\rceil+p_{2}}\Psi\|,$

where $\lceil p\rceil$ is the smallest integer larger or equal to $p$, in the
last line we estimated $1+|s-t|\leq 1+|s|+\ell\leq(1+|s|)(1+\ell)$ because
$(\psi_{1}(s-t)\otimes\mathbbm{1})\Psi=0$ when $t<-s-\ell$ or
$(\mathbbm{1}\otimes\psi_{2}(t))\Psi=0$ when $t>\ell$.

For a general $\Psi=\sum_{\ell}\Psi_{\ell}$, the
$\sum_{t}\psi_{1}(s-t)\otimes\psi_{2}(t)\Psi_{\ell}$ are orthogonal for
different $\ell$. Therefore, a polynomial energy bound follows for
$\psi_{1}(z)\otimes\psi_{2}(z)$. ∎

Clearly, this estimate is not optimal. It can happen, as we will see with
concrete examples, that a product of two charged primary fields whose Fourier
components are bounded $\|\psi(s)\|\leq C$ has again bounded Fourier
components.

######  Lemma 5.6.

Let $\psi_{1}(z),\psi_{2}(z)$ be charged primary fields for nets
$({\mathcal{A}}_{1},U_{1},\Omega_{1}),({\mathcal{A}}_{2},U_{2},\Omega_{2})$
with polynomial energy bounds. Then $\psi_{1}(z)\otimes\psi_{2}(z)$ is
diffeomorphism covariant with respect to
$\widetilde{U}_{1}\otimes\widetilde{U}_{2}$.

###### Proof.

By Lemma 5.5, $\psi_{1}(z)\otimes\psi_{2}(z)$ satisfies polynomial energy
bounds with respect to
$H=\hat{L}_{1,0}\otimes\mathbbm{1}+\mathbbm{1}\otimes\hat{L}_{2,0}$. Then the
diffeomorphism covariance follows from Lemma 5.4. ∎

As in Section 3, let
$\\{({\mathcal{A}}_{\kappa},U_{\kappa},\Omega_{\kappa})\\}_{\kappa\in K}$ be a
finite family of conformal nets on $S^{1}$ generated by fields
$\\{\phi_{\kappa,j}\\}_{\kappa\in K}$, with a collection $\Delta_{\kappa}$ of
localized automorphisms in some interval, with a common (finitely generated)
abelian group structure $G$ and a family of bijections
$\kappa:G\to\Delta_{\kappa}$ for every index $\kappa\in K$. Let
$\\{\psi_{\kappa}^{h}(\xi,z)\\}_{\kappa\in K,h\in G,\xi\in\Xi_{\kappa}}$ be a
family of charged primary fields satisfying the conditions of Section 5.3.

We assume that $\prod_{\kappa}\epsilon^{+}_{\kappa(h_{1}),\kappa(h_{2})}=1$
for all pairs $h_{1},h_{2}\in G$, as in Section 3. Let us pick $\xi_{\kappa}$
for each $\kappa$. Then we consider the formal series
$\psi_{K}^{h}(\xi_{K},z):=\bigotimes_{\kappa}\psi_{\kappa}^{h}(\xi_{\kappa},z)$
acting on the auxiliary space $\bigotimes_{\kappa}\hat{\mathcal{H}}_{\kappa}$,
where $\xi_{K}=(\xi_{1},\cdots,\xi_{K})$ is an index to label the formal
series for the tensor product of $K$ factors. This is a (normal) product of
two commuting formal series, hence this makes sense as a formal series, see
Section 5.2. It follows from Lemma 5.5 that it makes sense as an operator-
valued distribution if each of the fields satisfies polynomial energy bounds.

######  Theorem 5.7.

The fields $\psi_{K}^{h}$, $h\in G$, together with chiral fields
$\phi_{\kappa,j}$, is a conformal Wightman field theory acting on the Hilbert
space ${\mathcal{H}}_{K}=\bigoplus_{g\in
G}\bigotimes_{\kappa}{\mathcal{H}}_{\kappa}^{\kappa(g)}$. If, in addition,
$\psi_{K}^{h}$ satisfies linear energy bounds with respect to
$\sum_{\kappa}\mathbbm{1}\otimes\cdots\otimes\mathbbm{1}\otimes\underset{\kappa\text{-th}}{\hat{L}_{0}}\otimes\mathbbm{1}\otimes\cdots\otimes\mathbbm{1}$
for all $h\in G$, then $\\{\psi_{K}^{h}:h\in G\\}$ generates a conformal net
on $S^{1}$.

###### Proof.

The formal series $\psi_{K}^{h}$ is defined on the Hilbert space
$\bigotimes_{\kappa}\hat{\mathcal{H}}_{\kappa}$. It generates vectors in
${\mathcal{H}}_{K}=\bigoplus_{g\in
G}\bigotimes_{\kappa}{\mathcal{H}}_{\kappa}^{\kappa(g)}$ from the vacuum. Then
the $\phi_{\kappa,j}(f)$ generate a dense set of vectors in each of these
summands. On the other hand, no other vector is generated from $\psi_{K}^{h}$
and $\phi_{\kappa,j}$ acting on the vacuum.

Let us prove locality. For two test functions $f_{1},f_{2}$ with
$\arg\operatorname{supp}f_{1}<\arg\operatorname{supp}f_{2}$, we have by Lemma
5.3 that

$\displaystyle\psi_{K}^{h_{1}}(f_{1})\psi_{K}^{h_{2}}(f_{2})$
$\displaystyle=\prod_{\kappa}\epsilon^{+}_{\kappa(h_{1}),\kappa(h_{2})}\psi_{K}^{h_{2}}(f_{2})\psi_{K}^{h_{1}}(f_{1})$
$\displaystyle=\psi_{K}^{h_{2}}(f_{2})\psi_{K}^{h_{1}}(f_{1}),$

giving locality as operator-valued distributions. The fields $\psi_{K}^{h}$
and $\phi_{\kappa,j}$ are relatively local. The representation $U_{K}$ is a
tensor product of positive-energy representations, therefore it has itself
positive energy.

If we assume linear energy bounds, then the fields strongly commute and
generate a conformal net on $S^{1}$ by Theorem A.2. ∎

######  Remark 5.8.

We expect that the conformal net on $S^{1}$ in Theorem 5.7 generated by the
conformal Wightman field is unitarily equivalent to the net extension
${\mathcal{A}}_{K}$ constructed in Theorem 3.1. This, at least for $|K|=2$,
should follow by a similar argument as in the proof of Theorem 5.9. More
generally, assuming properties that assures that the VOA intertwiners
intertwine DHR sectors (strong integrability, strong intertwining property and
strong braiding [CWX, Gui20]), it should be possible to show that the two nets
coincide. We will address this question in a future publication.

###  5.5 Two-dimensional Wightman fields through braiding-cancelling map

From a pair of charged primary fields with the braiding satisfying certain
conditions, we can construct a two-dimensional conformal Wightman field. This
is in particular the case if we take two copies of the same conformal net
${\mathcal{A}}_{\mathrm{L}}={\mathcal{A}}_{\mathrm{R}}$, with a pointed
braided _fusion_ subcategory of automorphisms whose fusion ring is isomorphic
to a _finite_ abelian group $G$, and we take the tensor product of the left
charged fields $\psi_{\mathrm{L}}^{h}$ and $\psi_{\mathrm{L}}^{h^{-1}}$, where
$h\in G$, similarly to (4.1), namely
$\widetilde{\psi}^{h}:=\psi_{\mathrm{L}}^{h}\otimes\psi_{\mathrm{L}}^{h^{-1}}$
acting on ${\mathcal{H}}_{\mathrm{L}}\otimes{\mathcal{H}}_{\mathrm{L}}$.

If we assume that the braidings cancel in the sense777Here, instead of
assuming additionally that
$\epsilon^{+}_{{\mathrm{L}}(g)^{-1},{\mathrm{L}}(h)}=\overline{\epsilon^{-}_{{\mathrm{R}}(g)^{-1},{\mathrm{R}}(h)}}$
as we did in Section 4, we assume that the set of charged primary fields is
closed under conjugate and each of them satisfies the braiding relation. that
$\epsilon^{+}_{{\mathrm{L}}(g),{\mathrm{L}}(h)}=\overline{\epsilon^{-}_{{\mathrm{R}}(g),{\mathrm{R}}(h)}}$,
we can prove locality of the combined Wightman field. We show that this
corresponds to a (finite index) Longo–Rehren extension with respect to a
pointed braided fusion category, [LR95], see Section 4.2. The basic idea
behind these constructions is also present in the two-dimensional quantum
field theory context, e.g., in [Reh97], see in particular [Reh97, Section
2.3].

Let
$\\{({\mathcal{A}}_{{\mathrm{L}}/{\mathrm{R}}},U_{{\mathrm{L}}/{\mathrm{R}}},\Omega_{{\mathrm{L}}/{\mathrm{R}}})\\}$
be a pair of conformal nets on $S^{1}$ generated by fields
$\\{\phi_{{{\mathrm{L}}/{\mathrm{R}}},j}\\}$, with a collection
$\Delta_{{\mathrm{L}}/{\mathrm{R}}}$ of (mutually commuting inequivalent)
localized automorphisms in some interval (and closed under composition and
inverses up to unitary equivalence), as in Section 4, with a common finite
abelian group structure $G$ and bijections
${{\mathrm{L}}/{\mathrm{R}}}:G\to\Delta_{{\mathrm{L}}/{\mathrm{R}}}$. Let
$\\{\psi^{h}_{{\mathrm{L}}/{\mathrm{R}}}(\xi,z)\\}_{h\in
G,\xi\in\Xi_{{\mathrm{L}}/{\mathrm{R}}}}$ be a family of primary fields
satisfying the conditions of Section 5.3. Denote
$\epsilon^{+}:=\epsilon^{+}_{{\mathrm{L}}(g),{\mathrm{L}}(h)}$ and
$\epsilon^{-}:=\epsilon^{-}_{{\mathrm{R}}(g),{\mathrm{R}}(h)}$, for short.
Then our braiding cancellation assumption reads
$\epsilon^{+}=\overline{\epsilon^{-}}$. For $w_{1},w_{2}\in
S^{1}\setminus\\{-1\\},\arg w_{1}<\arg w_{2}$, we have by Lemma 5.3 that

$\displaystyle\psi^{g}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{1})\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{2})$
$\displaystyle=\epsilon^{+}\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{2})\psi^{g}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{1}),$
$\displaystyle\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{2})\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{1})$
$\displaystyle=\epsilon^{-}\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{1})\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{2})$
$\displaystyle=\overline{\epsilon^{+}}\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{1})\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},w_{2}).$

For fixed $\xi_{\mathrm{L}}$ and $\xi_{\mathrm{R}}$, let us introduce a two-
dimensional formal power series by

$\displaystyle\widetilde{\psi}^{h}(w,z):=\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w)\otimes\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z).$

######  Theorem 5.9.

The field $\widetilde{\psi}^{h}$ is a two-dimensional conformal Wightman field
on the Hilbert space $\widetilde{\mathcal{H}}=\bigoplus_{h\in
G}{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(h)}\otimes{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(h)}$.
If the field satisfies a linear energy bound with respect to
$\hat{L}_{0}\otimes\mathbbm{1}+\mathbbm{1}\otimes\hat{L}_{0}$, then it
generates a two-dimensional conformal net. If $G$ is finite, the net coincides
(up to unitary equivalence) with $\widetilde{\mathcal{A}}$ constructed in
Theorem 4.2 in the case
${\mathcal{A}}_{\mathrm{L}}={\mathcal{A}}_{\mathrm{R}}$.

###### Proof.

The field $\widetilde{\psi}^{h}$ satisfies two-dimensional locality: the point
$(w_{1},z_{1})$ is spacelike to $(w_{2},z_{2})$ if $\arg w_{1}<\arg w_{2}$ and
$\arg z_{1}>\arg z_{2}$ (or the reversed relations, and in this case, with
$\epsilon^{+}:=\epsilon^{+}_{{\mathrm{L}}(h),{\mathrm{L}}(g)}$),

$\displaystyle\widetilde{\psi}^{h}(w_{1},z_{1})\widetilde{\psi}^{g}(w_{2},z_{2})$
$\displaystyle=\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{1})\otimes\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{1})\cdot\psi^{g}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{2})\otimes\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{2})$
$\displaystyle=\epsilon^{+}\overline{\epsilon^{+}}\psi^{g}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{2})\otimes\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{2})\cdot\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{1})\otimes\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{1})$
$\displaystyle=\psi^{g}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{2})\otimes\psi^{g}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{2})\cdot\psi^{h}_{{\mathrm{L}}}(\xi_{\mathrm{L}},w_{1})\otimes\psi^{h}_{{\mathrm{R}}}(\xi_{\mathrm{R}},z_{1})$
$\displaystyle=\widetilde{\psi}^{h_{2}}(w_{2},z_{2})\widetilde{\psi}^{h_{1}}(w_{1},z_{1}).$

It is a two-dimensional operator-valued distribution because it is a tensor
product of two one-dimensional operator-valued distributions. Other axioms,
positivity of energy, diffeomorphism covariance, the cyclicity of vacuum in
$\widetilde{\mathcal{H}}$, can be proven as in Theorem 5.7. If we assume
linear energy bounds, strong commutativity of smeared fields follows.

Let $G$ be finite. As the decomposition of the vacuum representation
$\widetilde{\mathcal{H}}=\bigoplus_{h\in
G}{\mathcal{H}}_{\mathrm{L}}^{{\mathrm{L}}(h)}\otimes{\mathcal{H}}_{\mathrm{R}}^{{\mathrm{R}}(h)}$
with respect to the action of
${\mathcal{A}}_{\mathrm{L}}\otimes{\mathcal{A}}_{\mathrm{R}}$ is the same as
the decomposition of the vacuum representation of the Longo–Rehren extension
$\tilde{\mathcal{A}}_{\operatorname{LR}}$ (4.3), and the former determines the