# Construction of wedge-local nets of observables through Longo-Witten
endomorphisms

Yoh Tanimoto111Supported in part by the ERC Advanced Grant 227458 OACFT
“Operator Algebras and Conformal Field Theory”.
Dipartimento di Matematica, Università di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy.
e-mail<EMAIL_ADDRESS>
###### Abstract

A convenient framework to treat massless two-dimensional scattering theories
has been established by Buchholz. In this framework, we show that the
asymptotic algebra and the scattering matrix completely characterize the given
theory under asymptotic completeness and standard assumptions.

Then we obtain several families of interacting wedge-local nets by a purely
von Neumann algebraic procedure. One particular case of them coincides with
the deformation of chiral CFT by Buchholz-Lechner-Summers. In another case, we
manage to determine completely the strictly local elements. Finally, using
Longo-Witten endomorphisms on the $U(1)$-current net and the free fermion net,
a large family of wedge-local nets is constructed.

##  1 Introduction

Construction of interacting models of quantum field theory in physical four-
dimensional spacetime has been a long-standing open problem since the birth of
quantum theory. Recently, operator-algebraic methods have been applied to
construct models with weaker localization property [18, 19, 10, 7, 22]. It is
still possible to calculate the two-particle scattering matrix for these
weakly localized theories and they have been shown to be nontrivial. However,
the strict locality still remains difficult. Indeed, of these deformed
theories, strictly localized contents have been shown to be trivial in higher
dimensions [7]. In contrast, in two-dimensional spacetime, a family of
strictly local theories has been constructed and nontrivial scattering
matrices have been calculated [23]. The construction of local nets of
observables is split up into two procedures: construction of wedge-local nets
and determination of strictly local elements. In this paper we present a
purely von Neumann algebraic procedure to construct wedge-local nets based on
chiral CFT and completely determine strictly local elements for some of these
wedge-local nets. Furthermore, we show that the pair of the S-matrix and the
asymptotic algebra forms a complete invariant of the given net and give a
simple formula to recover the original net from these data.

In algebraic approach to quantum field theory, or algebraic QFT, theories are
realized as local nets of operator algebras. Principal examples are
constructed from local quantum fields, or in mathematical terms, from
operator-valued distributions which commute in spacelike regions. However,
recent years purely operator-algebraic constructions of such nets have been
found. A remarkable feature of these new constructions is that they first
consider a single von Neumann algebra (instead of a family of von Neumann
algebras) which is acted on by the spacetime symmetry group in an appropriate
way. The construction procedure relying on a single von Neumann algebra has
been proposed in [4] and resulted in some intermediate constructions [18, 19,
7, 22] and even in a complete construction of local nets [23]. This von
Neumann algebra is interpreted as the algebra of observables localized in a
wedge-shaped region. There is a prescription to recover the strictly localized
observables [4]. However, the algebras of strictly localized observables are
not necessarily large enough and they can be even trivial [7]. When it turned
out to be sufficiently large, one had to rely on the modular nuclearity
condition, a sophisticated analytic tool [8, 23].

Among above constructions, the deformation by Buchholz, Lechner and Summers
starts with an arbitrary wedge-local net. When one applies the BLS deformation
to chiral conformal theories in two dimensions, things get considerably
simplified. We have seen that the theory remains to be asymptotically complete
in the sense of waves [6] even after the deformation and the full S-matrix has
been computed [15]. In this paper we carry out a further construction of
wedge-local nets based on chiral conformal nets. It turns out that all these
construction are related with endomorphisms of the half-line algebra in the
chiral components recently studied by Longo and Witten [26]. Among such
endomorphisms, the simplest ones are translations and inner symmetries. We
show that the construction related to translations coincides with the BLS
deformation of chiral CFT. The construction related to inner symmetries is new
and we completely determine the strictly localized observables under some
technical conditions. Furthermore, by using the family of endomorphisms on the
$U(1)$-current net considered in [26], we construct a large family of wedge-
local nets parametrized by inner symmetric functions. All these wedge-local
nets have nontrivial S-matrix, but the strictly local part of the wedge-local
nets constructed through inner symmetries has trivial S-matrix. The strict
locality of the other constructions remains open. Hence, to our opinion, the
true difficulty lies in strict locality.

Another important question is how large the class of theories is obtained by
this procedure. The class of S-matrices so far obtained is considered rather
small, since any of such S-matrices is contained in the tensor product of
abelian algebras in chiral components, which corresponds to the notion of
local diagonalizability in quantum information. In this paper, however, we
show that a massless asymptotically complete theory is completely
characterized by its asymptotic behaviour and the S-matrix, and the whole
theory can be recovered with a simple formula. Hence we can say that this
formula is sufficiently general.

In Section 2 we recall standard notions of algebraic QFT and scattering
theory. In Section 3 we show that the pair of S-matrix and the asymptotic
algebra is a complete invariant of a massless asymptotically complete net. In
Section 4 we construct wedge-local nets using one-parameter endomorphisms of
Longo-Witten. It is shown that the case of translations coincides with the BLS
deformation of chiral CFT and the strictly local elements are completely
determined for the case of inner symmetries. A common argument is summarized
in Section 4.1. Section 5 is devoted to the construction of wedge-local nets
based on a specific example, the $U(1)$-current net. A similar construction is
obtained also for the free fermionic net. Section 6 summarizes our
perspectives.

##  2 Preliminaries

###  2.1 Poincaré covariant net

We recall the algebraic treatment of quantum field theory [20]. A (local)
Poincaré covariant net ${\mathcal{A}}$ on ${\mathbb{R}}^{2}$ assigns to each
open bounded region $O$ a von Neumann algebra ${\mathcal{A}}(O)$ on a
(separable) Hilbert space ${\mathcal{H}}$ satisfying the following conditions:

  1. (1)

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

  2. (2)

Locality. If $O_{1}$ and $O_{2}$ are causally disjoint, then
$[{\mathcal{A}}(O_{1}),{\mathcal{A}}(O_{2})]=0$.

  3. (3)

Poincaré covariance. There exists a strongly continuous unitary representation
$U$ of the (proper orthochronous) Poincaré group
${{\mathcal{P}}^{\uparrow}_{+}}$ such that for any open region $O$ it holds
that

$U(g){\mathcal{A}}(O)U(g)^{*}={\mathcal{A}}(gO),\mbox{ for
}g\in{{\mathcal{P}}^{\uparrow}_{+}}.$

  4. (4)

Positivity of energy. The joint spectrum of the translation subgroup
${\mathbb{R}}^{2}$ of ${{\mathcal{P}}^{\uparrow}_{+}}$ of $U$ is contained in
the forward lightcone
$V_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}^{2}:p_{0}\geq|p_{1}|\\}$.

  5. (5)

Existence of the vacuum. There is a unique (up to a phase) unit vector
$\Omega$ in ${\mathcal{H}}$ which is invariant under the action of $U$, and
cyclic for $\bigvee_{O\Subset{\mathbb{R}}^{2}}{\mathcal{A}}(O)$.

  6. (6)

Additivity. If $O=\bigcup_{i}O_{i}$, then
${\mathcal{A}}(O)=\bigvee_{i}{\mathcal{A}}(O_{i})$.

From these axioms, the following property automatically follows (see [2])

  1. (7)

Reeh-Schlieder property. The vector $\Omega$ is cyclic and separating for each
${\mathcal{A}}(O)$.

It is convenient to extend the definition of net also to a class of unbounded
regions called wedges. By definition, the standard left and right wedges are
as follows:

$\displaystyle W_{\mathrm{L}}$ $\displaystyle:=$
$\displaystyle\\{(t_{0},t_{1}):t_{0}>t_{1},t_{0}<-t_{1}\\}$ $\displaystyle
W_{\mathrm{R}}$ $\displaystyle:=$
$\displaystyle\\{(t_{0},t_{1}):t_{0}<t_{1},t_{0}>-t_{1}\\}$

The wedges $W_{\mathrm{L}}$, $W_{\mathrm{R}}$ are invariant under Lorentz
boosts. They are causal complements of each other. All the regions obtained by
translations of standard wedges are still called left- and right-wedges,
respectively. Moreover, a bounded region obtained as the intersection of a
left wedge and a right wedge is called a double cone. Let $O^{\prime}$ denote
the causal complement of $O$. It holds that
$W_{\mathrm{L}}^{\prime}=W_{\mathrm{R}}$, and if
$D=(W_{\mathrm{L}}+a)\cap(W_{\mathrm{R}}+b)$ is a double cone,
$a,b\in{\mathbb{R}}^{2}$, then
$D^{\prime}=(W_{\mathrm{R}}+a)\cup(W_{\mathrm{L}}+b)$. It is easy to see that
$\Omega$ is still cyclic and separating for ${\mathcal{A}}(W_{\mathrm{L}})$
and ${\mathcal{A}}(W_{\mathrm{R}})$.

We assume the following properties as natural conditions.

  * •

Bisognano-Wichmann property. The modular group $\Delta^{it}$ of
${\mathcal{A}}(W_{\mathrm{R}})$ with respect to $\Omega$ is equal to
$U(\Lambda(-2\pi t))$, where $\Lambda(t)=\left(\begin{matrix}\cosh t&\sinh
t\\\ \sinh t&\cosh t\end{matrix}\right)$ denotes the Lorentz boost.

  * •

Haag duality. If $O$ is a wedge or a double cone, then it holds that
${\mathcal{A}}(O)^{\prime}={\mathcal{A}}(O^{\prime})$.

If ${\mathcal{A}}$ is Möbius covariant (conformal), then the Bisognano-
Wichmann property is automatic [5], and Haag duality is equivalent to strong
additivity ([29], see also Section 2.2). These properties are valid even in
massive interacting models [23]. Duality for wedge regions (namely
${\mathcal{A}}(W_{\mathrm{L}})^{\prime}={\mathcal{A}}(W_{\mathrm{R}})$)
follows from Bisognano-Wichmann property [31], and it implies that the dual
net indeed satisfies the Haag duality [2].

###  2.2 Chiral conformal nets

In this Section we introduce a fundamental class of examples of Poincaré
covariant nets. For this purpose, first we explain nets on the one-dimensional
circle $S^{1}$. An open nonempty connected nondense subset $I$ of the circle
$S^{1}$ is called an interval. A (local) Möbius covariant net
${\mathcal{A}}_{0}$ on $S^{1}$ assigns to each interval a von Neumann algebra
${\mathcal{A}}_{0}(I)$ on a (separable) Hilbert space ${\mathcal{H}}_{0}$
satisfying the following conditions:

  1. (1)

Isotony. If $I_{1}\subset I_{2}$, then
${\mathcal{A}}_{0}(I_{1})\subset{\mathcal{A}}_{0}(I_{2})$.

  2. (2)

Locality. If $I_{1}\cap I_{2}=\emptyset$, then
$[{\mathcal{A}}_{0}(I_{1}),{\mathcal{A}}_{0}(I_{2})]=0$.

  3. (3)

Möbius covariance. There exists a strongly continuous unitary representation
$U_{0}$ of the Möbius group ${\rm PSL}(2,{\mathbb{R}})$ such that for any
interval $I$ it holds that

$U_{0}(g){\mathcal{A}}_{0}(I)U_{0}(g)^{*}={\mathcal{A}}_{0}(gI),\mbox{ for
}g\in{\rm PSL}(2,{\mathbb{R}}).$

  4. (4)

Positivity of energy. The generator of the one-parameter subgroup of rotations
in the representation $U_{0}$ is positive.

  5. (5)

Existence of the vacuum. There is a unique (up to a phase) unit vector
$\Omega_{0}$ in ${\mathcal{H}}_{0}$ which is invariant under the action of
$U_{0}$, and cyclic for $\bigvee_{I\Subset S^{1}}{\mathcal{A}}_{0}(I)$.

We identify the circle $S^{1}$ as the one-point compactification of the real
line ${\mathbb{R}}$ by the Cayley transform:

$t=i\frac{z-1}{z+1}\Longleftrightarrow
z=-\frac{t-i}{t+i},\phantom{...}t\in{\mathbb{R}},\phantom{..}z\in
S^{1}\subset{\mathbb{C}}.$

Under this identification, we refer to translations $\tau$ and dilations
$\delta$ of ${\mathbb{R}}$ and these are contained in ${\rm
PSL}(2,{\mathbb{R}})$. It is known that the positivity of energy is equivalent
to the positivity of the generator of translations [25].

From the axioms above, the following properties automatically follow (see
[17])

  1. (6)

Reeh-Schlieder property. The vector $\Omega_{0}$ is cyclic and separating for
each ${\mathcal{A}}_{0}(I)$.

  2. (7)

Additivity. If $I=\bigcup_{i}I_{i}$, then
${\mathcal{A}}_{0}(I)=\bigvee_{i}{\mathcal{A}}_{0}(I_{i})$.

  3. (8)

Haag duality on $S^{1}$. For an interval $I$ it holds that
${\mathcal{A}}_{0}(I)^{\prime}={\mathcal{A}}_{0}(I^{\prime})$, where
$I^{\prime}$ is the interior of the complement of $I$ in $S^{1}$.

  4. (9)

Bisognano-Wichmann property. The modular group $\Delta_{0}^{it}$ of
${\mathcal{A}}_{0}({\mathbb{R}}_{+})$ with respect to $\Omega_{0}$ is equal to
$U_{0}(\delta(-2\pi t))$, where $\delta$ is the one-parameter group of
dilations.

######  Example 2.1.

At this level, we have a plenty of examples: The simplest one is the
$U(1)$-current net which will be explained in detail in Section 5.1. Among
others, the most important family is the loop group nets [17, 32]. Even a
classification result has been obtained for a class of nets on $S^{1}$ [21].

A net ${\mathcal{A}}_{0}$ on $S^{1}$ is said to be strongly additive if it
holds that
${\mathcal{A}}_{0}(I)={\mathcal{A}}_{0}(I_{1})\vee{\mathcal{A}}_{0}(I_{2})$,
where $I_{1}$ and $I_{2}$ are intervals obtained by removing an interior point
of $I$.

Let us denote by ${\rm Diff}(S^{1})$ the group of orientation-preserving
diffeomorphisms of the circle $S^{1}$. This group naturally includes ${\rm
PSL}(2,{\mathbb{R}})$. A Möbius covariant net ${\mathcal{A}}_{0}$ on $S^{1}$
is said to be conformal or diffeomorphism covariant if the representation
$U_{0}$ of ${\rm PSL}(2,{\mathbb{R}})$ associated to ${\mathcal{A}}_{0}$
extends to a projective unitary representation of ${\rm Diff}(S^{1})$ such
that for any interval $I$ and $x\in{\mathcal{A}}_{0}(I)$ it holds that

$\displaystyle
U_{0}(g){\mathcal{A}}_{0}(I)U_{0}(g)^{*}={\mathcal{A}}_{0}(gI),\mbox{ for
}g\in{\rm Diff}(S^{1}),$ $\displaystyle U_{0}(g)xU_{0}(g)^{*}=x,\mbox{ if
}{\rm supp}(g)\subset I^{\prime},$

where ${\rm supp}(g)\subset I^{\prime}$ means that $g$ acts identically on
$I$.

Let ${\mathcal{A}}_{0}$ be a Möbius covariant net on $S^{1}$. If a unitary
operator $V_{0}$ commutes with the translation unitaries
$T_{0}(t)=U_{0}(\tau(t))$ and it holds that
$V_{0}{\mathcal{A}}_{0}({\mathbb{R}}_{+})V_{0}^{*}\subset{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
then we say that $V_{0}$ implements a Longo-Witten endomorphism of
${\mathcal{A}}_{0}$. In particular, $V_{0}$ preserves $\Omega_{0}$ up to a
scalar since $\Omega_{0}$ is the unique invariant vector under $T_{0}(t)$.
Such endomorphisms have been studied first in [26] and they found a large
family of endomorphisms for the $U(1)$-current net, its extensions and the
free fermion net.

Let us denote two lightlines by
${\mathrm{L}}_{\pm}:=\\{(t_{0},t_{1})\in{\mathbb{R}}^{2}:t_{0}\pm t_{1}=0\\}$.
Note that any double cone $D$ can be written as a direct product of intervals
$D=I_{+}\times I_{-}$ where $I_{+}\subset{\mathrm{L}}_{+}$ and
$I_{-}\subset{\mathrm{L}}_{-}$. Let ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ be
two Möbius covariant nets on $S^{1}$ defined on the Hilbert spaces
${\mathcal{H}}_{1},{\mathcal{H}}_{2}$ with the vacuum vectors
$\Omega_{1},\Omega_{2}$ and the representations $U_{1},U_{2}$ of ${\rm
PSL}(2,{\mathbb{R}})$. From this pair, we can construct a two-dimensional net
${\mathcal{A}}$ as follows: For a double cone $D=I_{+}\times I_{-}$, we set
${\mathcal{A}}(D)={\mathcal{A}}_{1}(I_{+})\otimes{\mathcal{A}}_{2}(I_{-})$.
For a general open region $O\subset{\mathbb{R}}$, we set
${\mathcal{A}}(O):=\bigvee_{D\subset O}{\mathcal{A}}(D)$. We set
$\Omega:=\Omega_{1}\otimes\Omega_{2}$ and define the representation $U$ of
${\rm PSL}(2,{\mathbb{R}})\times{\rm PSL}(2,{\mathbb{R}})$ by $U(g_{1}\times
g_{2}):=U_{1}(g_{1})\otimes U_{2}(g_{2})$. By recalling that ${\rm
PSL}(2,{\mathbb{R}})\times{\rm PSL}(2,{\mathbb{R}})$ contains the Poincaré
group ${{\mathcal{P}}^{\uparrow}_{+}}$, it is easy to see that ${\mathcal{A}}$
together with $U$ and $\Omega$ is a Poincaré covariant net. We say that such
${\mathcal{A}}$ is chiral and ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ are
referred to as the chiral components. If ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$
are conformal, then the representation $U$ naturally extends to a projective
representation of ${\rm Diff}(S^{1})\times{\rm Diff}(S^{1})$.

###  2.3 Scattering theory for Borchers triples

A Borchers triple on a Hilbert space ${\mathcal{H}}$ is a triple
$({\mathcal{M}},T,\Omega)$ of a von Neumann algebra ${\mathcal{M}}\subset
B({\mathcal{H}})$, a unitary representation $T$ of ${\mathbb{R}}^{2}$ on
${\mathcal{H}}$ and a vector $\Omega\in{\mathcal{H}}$ such that

  * •

${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathcal{M}})\subset{\mathcal{M}}$ for
$(t_{0},t_{1})\in W_{\mathrm{R}}$, the standard right wedge.

  * •

The joint spectrum ${\rm sp}\,T$ is contained in the forward lightcone
$V_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}_{2}:p_{0}\geq|p_{1}|\\}$.

  * •

$\Omega$ is a unique (up to scalar) invariant vector under $T$, and cyclic and
separating for ${\mathcal{M}}$.

By the theorem of Borchers [4, 16], the representation $T$ extends to the
Poincaré group ${{\mathcal{P}}^{\uparrow}_{+}}$, with Lorentz boosts
represented by the modular group of ${\mathcal{M}}$ with respect to $\Omega$.
With this extension $U$, ${\mathcal{M}}$ is Poincaré covariant in the sense
that if $gW_{\mathrm{R}}\subset W_{\mathrm{R}}$ for
$g\in{{\mathcal{P}}^{\uparrow}_{+}}$, then
$U(g){\mathcal{M}}U(g)^{*}\subset{\mathcal{M}}$.

The relevance of Borchers triples comes from the fact that we can construct
wedge-local nets from them: Let ${\mathcal{W}}$ be the set of wedges, i.e. the
the set of all $W=gW_{\mathrm{R}}$ or $W=gW_{\mathrm{L}}$ where $g$ is a
Poincaré transformation. A wedge-local net ${\mathcal{W}}\ni
W\mapsto{\mathcal{A}}(W)$ is a map from ${\mathcal{W}}$ to the set of von
Neumann algebras which satisfy isotony, locality, Poincaré covariance,
positivity of energy, and existence of vacuum, restricted to ${\mathcal{W}}$.
A wedge-local net associated with the Borchers triple
$({\mathcal{M}},T,\Omega)$ is the map defined by
${\mathcal{A}}(W_{R}+a)=T(a){\mathcal{M}}T(a)^{\ast}$ and
${\mathcal{A}}(W_{R}^{\prime}+a)=T(a){\mathcal{M}}^{\prime}T(a)^{\ast}$. This
can be considered as a notion of nets with a weaker localization property. It
is clear that there is a one-to-one correspondence between Borchers triples
and wedge-local nets. A further relation with local nets will be explained at
the end of this section. For simplicity, we study always Borchers triples,
which involve only a single von Neumann algebra.

We denote by ${\mathcal{H}}_{+}$ (respectively by ${\mathcal{H}}_{-}$) the
space of the single excitations with positive momentum, (respectively with
negative momentum) i.e.,
${\mathcal{H}}_{+}=\\{\xi\in{\mathcal{H}}:T(t,t)\xi=\xi\mbox{ for
}t\in{\mathbb{R}}\\}$ (respectively
${\mathcal{H}}_{-}=\\{\xi\in{\mathcal{H}}:T(t,-t)\xi=\xi\mbox{ for
}t\in{\mathbb{R}}\\}$).

Our fundamental examples come from Poincaré covariant nets. For a Poincaré
covariant net ${\mathcal{A}}$, we can construct a Borchers triple as follows:

  * •

${\mathcal{M}}={\mathcal{A}}(W_{\mathrm{R}})$

  * •

$T:=U|_{{\mathbb{R}}^{2}}$, the restriction of $U$ to the translation
subgroup.

  * •

$\Omega$: the vacuum vector.

Indeed, the first condition follows from the Poincaré (in particular,
translation) covariance of the nets and the other conditions are assumed
properties of $U$ and $\Omega$ of the net. If $({\mathcal{M}},T,\Omega)$ comes
from a chiral conformal net
${\mathcal{A}}={\mathcal{A}}_{1}\otimes{\mathcal{A}}_{2}$, then we say this
triple is chiral, as well. This simple construction by tensor product of
chiral nets is considered to be the “undeformed net”. We will exhibit later
different constructions.

Given a Borchers triple $({\mathcal{M}},T,\Omega)$, we can consider the
scattering theory with respect to massless particles [15], which is an
extension of [6]: For a bounded operator $x\in B({\mathcal{H}})$ we write
$x(a)={\hbox{\rm Ad\,}}T(a)(x)$ for $a\in{\mathbb{R}}^{2}$. Furthermore, we
define a family of operators parametrized by ${\mathcal{T}}$:

$x_{\pm}(h_{\mathcal{T}}):=\int dt\,h_{\mathcal{T}}(t)x(t,\pm t),$

where
$h_{\mathcal{T}}(t)=|{\mathcal{T}}|^{-\varepsilon}h(|{\mathcal{T}}|^{-\varepsilon}(t-{\mathcal{T}}))$,
$0<\varepsilon<1$ is a constant, ${\mathcal{T}}\in{\mathbb{R}}$ and $h$ is a
nonnegative symmetric smooth function on ${\mathbb{R}}$ such that $\int
dt\,h(t)=1$.

######  Lemma 2.2 ([6] Lemma 2(b), [15] Lemma 2.1).

Let $x\in{\mathcal{M}}$, then the limits
$\Phi^{\mathrm{out}}_{+}(x):=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{+}(h_{\mathcal{T}})$
and
$\Phi^{\mathrm{in}}_{-}(x):=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{-}(h_{\mathcal{T}})$
exist and it holds that

  * •

$\Phi^{\mathrm{out}}_{+}(x)\Omega=P_{+}x\Omega$ and
$\Phi^{\mathrm{in}}_{-}(x)\Omega=P_{-}x\Omega$

  * •

$\Phi^{\mathrm{out}}_{+}(x){\mathcal{H}}_{+}\subset{\mathcal{H}}_{+}$ and
$\Phi^{\mathrm{in}}_{-}(x){\mathcal{H}}_{-}\subset{\mathcal{H}}_{-}$.

  * •

${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}_{+}(x))=\Phi^{\mathrm{out}}_{+}({\hbox{\rm
Ad\,}}U(g)(x))$ and ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{in}}_{-}(x))=\Phi^{\mathrm{in}}_{-}({\hbox{\rm
Ad\,}}U(g)(x))$ for $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that
$gW_{\mathrm{R}}\subset W_{\mathrm{R}}$.

Furthermore, the limits $\Phi^{\mathrm{out}}_{+}(x)$ (respectively
$\Phi^{\mathrm{in}}_{-}(x)$) depends only on $P_{+}x\Omega$ (respectively on
$P_{-}x\Omega$).

Similarly we define asymptotic objects for the left wedge $W_{\mathrm{L}}$.
Since $J{\mathcal{M}}^{\prime}J={\mathcal{M}}$, where $J$ is the modular
conjugation for ${\mathcal{M}}$ with respect to $\Omega$, we can define for
any $y\in{\mathcal{M}}^{\prime}$

$\Phi_{+}^{\mathrm{in}}(y):=J\Phi_{+}^{\mathrm{out}}(JyJ)J,\,\,\Phi_{-}^{\mathrm{out}}(y):=J\Phi_{-}^{\mathrm{in}}(JyJ)J.$

Then we have the following.

######  Lemma 2.3 ([15], Lemma 2.2).

Let $y\in{\mathcal{M}}^{\prime}$. Then

$\Phi_{+}^{\mathrm{in}}(y)=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y_{+}(h_{\mathcal{T}}),\,\,\Phi_{-}^{\mathrm{out}}(y)=\underset{{\mathcal{T}}\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y_{-}(h_{\mathcal{T}}).$

These operators depend only on the respective vectors
$\Phi_{+}^{\mathrm{in}}(y)\Omega=P_{+}y\Omega$,
$\Phi_{-}^{\mathrm{out}}(y)\Omega=P_{-}y\Omega$ and we have

  1. (a)

$\Phi_{+}^{\mathrm{in}}(y){\mathcal{H}}_{+}\subset{\mathcal{H}}_{+},\,\,\Phi_{-}^{\mathrm{out}}(y){\mathcal{H}}_{-}\subset{\mathcal{H}}_{-}$,

  2. (b)

${\hbox{\rm
Ad\,}}U(g)(\Phi_{+}^{\mathrm{in}}(y))=\Phi_{+}^{\mathrm{in}}({\hbox{\rm
Ad\,}}U(g)(y))$ and ${\hbox{\rm
Ad\,}}U(g)(\Phi_{-}^{\mathrm{out}}(y))=\Phi_{-}^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(y))$ for $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that
$gW_{\mathrm{L}}\subset W_{\mathrm{L}}$.

For $\xi_{+}\in{\mathcal{H}}_{+}$, $\xi_{-}\in{\mathcal{H}}_{-}$, there are
sequences of local operators $\\{x_{n}\\}\subset{\mathcal{M}}$ and
$\\{y_{n}\\}\subset{\mathcal{M}}^{\prime}$ such that
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x_{n}\Omega=\xi_{+}$
and
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{-}y_{n}\Omega=\xi_{-}$.
With these sequences we define collision states as in [15]:

$\displaystyle\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{in}}_{+}(x_{n})\Phi^{\mathrm{in}}_{-}(y_{n})\Omega$
$\displaystyle\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{out}}_{+}(x_{n})\Phi^{\mathrm{out}}_{-}(y_{n})\Omega.$

We interpret $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ (respectively
$\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$) as the incoming
(respectively outgoing) state which describes two non-interacting waves
$\xi_{+}$ and $\xi_{-}$. These asymptotic states have the following natural
properties.

######  Lemma 2.4 ([15], Lemma 2.3).

For the collision states $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ and
$\eta_{+}{\overset{{\mathrm{in}}}{\times}}\eta_{-}$ it holds that

  1. 1.

$\langle\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-},\eta_{+}{\overset{{\mathrm{in}}}{\times}}\eta_{-}\rangle=\langle\xi_{+},\eta_{+}\rangle\cdot\langle\xi_{-},\eta_{-}\rangle$.

  2. 2.

$U(g)(\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-})=(U(g)\xi_{+}){\overset{{\mathrm{in}}}{\times}}(U(g)\xi_{-})$
for all $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that $gW_{\mathrm{R}}\subset
W_{\mathrm{R}}$.

And analogous formulae hold for outgoing collision states.

Furthermore, we set the spaces of collision states: Namely, we let
${\mathcal{H}}^{\mathrm{in}}$ (respectively ${\mathcal{H}}^{\mathrm{out}}$) be
the subspace generated by $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$
(respectively $\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$). From Lemma
2.4, we see that the following map

$S:\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}\longmapsto\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$

is an isometry. The operator
$S:{\mathcal{H}}^{\mathrm{out}}\to{\mathcal{H}}^{\mathrm{in}}$ is called the
scattering operator or the S-matrix of the Borchers triple
$({\mathcal{M}},U,\Omega)$. We say the waves in the triple are interacting if
$S$ is not a constant multiple of the identity operator on
${\mathcal{H}}^{\mathrm{out}}$. We say that the Borchers triple is
asymptotically complete (and massless) if it holds that
${\mathcal{H}}^{\mathrm{in}}={\mathcal{H}}^{\mathrm{out}}={\mathcal{H}}$. We
have seen that a chiral net and its BLS deformations (see Section 4.2.2) are
asymptotically complete [15]. If the Borchers triple
$({\mathcal{M}},T,\Omega)$ is constructed from a Poincaré covariant net
${\mathcal{A}}$, then we refer to these objects and notions as
$S,{\mathcal{H}}_{\pm}$ and asymptotic completeness of ${\mathcal{A}}$, etc.
This notion of asymptotic completeness concerns only massless excitations.
Indeed, if one considers the massive free model for example, then it is easy
to see that all the asymptotic fields are just the vacuum expectation (mapping
to ${\mathbb{C}}{\mathbbm{1}}$).

To conclude this section, we put a remark on the term “wedge-local net”. If a
Borchers triple $({\mathcal{M}},T,\Omega)$ comes from a Haag dual Poincaré
covariant net ${\mathcal{A}}$, then the local algebras are recovered by the
formula ${\mathcal{A}}(D)=T(a){\mathcal{M}}T(a)^{*}\cap
T(b){\mathcal{M}}^{\prime}T(b)^{*}={\mathcal{A}}(W_{\mathrm{R}}+a)\cap{\mathcal{A}}(W_{\mathrm{L}}+b)$,
where $D=(W_{\mathrm{R}}+a)\cap(W_{\mathrm{L}}+b)$ is a double cone.
Furthermore, if ${\mathcal{A}}$ satisfies Bisognano-Wichmann property, then
the Lorentz boost is obtained from the modular group, hence all the components
of the net are regained from the triple. Conversely, for a given Borchers
triple, one can define a “local net” by the same formula above. In general,
this “net” satisfies isotony, locality, Poincaré covariance and positivity of
energy, but not necessarily satisfies additivity and cyclicity of vacuum [4].
Addivity is usually used only in the proof of Reeh-Schlieder property, thus we
do not consider it here. If the “local net” constructed from a Borchers triple
satisfies cyclicity of vacuum, we say that the original Borchers triple is
strictly local. In this respect, a Borchers triple or a wedge-local net is
considered to have a weaker localization property. Hence the search for
Poincaré covariant nets reduces to the search for strictly local nets. Indeed,
by this approach a family of (massive) interacting Poincaré nets has been
obtained [23].

##  3 Asymptotic chiral algebra and S-matrix

###  3.1 Complete invariant of nets

Here we observe that asymptotically complete (massless) net ${\mathcal{A}}$ is
completely determined by its behaviour at asymptotic times. This is
particularly nice, since the search for Poincaré covariant nets is reduced to
the search for appropriate S-matrices. Having seen the classification of a
class of chiral components [21], one would hope even for a similar
classification result for massless asymptotically complete nets.

Specifically, we construct a complete invariant of a net with Bisognano-
Wichmann property consisting of two elements. We already know the first
element, the S-matrix. Let us construct the second element, the asymptotic
algebra. An essential tool is half-sided modular inclusion (see [33, 1] for
the original references). Indeed, we use an analogous argument as in [31,
Lemma 5.5]. Let ${\mathcal{N}}\subset{\mathcal{M}}$ be an inclusion of von
Neumann algebras. If there is a cyclic and separating vector $\Omega$ for
${\mathcal{N}},{\mathcal{M}}$ and ${\mathcal{M}}\cap{\mathcal{N}}^{\prime}$,
then the inclusion ${\mathcal{N}}\subset{\mathcal{M}}$ is said to be standard
in the sense of [13]. If
$\sigma^{\mathcal{M}}_{t}({\mathcal{N}})\subset{\mathcal{N}}$ for
$t\in{\mathbb{R}}_{\pm}$ where $\sigma^{\mathcal{M}}_{t}$ is the modular
automorphism of ${\mathcal{M}}$ with respect to $\Omega$, then it is called a
$\pm$half-sided modular inclusion.

######  Theorem 3.1 ([33, 1]).

If $({\mathcal{N}}\subset{\mathcal{M}},\Omega)$ is a standard $+$(respectively
$-$)half-sided modular inclusion, then there is a Möbius covariant net
${\mathcal{A}}_{0}$ on $S^{1}$ such that
${\mathcal{A}}_{0}({\mathbb{R}}_{-})={\mathcal{M}}$ and
${\mathcal{A}}_{0}({\mathbb{R}}_{-}-1)={\mathcal{N}}$ (respectively
${\mathcal{A}}_{0}({\mathbb{R}}_{+})={\mathcal{M}}$ and
${\mathcal{A}}_{0}({\mathbb{R}}_{+}+1)={\mathcal{N}}$).

If a unitary representation $T_{0}$ of ${\mathbb{R}}$ with positive spectrum
satisfies $T_{0}(t)\Omega=\Omega$ for $t\in{\mathbb{R}}$, ${\hbox{\rm
Ad\,}}T_{0}(t)({\mathcal{M}})\subset{\mathcal{M}}$ for $t\leq 0$ (respectively
$t\geq 0$) and ${\hbox{\rm Ad\,}}T_{0}(-1)({\mathcal{M}})={\mathcal{N}}$
(respectively ${\hbox{\rm Ad\,}}T_{0}(1)({\mathcal{M}})={\mathcal{N}}$), then
$T_{0}$ is the representation of the translation group of the Möbius covariant
net constructed above.

We put
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(O):=\\{\Phi^{\mathrm{out}}_{+}(x),x\in{\mathcal{A}}(O)\\}$.
We will show that
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))\subset\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})$
is a standard $+$half-sided modular inclusion when restricted to
${\mathcal{H}}_{+}$. Indeed, $\Phi^{\mathrm{out}}_{+}$ commutes with
${\hbox{\rm Ad\,}}U(g_{t})$ where $g_{t}=\Lambda(-2\pi t)$ is a Lorentz boost
(Lemma 2.2), and $\tilde{{\mathcal{A}}}(W_{\mathrm{R}}+(-1,1))$ is sent into
itself under ${\hbox{\rm Ad\,}}U(g_{t})$ for $t\geq 0$. Hence by Bisognano-
Wichmann property,
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))\subset\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})$
is a $+$half-sided modular inclusion. In addition, when restricted to
${\mathcal{H}}_{+}$, this inclusion is standard. To see this, note that
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1)+(1,1))=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(0,2))$
because $\Phi^{\mathrm{out}}_{+}$ is invariant under $T(1,1)$, and hence
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(D)\subset\left(\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))^{\prime}\cap\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})\right)$,
where $D=W_{\mathrm{R}}\cap(W_{\mathrm{L}}+(0,2))$. It follows that

$\overline{\left(\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}\left(W_{\mathrm{R}}+(-1,1)\right)^{\prime}\cap\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})\right)\Omega}\supset\overline{\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(D)\Omega}=\overline{P_{+}{\mathcal{A}}(D)\Omega}={\mathcal{H}}_{+},$

which is the standardness on ${\mathcal{H}}_{+}$. Then we obtain a Möbius
covariant net on $S^{1}$ acting on ${\mathcal{H}}_{+}$, which we denote by
${\mathcal{A}}^{\mathrm{out}}_{+}$. Similarly we get a Möbius covariant net
${\mathcal{A}}^{\mathrm{out}}_{-}$ on ${\mathcal{H}}_{-}$. Two nets
${\mathcal{A}}^{\mathrm{out}}_{+}$ and ${\mathcal{A}}^{\mathrm{out}}_{-}$ act
like tensor product by Lemma 2.4, and span the whole space ${\mathcal{H}}$
from the vacuum $\Omega$ by asymptotic completeness. In other words,
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ is a
chiral Möbius covariant net on ${\mathbb{R}}^{2}$ acting on ${\mathcal{H}}$.
We call this chiral net
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ the
(out-)asymptotic algebra of the given net ${\mathcal{A}}$. Similarly one
defines ${\mathcal{A}}^{\mathrm{in}}_{+}$ and
${\mathcal{A}}^{\mathrm{in}}_{-}$.

Let $({\mathcal{M}},T,\Omega)$, where
${\mathcal{M}}:={\mathcal{A}}(W_{\mathrm{R}})$, be the Borchers triple
associated to an asymptotically complete Poincare covariant net
${\mathcal{A}}$ which satisfies Bisognano-Wichmann property and Haag duality.
Our next observation is that ${\mathcal{M}}$ can be recovered from asymptotic
fields.

######  Proposition 3.2.

It holds that
${\mathcal{M}}=\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}({\mathbb{R}}_{-})\vee\tilde{{\mathcal{A}}}^{\mathrm{in}}_{-}({\mathbb{R}}_{+})$.

###### Proof.

The inclusion
${\mathcal{M}}\supset\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}$
is obvious from the definition of asymptotic fields. The converse inclusion is
established by the modular theory: From the assumption of Bisognano-Wichmann
property, the modular automorphism of ${\mathcal{M}}$ with respect to $\Omega$
is the Lorentz boosts $U(\Lambda(-2\pi t))$. Furthermore, it holds that
${\hbox{\rm Ad\,}}U(\Lambda(-2\pi
t))(\Phi^{\mathrm{out}}_{+}(x))=\Phi^{\mathrm{out}}_{+}({\hbox{\rm
Ad\,}}U(\Lambda(-2\pi t))(x))$ by Lemma 2.2. An analogous formula holds for
$\Phi^{\mathrm{in}}$. Namely, the algebra in the middle term of the statement
is invariant under the modular group.

By the assumed asymptotic completeness, the algebra in the middle term spans
the whole space ${\mathcal{H}}$ from the vacuum $\Omega$ as well. Hence by a
simple consequence of Takesaki’s theorem [30, Theorem IX.4.2] [31, Theorem
A.1], these two algebras coincide.

The last equation follows by the definition of asymptotic algebra and their
invariance under translations in respective directions. ∎

######  Proposition 3.3.

It holds that $S\cdot\Phi^{\mathrm{out}}_{\pm}(x)\cdot
S^{*}=\Phi^{\mathrm{in}}_{\pm}(x)$ and
$S\cdot\tilde{{\mathcal{A}}}^{\mathrm{out}}_{\pm}({\mathbb{R}}_{\mp})\cdot
S^{*}=\tilde{{\mathcal{A}}}^{\mathrm{in}}_{\pm}({\mathbb{R}}_{\mp})$.

###### Proof.

This follows from the calculation, using Lemmata 2.2, 2.3 and 2.4,

$\displaystyle\Phi^{\mathrm{in}}_{+}(x)(\xi{\overset{{\mathrm{in}}}{\times}}\eta)$
$\displaystyle=$
$\displaystyle(P_{+}x\xi){\overset{{\mathrm{in}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle
S\left((P_{+}x\xi){\overset{{\mathrm{out}}}{\times}}\eta\right)$
$\displaystyle=$ $\displaystyle
S\cdot\Phi^{\mathrm{out}}_{+}(x)(\xi{\overset{{\mathrm{out}}}{\times}}\eta)$
$\displaystyle=$ $\displaystyle S\cdot\Phi^{\mathrm{out}}_{+}(x)\cdot
S^{*}(\xi{\overset{{\mathrm{in}}}{\times}}\eta),$

and asymptotic completeness. The equation for “$-$” fields is proved
analogously. The last equalities are simple consequences of the formulae for
asymptotic fields. ∎

######  Theorem 3.4.

The out-asymptotic net
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ and
the S-matrix $S$ completely characterizes the original net ${\mathcal{A}}$ if
it satisfies Bisognano-Wichmann property, Haag duality and asymptotic
completeness.

###### Proof.

The wedge algebra is recovered by
${\mathcal{M}}={\mathcal{A}}(W_{\mathrm{R}})=\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}$
by Proposition 3.2. In the right-hand side, $\Phi^{\mathrm{in}}_{-}$ is
recovered from $\Phi^{\mathrm{out}}_{-}$ and $S$ by Proposition 3.3. Hence the
wedge algebra is completely recovered from the data
$\Phi^{\mathrm{out}}_{\pm}$ and $S$, or ${\mathcal{A}}^{\mathrm{in}}_{\pm}$
and $S$ by Proposition 3.2. By Haag duality, the data of wedge algebras are
enough to recover the local algebras. By Bisognano-Wichmann property, the
representation $U$ of the whole Poincaré group is recovered from the modular
data. ∎

######  Remark 3.5.

Among the conditions on ${\mathcal{A}}$, Bisognano-Wichmann property is
satisfied in almost all known examples. Haag duality can be satisfied by
extending the net [2] without changing the S-matrix. Hence we consider them as
standard assumptions. On the other hand, asymptotic completeness is in fact a
very strong condition. For example, a conformal net is asymptotically complete
if and only if it is chiral [31]. Hence the class of asymptotically complete
nets could be very small even among Poincaré covariant nets. But a clear-cut
scattering theory is available only for asymptotically complete cases. The
general case is under investigation [14].

###  3.2 Chiral nets as asymptotic nets

We can express the modular objects of the interacting net in terms of the ones
of the asymptotic chiral net.

######  Proposition 3.6.

Let $\Delta^{\mathrm{out}}$ and $J^{\mathrm{out}}$ be the modular operator and
the modular conjugation of
${\mathcal{A}}_{+}^{\mathrm{out}}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{-}^{\mathrm{out}}({\mathbb{R}}_{+})$
with respect to $\Omega$. Then it holds that $\Delta=\Delta^{\mathrm{out}}$
and $J=SJ^{\mathrm{out}}$.

###### Proof.

First we note that the modular objects of ${\mathcal{A}}(W_{\mathrm{R}})$
restrict to ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$ by Takesaki’s theorem
[30, Theorem IX.4.2]. Indeed,
${\mathcal{A}}^{\mathrm{out}}_{+}({\mathbb{R}}_{+})$ and
${\mathcal{A}}^{\mathrm{out}}_{-}({\mathbb{R}}_{-})$ are subalgebras of
${\mathcal{A}}(W_{\mathrm{R}})$ and invariant under ${\hbox{\rm
Ad\,}}\Delta^{it}$, or equivalently under the Lorentz boosts ${\hbox{\rm
Ad\,}}U(\Lambda(-2\pi t))$ by Bisognano-Wichmann property, as we saw in the
proof of Proposition 3.2, then the projections onto the respective subspaces
commute with the modular objects. Let us denote these restrictions by
$\Delta_{+}^{it},J_{+},\Delta_{-}^{it}$ and $J_{-}$, respectively.

We identify ${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ and the full Hilbert
space ${\mathcal{H}}$ by the action of
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$. By
Bisognano-Wichmann property and Lemma 2.4, we have

$\displaystyle\Delta^{it}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle(U(\Lambda(-2\pi
t))\xi){\overset{{\mathrm{out}}}{\times}}(U(\Lambda(-2\pi t))\eta)$
$\displaystyle=$ $\displaystyle\Delta_{+}^{it}\xi\otimes\Delta_{-}^{it}\eta$
$\displaystyle=$
$\displaystyle(\Delta_{+}\otimes\Delta_{-})^{it}\cdot\xi\otimes\eta,$

which implies that $\Delta=\Delta_{+}\otimes\Delta_{-}=\Delta^{\mathrm{out}}$.

As for modular conjugations, we take $x\in{\mathcal{A}}(W_{\mathrm{R}})$ and
$y\in{\mathcal{A}}(W_{\mathrm{R}})^{\prime}={\mathcal{A}}(W_{\mathrm{L}})$ and
set $\xi=\Phi^{\mathrm{out}}_{+}(x)\Omega$ and
$\eta=\Phi^{\mathrm{out}}_{-}(y)\Omega$. Then we use Lemma 2.3 to see

$\displaystyle J\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle
J\cdot\Phi^{\mathrm{out}}_{+}(x)\Phi^{\mathrm{out}}_{-}(y)\Omega$
$\displaystyle=$
$\displaystyle\Phi^{\mathrm{in}}_{+}(JxJ)\Phi^{\mathrm{in}}_{-}(JyJ)\Omega$
$\displaystyle=$ $\displaystyle(J\xi){\overset{{\mathrm{in}}}{\times}}(J\eta)$
$\displaystyle=$ $\displaystyle
S\cdot(J_{+}\xi){\overset{{\mathrm{out}}}{\times}}(J_{-}\eta)$
$\displaystyle=$ $\displaystyle S\cdot(J_{+}\otimes J_{-})\cdot\xi\otimes\eta$

from which one infers that $J=S\cdot(J_{+}\otimes J_{-})=S\cdot
J^{\mathrm{out}}$. ∎

Theorem 3.4 tells us that chiral conformal nets can be viewed as free field
nets for massless two-dimensional theory (cf. [31]). Let us formulate the
situation the other way around. Let
${\mathcal{A}}_{+}\otimes{\mathcal{A}}_{-}$ be a chiral CFT, then it is an
interesting open problem to characterize unitary operators which can be
interpreted as a S-matrix of a net whose asymptotic net is the given
${\mathcal{A}}_{+}\otimes{\mathcal{A}}_{-}$. We restrict ourselves to point
out that there are several immediate necessary conditions: For example, $S$
must commute with the Poincaré symmetry of the chiral net since it coincides
with the one of the interacting net. Analogously it must hold that
$(J_{+}\otimes J_{-})S(J_{+}\otimes J_{-})=S^{*}$. Furthermore, the algebra of
the form as in Proposition 3.2 must be strictly local.

If one has an appropriate operator $S$, an interacting Borchers triple can be
constructed by (cf. Propositions 3.2, 3.3)

  * •

${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$U:=U_{+}\otimes U_{-}$,

  * •

$\Omega:=\Omega_{+}\otimes\Omega_{-}$.

By the formula for the modular conjugation in Proposition 3.6, it is immediate
to see that

${\mathcal{M}}_{S}^{\prime}:=\\{{\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\mathcal{A}}_{+}({\mathbb{R}}_{+}),y^{\prime}\in{\mathcal{A}}_{-}({\mathbb{R}}_{-})\\}^{\prime\prime}.$

Then for
$x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})$
it holds that
$\Phi^{\mathrm{out}}_{+}(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$ and
$\Phi^{\mathrm{in}}_{-}({\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y))={\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes y)$. Similarly, we have
$\Phi^{\mathrm{out}}_{-}({\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}))={\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ and
$\Phi^{\mathrm{in}}_{+}({\mathbbm{1}}\otimes y^{\prime})={\mathbbm{1}}\otimes
y^{\prime}$ for $x^{\prime}\in{\mathcal{A}}_{+}({\mathbb{R}}_{+})$ and
$y^{\prime}\in{\mathcal{A}}_{-}({\mathbb{R}}_{-})$. From this it is easy to
see that $S$ is indeed the S-matrix of the constructed Borchers triple.

In the following Sections we will construct unitary operators which comply
with these conditions except strict locality. To my opinion, however, the true
difficulty is the strict locality, which has been so far established only for
“regular” massive models [23]. But it is also true that the class of
S-matrices constructed in the present paper can be seen rather small (see the
discussion in Section 6).

##  4 Construction through one-parameter semigroup of endomorphisms

In this Section, we construct families of Borchers triples using one-parameter
semigroup of endomorphisms of Longo-Witten type. The formula to define the von
Neumann algebra is very simple and the proofs use a common argument based on
spectral decomposition.

Our construction is based on chiral conformal nets on $S^{1}$, and indeed one
family can be identified as the BLS deformation of chiral nets (see Section
4.2). But in our construction, the meaning of the term “deformation” is not
clear and we refrain from using it. From now on, we consider only chiral net
with the identical components
${\mathcal{A}}_{1}={\mathcal{A}}_{2}={\mathcal{A}}_{0}$ for simplicity. It is
not difficult to generalize it to “heterotic case” where
${\mathcal{A}}_{1}\neq{\mathcal{A}}_{2}$.

###  4.1 The commutativity lemma

The following Lemma is the key of all the arguments and will be used later in
this Section concerning one-parameter endomorphisms. Typical examples of the
operator $Q_{0}$ in Lemma will be the generator of one-dimensional
translations $P_{0}$ (Section 4.2), or of one-parameter inner symmetries of
the chiral component (Section 4.4).

As a preliminary, we give a remark on tensor product. See [12] for a general
account on spectral measure and measurable family. Let $E_{0}$ be a
projection-valued measure on $Z$ (typically, the spectral measure of some
self-adjoint operator) and $V(\lambda)$ be a measurable family of operators
(bounded or not). Then one can define an operator

$\int_{Z}V(\lambda)\otimes
dE_{0}(\lambda)(\xi\otimes\eta):=\int_{Z}V(\lambda)\xi\otimes
dE_{0}(\lambda)\eta.$

If $V(\lambda)$ is unbounded, the vector $\xi$ should be in a common domain of
$\\{V(\lambda)\\}$. As we will see, this will not matter in our cases. For two
bounded measurable families $V,V^{\prime}$, it is easy to see that

$\int_{Z}V(\lambda)\otimes
dE_{0}(\lambda)\cdot\int_{Z}V^{\prime}(\lambda)\otimes
dE_{0}(\lambda)=\int_{Z}V(\lambda)V^{\prime}(\lambda)\otimes dE_{0}(\lambda).$

######  Lemma 4.1.

We fix a parameter $\kappa\in{\mathbb{R}}$. Let $Q_{0}$ be a self-adjoint
operator on ${\mathcal{H}}_{0}$ and Let $Z\subset{\mathbb{R}}$ be the spectral
supports of $Q_{0}$. If it holds that $[x,{\hbox{\rm Ad\,}}e^{is\kappa
Q_{0}}(x^{\prime})]=0$ for $x,x^{\prime}\in B({\mathcal{H}}_{0})$ and $s\in
Z$, then we have that

$\displaystyle[x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})]$ $\displaystyle=0,$
$\displaystyle[{\mathbbm{1}}\otimes x,{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}({\mathbbm{1}}\otimes x^{\prime})]$ $\displaystyle=0.$

###### Proof.

We prove only the first commutation relation, since the other is analogous.
Let $Q_{0}=\int_{Z}s\cdot dE_{0}(s)$ be the spectral decomposition of $Q_{0}$.
According to this spectral decomposition, we can decompose only the second
component:

$Q_{0}\otimes Q_{0}=Q_{0}\otimes\int_{Z}s\cdot dE_{0}(x)=\int_{Z}sQ_{0}\otimes
dE_{0}(s).$

Hence we can describe the adjoint action of $e^{i\kappa Q_{0}\otimes Q_{0}}$
explicitly:

$\displaystyle{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})$ $\displaystyle=$
$\displaystyle\int_{Z}e^{is\kappa Q_{0}}\otimes
dE(s)\cdot(x^{\prime}\otimes{\mathbbm{1}})\cdot\int_{Z}e^{-is\kappa
Q_{0}}\otimes dE_{0}(s)$ $\displaystyle=$
$\displaystyle\int_{Z}\left({\hbox{\rm Ad\,}}e^{is\kappa
Q_{0}}(x^{\prime})\right)\otimes dE_{0}(s)$

Then it is easy to see that this commutes with $x\otimes{\mathbbm{1}}$ by the
assumed commutativity. ∎

###  4.2 Construction of Borchers triples with respect to translation

The objective here is to apply the commutativity lemma in Section 4.1 to the
endomorphism of translation. Then it turns out that the Borchers triples
obtained by the BLS deformation of a chiral triple coincide with this
construction. A new feature is that our construction involves only von Neumann
algebras.

####  4.2.1 Construction of Borchers triples

Let $({\mathcal{M}},T,\Omega)$ be a chiral Borchers triple with chiral
component ${\mathcal{A}}_{0}$ and $T_{0}(t)=e^{itP_{0}}$ the chiral
translation: Namely,
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
$T(t_{0},t_{1})=T_{0}\left(\frac{t_{0}-t_{1}}{\sqrt{2}}\right)\otimes
T_{0}\left(\frac{t_{0}+t_{1}}{\sqrt{2}}\right)$ and
$\Omega=\Omega_{0}\otimes\Omega_{0}$.

Note that $T_{0}(t)$ implements a Longo-Witten endomorphism of
${\mathcal{A}}_{0}$ for $t\geq 0$. In this sense, the construction of this
Section is considered to be based on the endomorphisms $\\{{\hbox{\rm
Ad\,}}T_{0}(t)\\}$. A nontrivial family of endomorphisms will be featured in
Section 5.

We construct a new Borchers triple on the same Hilbert space
${\mathcal{H}}={\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$ as follows. Let us
fix $\kappa\in{\mathbb{R}}_{+}$.

  * •

${\mathcal{M}}_{P_{0},\kappa}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

the same $T$ from the chiral triple,

  * •

the same $\Omega$ from the chiral triple.

######  Theorem 4.2.

Let $\kappa\geq 0$. Then the triple $({\mathcal{M}}_{P_{0},\kappa},T,\Omega)$
is a Borchers triple with the S-matrix $S_{P_{0},\kappa}=e^{i\kappa
P_{0}\otimes P_{0}}$.

###### Proof.

The vector $\Omega_{0}\otimes\Omega_{0}$ is obviously invariant under $T$ and
$T$ has the spectrum contained in $V_{+}$. The generator $P_{0}$ of one-
dimensional translations obviously commutes with one-dimensional translation
$T_{0}$, hence $P_{0}\otimes P_{0}$ commutes with $T=T_{0}\otimes T_{0}$, so
does $e^{i\kappa P_{0}\otimes P_{0}}$. We claim that
${\mathcal{M}}_{P_{0},\kappa}$ is preserved under translations in the right
wedge. Indeed, if $(t_{0},t_{1})\in W_{\mathrm{R}}$, then we have

${\hbox{\rm Ad\,}}T(t_{0},t_{1})\left({\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}({\mathbbm{1}}\otimes y)\right)={\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}\left({\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathbbm{1}}\otimes y)\right).$

and ${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathbbm{1}}\otimes
y)\in{\mathbbm{1}}\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$ and it is
obvious that ${\hbox{\rm
Ad\,}}T(t_{0},t_{1})(x\otimes{\mathbbm{1}})\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathbbm{1}}$,
hence the generators of the von Neumann algebra ${\mathcal{M}}_{P_{0},\kappa}$
are preserved.

We have to show that $\Omega$ is cyclic and separating for
${\mathcal{M}}_{P_{0},\kappa}$. Note that it holds that $e^{i\kappa
P_{0}\otimes P_{0}}\cdot\xi\otimes\Omega_{0}=\xi\otimes\Omega_{0}$ for any
$\kappa\in{\mathbb{R}},\xi\in{\mathcal{H}}_{0}$, by the spectral calculus. Now
cyclicity is seen by noting that

$\displaystyle(x\otimes{\mathbbm{1}})\cdot{\hbox{\rm Ad\,}}e^{i\kappa
P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes y)\cdot\Omega$ $\displaystyle=$
$\displaystyle(x\otimes{\mathbbm{1}})\cdot e^{i\kappa P_{0}\otimes
P_{0}}\cdot(x\Omega_{0})\otimes\Omega_{0}$ $\displaystyle=$
$\displaystyle(x\Omega_{0})\otimes(y\Omega_{0})$

and by the cyclicity of $\Omega$ for the original algebra
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.

Finally we show separating property as follows: we set

${\mathcal{M}}^{1}_{P_{0},\kappa}=\\{{\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}),y^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\\}^{\prime\prime}.$

Note that $\Omega$ is cyclic for ${\mathcal{M}}^{1}_{P_{0},\kappa}$ by an
analogous proof for ${\mathcal{M}}_{P_{0},\kappa}$, thus for the separating
property, it suffices to show that ${\mathcal{M}}_{P_{0},\kappa}$ and
${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute. Let
$x,y^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
First, $x\otimes{\mathbbm{1}}$ and ${\mathbbm{1}}\otimes y^{\prime}$ obviously
commute. Next, we apply Lemma 4.1 to $x,x^{\prime}$ and $Q_{0}=P_{0}$ to see
that $x\otimes{\mathbbm{1}}$ and ${\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}(x^{\prime}\otimes{\mathbbm{1}})$ commute: Indeed, the spectral support
of $P_{0}$ is ${\mathbb{R}}_{+}$, and for $s\in{\mathbb{R}}_{+}$, $x$ and
${\hbox{\rm Ad\,}}e^{is\kappa P_{0}}(x^{\prime})$ commute since $P_{0}$ is the
generator of one-dimensional translations and since
$x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
Similarly, for $y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$, ${\hbox{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes y)$ and
${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute. This implies that
${\mathcal{M}}_{P_{0},\kappa}$ and ${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute.

The S-matrix corresponds to the unitary used to twist the chiral net as we saw
in the discussion at the end of Section 3.2. ∎

Now that we have constructed a Borchers triple, it is possible to express its
modular objects in terms of the ones of the chiral triple by an analogous
argument as Proposition 3.6. Then one sees that
${\mathcal{M}}^{1}_{P_{0},\kappa}$ is indeed the commutant
${\mathcal{M}}_{P_{0},\kappa}^{\prime}$.

####  4.2.2 BLS deformation

We briefly review the BLS deformation [7]. Let $({\mathcal{M}},T,\Omega)$ be a
Borchers triple. We denote by ${\mathcal{M}}^{\infty}$ the subset of elements
of ${\mathcal{M}}$ which are smooth under the action of $\alpha={\hbox{\rm
Ad\,}}T$ in the norm topology. Then one can define for any
$x\in{\mathcal{M}}^{\infty}$, and a matrix
$\Theta_{\kappa}=\left(\begin{matrix}0&\kappa\\\ \kappa&0\end{matrix}\right)$,
the warped convolution

$x_{\kappa}=\int
dE(a)\,\alpha_{\Theta_{\kappa}a}(x):=\lim_{\varepsilon\searrow
0}(2\pi)^{-2}\int d^{2}a\,d^{2}b\,f(\varepsilon a,\varepsilon b)e^{-ia\cdot
b}\alpha_{\Theta_{\kappa}a}(x)T(b)$

on a suitable domain, where $dE$ is the spectral measure of $T$ and
$f\in\mathscr{S}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$ satisfies
$f(0,0)=1$. We set

${\mathcal{M}}_{\kappa}:=\\{x_{\kappa}:x\in{\mathcal{M}}^{\infty}\\}^{\prime\prime}.$

For $\kappa>0$, the following holds.

######  Theorem 4.3 ([7]).

If $({\mathcal{M}},T,\Omega)$ is a Borchers triple, then
$({\mathcal{M}}_{\kappa},T,\Omega)$ is also a Borchers triple.

We call the latter the BLS deformation of the original triple
$({\mathcal{M}},T,\Omega)$. One of the main results of this paper is to obtain
the BLS deformation by a simple procedure.

We have determined the property of deformed scattering theory in [15]. In our
notation $M^{2}=P_{0}\otimes P_{0}$ we have the following.

######  Theorem 4.4.

For any $\xi\in{\mathcal{H}}_{+}$ and $\eta\in{\mathcal{H}}_{-}$, the
following relations hold:

$\displaystyle\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta=e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(\xi\otimes\eta),$
$\displaystyle\xi{\overset{{\mathrm{in}}}{\times}}_{\kappa}\eta=e^{\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(\xi\otimes\eta),$

where on the left-hand sides there appear the collision states of the deformed
theory.

####  4.2.3 Reproduction of BLS deformation

Let $({\mathcal{M}},T,\Omega)$ be a chiral Borchers triple. In this Section we
show that the Borchers triple $({\mathcal{M}}_{P_{0},\kappa},T,\Omega)$
obtained above is unitarily equivalent to the BLS deformation
$({\mathcal{M}}_{\kappa},T,\Omega)$. Then we can calculate the asymptotic
fields very simply. We use symbols ${\overset{{\mathrm{out}}}{\times}}$ and
${\overset{{\mathrm{out}}}{\times}}_{\kappa}$ to denote collision states with
respect to the corresponding Borchers triples with ${\mathcal{M}}$
(undeformed) and ${\mathcal{M}}_{\kappa}$, respectively. Recall that for the
undeformed chiral triple, all these products
${\overset{{\mathrm{out}}}{\times}}$, ${\overset{{\mathrm{in}}}{\times}}$ and
$\otimes$ coincide [15].

######  Theorem 4.5.

Let us put ${\mathcal{N}}_{P_{0},\kappa}:={\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}{\mathcal{M}}_{P_{0},\kappa}$.
Then it holds that ${\mathcal{N}}_{P_{0},\kappa}={\mathcal{M}}_{\kappa}$,
hence we have the coincidence of two Borchers triples
$({\mathcal{N}}_{P_{0},\kappa},T,\Omega)=({\mathcal{M}}_{\kappa},T,\Omega)$.

###### Proof.

In [15], we have seen that the deformed BLS triple is asymptotically complete.
Furthermore, we have

$\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta=e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\xi{\overset{{\mathrm{out}}}{\times}}\eta.$

As for observables, let $x\in A_{0}({\mathbb{R}}_{-})$ and we use the notation
$x_{\Theta_{\kappa}}$ from [7] 222The reader is suggested to look at the
notation $F_{Q}$ in [7], where $F$ is an observable in ${\mathcal{M}}$ and $Q$
is a $2\times 2$ matrix. We keep the symbol $Q$ for a generator of one-
parameter automorphisms, hence we changed the notation to avoid confusions..
For the asymptotic field $\Phi^{\mathrm{out}}_{\kappa,+}$ of BLS deformation,
we have

$\displaystyle\Phi_{\kappa,+}^{\mathrm{out}}((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}})\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$
$\displaystyle((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}\xi){\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$
$\displaystyle(x\xi){\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$ $\displaystyle e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\cdot(x\xi){\overset{{\mathrm{out}}}{\times}}\eta$ $\displaystyle=$
$\displaystyle e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}\cdot
x\otimes{\mathbbm{1}}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle{\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}(x\otimes{\mathbbm{1}})\cdot
e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$ $\displaystyle=$
$\displaystyle{\hbox{\rm Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(x\otimes{\mathbbm{1}})\cdot\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta,$

(see Appendix A for the second equality) hence, we have
$\Phi_{\kappa,+}^{\mathrm{out}}((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}})={\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}(x\otimes{\mathbbm{1}})$.
Analogously we have $\Phi_{\kappa,-}^{\mathrm{in}}(({\mathbbm{1}}\otimes
y)_{\Theta_{\kappa}})={\hbox{\rm Ad\,}}e^{\frac{i\kappa}{2}P_{0}\otimes
P_{0}}({\mathbbm{1}}\otimes y)$ for $y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.

Note that by definition we have

${\mathcal{N}}_{P_{0},\kappa}=\\{{\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(x\otimes{\mathbbm{1}}),{\hbox{\rm
Ad\,}}e^{\frac{i\kappa}{2}P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}.$

Since the image of the right-wedge algebra by $\Phi^{\mathrm{out}}_{+}$ and
$\Phi^{\mathrm{in}}_{-}$ remains in the right-wedge algebra, from the above
observation, we see that
${\mathcal{N}}_{P_{0},\kappa}\subset{\mathcal{M}}_{\kappa}$ [15]. To see the
converse inclusion, recall that it has been proved that the modular group
$\Delta^{it}$ of the right-wedge algebra with respect to $\Omega$ remains
unchanged under the BLS deformation. We have that ${\hbox{\rm
Ad\,}}\Delta^{it}(e^{i\kappa P_{0}\otimes P_{0}})=e^{i\kappa P_{0}\otimes
P_{0}}$, hence it is easy to see that ${\mathcal{N}}_{P_{0},\kappa}$ is
invariant under ${\hbox{\rm Ad\,}}\Delta^{it}$. By the theorem of Takesaki
[30, Theorem IX.4.2], there is a conditional expectation from
${\mathcal{M}}_{\kappa}$ onto ${\mathcal{N}}_{P_{0},\kappa}$ which preserves
the state $\langle\Omega,\cdot\Omega\rangle$ and in particular,
${\mathcal{M}}_{\kappa}={\mathcal{N}}_{P_{0},\kappa}$ if and only if $\Omega$
is cyclic for ${\mathcal{N}}_{P_{0},\kappa}$. We have already seen the
cyclicity in Theorem 4.2, thus we obtain the thesis.

The translation $T$ and $\Omega$ remain unchanged under
$e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}$, which established the unitary
equivalence between two Borchers triples. ∎

######  Remark 4.6.

It is also possible to formulate Theorem 3.4 for Borchers triple, although the
asymptotic algebra will be neither local nor conformal in general. From this
point of view, Theorem 4.5 is just a corollary of the coincidence of S-matrix.
Here we preferred a direct proof, instead of formulating non local net on
${\mathbb{R}}$.

###  4.3 Endomorphisms with asymmetric spectrum

Here we briefly describe a generalization of the construction in previous
Sections. Let ${\mathcal{A}}_{0}$ be a local net on $S^{1}$, $T_{0}$ be the
representation of the translation. We assume that there is a one-parameter
family $V_{0}(t)=e^{iQ_{0}t}$ of unitary operators with a positive or negative
generator $Q_{0}$ such that $V_{0}(t)$ and $T_{0}(s)$ commute and ${\hbox{\rm
Ad\,}}V_{0}(t)({\mathcal{A}}_{0}({\mathbb{R}}_{+}))\subset{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
for $t\geq 0$. With these ingredients, we have the following:

######  Theorem 4.7.

The triple

  * •

${\mathcal{M}}_{Q_{0},\kappa}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}e^{\pm i\kappa Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$,

  * •

$T:=T_{0}\otimes T_{0}$,

  * •

$\Omega:=\Omega_{0}\otimes\Omega_{0}$,

where $\pm$ corresponds to ${\rm sp}\,Q_{0}\subset{\mathbb{R}}_{\pm}$, is a
Borchers triple with the S-matrix $e^{\pm i\kappa Q_{0}\otimes Q_{0}}$ for
$\kappa\geq 0$.

The proof is analogous to Theorem 4.2 and we refrain from repeating it here.

The construction looks very simple, but to our knowledge, there are only few
examples. The one-parameter group of translation itself has been studied in
the previous Sections. Another one-parameter family of unitaries with a
negative generator $\\{\Gamma(e^{-{\frac{\kappa}{P_{1}}}})\\}$ has been found
for the $U(1)$-current [26], where $P_{1}$ is the generator on the one-
particle space, $\kappa\geq 0$ and $\Gamma$ denotes the second quantization.
Indeed, by Borchers’ theorem [4, 16], such one-parameter group together with
the modular group forms a representation of the “$ax+b$” group, thus it is
related somehow with translation.

###  4.4 Construction of Borchers triples through inner symmetry in chiral CFT

####  4.4.1 Inner symmetry

Let ${\mathcal{A}}_{0}$ be a conformal (Möbius) net on $S^{1}$. An
automorphism of ${\mathcal{A}}_{0}$ is a family of automorphisms
$\\{\alpha_{0,I}\\}$ of local algebras $\\{{\mathcal{A}}_{0}(I)\\}$ with the
consistency condition $\alpha_{0,J}|_{{\mathcal{A}}_{0}(I)}=\alpha_{0,I}$ for
$I\subset J$. If each $\alpha_{0,I}$ preserves the vacuum state $\omega$, then
$\alpha_{0}$ is said to be an inner symmetry. An inner symmetry $\alpha_{0}$
is implemented by a unitary $V_{\alpha_{0}}$ defined by
$V_{\alpha_{0}}x\Omega=\alpha_{0,I}(x)\Omega$, where
$x\in{\mathcal{A}}_{0}(I)$. This definition does not depend on the choice of
$I$ by the consistency condition. If $\alpha_{0,t}$ is a one-parameter family
of weakly continuous automorphisms, then the implementing unitaries satisfy
$V_{\alpha_{0}}(t)V_{\alpha_{0}}(s)=V_{\alpha_{0}}(t+s)$ and
$V_{\alpha_{0}}(0)={\mathbbm{1}}$, hence there is a self-adjoint operator
$Q_{0}$ such that $V_{\alpha_{0}}(t)=e^{itQ_{0}}$ and $Q_{0}\Omega=0$.
Furthermore, $e^{itQ_{0}}$ commutes with modular objects [30]:
$J_{0}e^{itQ_{0}}J_{0}=e^{itQ_{0}}$, or $J_{0}Q_{0}J_{0}=-Q_{0}$ (note that
$J_{0}$ is an anti-unitary involution). If $\alpha_{t}$ is periodic with
period $2\pi$, namely $a_{0,t}=a_{0,t+2\pi}$ then it holds that
$V_{\alpha_{0}}(t)=V_{\alpha_{0}}(t+2\pi)$ and the generator $Q_{0}$ has a
discrete spectrum ${\rm sp}\,Q_{0}\subset{\mathbb{Z}}$. For the technical
simplicity, we restrict ourselves to the study of periodic inner symmetries.
We may assume that the period is $2\pi$ by a rescaling of the parameter.

######  Example 4.8.

We consider the loop group net ${\mathcal{A}}_{G,k}$ of a (simple, simply
connected) compact Lie group $G$ at level $k$ [17, 32], the net generated by
vacuum representations of loop groups $LG$ [28]. On this net, the original
group $G$ acts as a group of inner symmetries. We fix a maximal torus in $G$
and choose a one-parameter group in the maximal torus with a rational
direction, then it is periodic. Any one-parameter group is contained in a
maximal torus, so there are a good proportion of periodic one-parameter groups
in $G$ (although generic one-parameter groups have irrational direction, hence
not periodic). In particular, in the $SU(2)$-loop group net
${\mathcal{A}}_{SU(2),k}$, any one-parameter group in $SU(2)$ is periodic
since $SU(2)$ has rank $1$.

An inner automorphism $\alpha_{0}$ commutes with Möbius symmetry because of
Bisognano-Wichmann property. Hence it holds that
$U_{0}(g)Q_{0}U_{0}(g)^{*}=Q_{0}$. Furthermore, if the net ${\mathcal{A}}_{0}$
is conformal, then $\alpha_{0}$ commutes also with the diffeomorphism symmetry
[11]. Let $G$ be a group of inner symmetries and ${\mathcal{A}}_{0}^{G}$ be
the assignment: $I\mapsto{\mathcal{A}}_{0}(I)^{G}|_{{\mathcal{H}}_{0}^{G}}$,
where ${\mathcal{A}}_{0}(I)^{G}$ denotes the fixed point algebra of
${\mathcal{A}}_{0}(I)$ with respect to $G$ and
${\mathcal{H}}_{0}^{G}:=\overline{\\{x\Omega_{0}:x\in{\mathcal{A}}_{0}^{G}(I),I\subset
S^{1}\\}}$. Then it is easy to see that ${\mathcal{A}}_{0}^{G}$ is a Möbius
covariant net and it is referred to as the fixed point subnet of
${\mathcal{A}}_{0}$ with respect to $G$.

We can describe the action $\alpha_{0}$ of a periodic one-parameter group of
inner symmetries in a very explicit way, which can be considered as the
“spectral decomposition” of $\alpha_{0}$. Although it is well-known, we
summarize it here with a proof for the later use. This will be the basis of
the subsequent analysis.

######  Proposition 4.9.

Any element $x\in{\mathcal{A}}_{0}(I)$ can be written as $x=\sum_{n}x_{n}$,
where $x_{n}\in{\mathcal{A}}_{0}(I)$ and $\alpha_{0,t}(x_{n})=e^{int}x_{n}$.
We denote
${\mathcal{A}}_{0}(I)_{n}=\\{x\in{\mathcal{A}}_{0}(I):\alpha_{0,t}(x)=e^{int}x\\}$.
It holds that
${\mathcal{A}}_{0}(I)_{m}{\mathcal{A}}_{0}(I)_{n}\subset{\mathcal{A}}_{0}(I)_{m+n}$
and ${\mathcal{A}}_{0}(I)_{m}E_{0}(n){\mathcal{H}}_{0}\subset
E_{0}(m+n){\mathcal{H}}_{0}$, where $E_{0}(n)$ denotes the spectral projection
of $Q_{0}$ corresponding to the eigenvalue $n\in{\mathbb{Z}}$.

###### Proof.

Let us fix an element $x\in{\mathcal{A}}_{0}(I)$. The Fourier transform

$x_{n}:=\int_{0}^{2\pi}\alpha_{s}(x)e^{-ins}\,ds$

(here we consider the weak integral using the local normality of
$\alpha_{0,t}$) is again an element of ${\mathcal{A}}_{0}(I)$, since
${\mathcal{A}}_{0}(I)$ is invariant under $\alpha_{0,t}$. Furthermore it is
easy to see that

$\displaystyle\alpha_{0,t}(x_{n})$ $\displaystyle=$
$\displaystyle\alpha_{0,t}\left(\int_{0}^{2\pi}\alpha_{0,s}(x)e^{-ins}\,ds\right)$
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}\alpha_{0,s+t}(x)e^{-ins}\,ds$
$\displaystyle=$ $\displaystyle
e^{int}\int_{0}^{2\pi}\alpha_{0,s}(x)e^{-ins}\,ds=e^{int}x_{n},$

hence we have $x_{n}\in{\mathcal{A}}_{0}(I)_{n}$.

By assumption, $\alpha_{0,t}(x)={\hbox{\rm Ad\,}}e^{itQ_{0}}(x)$ and ${\rm
sp}\,Q_{0}\subset{\mathbb{Z}}$. If we define $x_{l,m}=E_{0}(l)xE_{0}(m)$, it
holds that ${\hbox{\rm Ad\,}}e^{itQ_{0}}x_{l,m}=e^{i(l-m)t}x_{l,m}$. The
integral and this decomposition into matrix elements are compatible, hence for
$x\in{\mathcal{A}}_{0}(I)$ we have

$x_{n}=\sum_{l-m=n}x_{l,m}.$

Now it is clear that $x=\sum_{n}x_{n}$ where each summand is a different
matrix element, hence the sum is strongly convergent. Furthermore from this
decomposition we see that
${\mathcal{A}}_{0}(I)_{m}{\mathcal{A}}_{0}(I)_{n}\subset{\mathcal{A}}_{0}(I)_{m+n}$
and ${\mathcal{A}}_{0}(I)_{m}E_{0}(n){\mathcal{H}}_{0}\subset
E_{0}(m+n){\mathcal{H}}_{0}$. ∎

At the end of this Section, we exhibit a simple formula for the adjoint action
${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}$ on the tensor product
Hilbert space ${\mathcal{H}}:={\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$.

######  Lemma 4.10.

For $x_{m}\in{\mathcal{A}}_{0}(I)_{m},y_{n}\in{\mathcal{A}}_{0}(I)_{n}$, it
holds that ${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x_{m}\otimes{\mathbbm{1}})=x_{m}\otimes e^{im\kappa Q_{0}}$ and
${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y_{n})=e^{in\kappa Q_{0}}\otimes y_{n}$.

###### Proof.

Recall that ${\rm sp}\,Q_{0}\in{\mathbb{Z}}$. Let $Q_{0}=\sum_{l}l\cdot
E_{0}(l)$ be the spectral decomposition of $Q_{0}$. As in the proof of Lemma
4.1, we decompose only the second component of $Q_{0}\otimes Q_{0}$ to see
that

$\displaystyle Q_{0}\otimes Q_{0}$ $\displaystyle=$ $\displaystyle
Q_{0}\otimes\left(\sum_{l}l\cdot E_{0}(l)\right)=\sum_{l}lQ_{0}\otimes
E_{0}(l)$ $\displaystyle e^{i\kappa Q_{0}\otimes Q_{0}}$ $\displaystyle=$
$\displaystyle\sum_{l}e^{il\kappa Q_{0}}\otimes E_{0}(l)$
$\displaystyle{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x_{m}\otimes{\mathbbm{1}})$ $\displaystyle=$
$\displaystyle\sum_{l}{\hbox{\rm Ad\,}}e^{il\kappa Q_{0}}(x_{m})\otimes
E_{0}(l)$ $\displaystyle=$ $\displaystyle\sum_{l}e^{iml\kappa}x_{m}\otimes
E_{0}(l)$ $\displaystyle=$ $\displaystyle x_{m}\otimes e^{im\kappa Q_{0}}.$

∎

######  Proposition 4.11.

For each $l\in{\mathbb{Z}}$ there is a cyclic and separating vector $v\in
E_{0}(l){\mathcal{H}}_{0}$ for a local algebra ${\mathcal{A}}_{0}(I)$.

###### Proof.

It is enough to note that the decomposition ${\mathbbm{1}}=\sum_{l}E_{0}(l)$
is compatible with the decomposition of the whole space with respect to
rotations, since inner symmetries commute with any Möbius transformation.
Hence each space $E_{0}(l){\mathcal{H}}_{0}$ is a direct sum of eigenspace of
rotation. It is a standard fact that a eigenvector of rotation which has
positive spectrum is cyclic and separating for each local algebra (see the
standard proof of Reeh-Schlieder property, e.g. [2]). ∎

We put $E(l,l^{\prime}):=E_{0}(l)\otimes E_{0}(l^{\prime})$.

######  Corollary 4.12.

Each space $E(l,l^{\prime}){\mathcal{H}}$ contains a cyclic and separating
vector $v$ for ${\mathcal{A}}_{0}(I)\otimes{\mathcal{A}}_{0}(J)$ for any pair
of intervals $I,J$.

####  4.4.2 Construction of Borchers triples and their intersection property

Let ${\mathcal{A}}_{0}$ be a Möbius covariant net and $\alpha_{0,t}$ be a
periodic one-parameter group of inner symmetries. The automorphisms can be
implemented as $\alpha_{0,t}={\hbox{\rm Ad\,}}e^{itQ_{0}}$ as explained in
Section 4.4.1. The self-adjoint operator $Q_{0}$ is referred to as the
generator of the inner symmetry.

We construct a Borchers triple as in Section 4.2.1. Let
$\kappa\in{\mathbb{R}}$ be a real parameter (this time $\kappa$ can be
positive or negative) and we put

$\displaystyle{\mathcal{M}}_{Q_{0},\kappa}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa
Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$
$\displaystyle T(t_{0},t_{1})$ $\displaystyle:=$ $\displaystyle
T_{0}\left(\frac{t_{0}-t_{1}}{\sqrt{2}}\right)\otimes
T_{0}\left(\frac{t_{0}+t_{1}}{\sqrt{2}}\right)$ $\displaystyle\Omega$
$\displaystyle:=$ $\displaystyle\Omega_{0}\otimes\Omega_{0}$

######  Theorem 4.13.

The triple $({\mathcal{M}}_{Q_{0},\kappa},T,\Omega)$ above is a Borchers
triple with a nontrivial scattering operator $S_{Q_{0},\kappa}=e^{i\kappa
Q_{0}\otimes Q_{0}}$.

###### Proof.

As remarked in Section 4.4.1, $Q_{0}$ commutes with Möbius symmetry $U_{0}$,
hence $Q_{0}\otimes Q_{0}$ and the translation $T=T_{0}\otimes T_{0}$ commute.
Since
$({\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+}),T,\Omega)$
is a Borchers triple (see Section 2), it holds that ${\hbox{\rm
Ad\,}}T(t_{0},t_{1}){\mathcal{M}}\subset{\mathcal{M}}$ for $(t_{0},t_{1})\in
W_{\mathrm{R}}$ and $T(t_{0},t_{1})\Omega=\Omega$ and $T$ has the joint
spectrum contained in $V_{+}$.

Since $\alpha_{0,t}$ is a one-parameter group of inner symmetries, it holds
that
$\alpha_{0,s}({\mathcal{A}}_{0}({\mathbb{R}}_{-}))={\mathcal{A}}_{0}({\mathbb{R}}_{-})$
and
$\alpha_{0,t}({\mathcal{A}}_{0}({\mathbb{R}}_{+}))={\mathcal{A}}_{0}({\mathbb{R}}_{+})$
for $s,t\in{\mathbb{R}}$. By Lemma 4.1, for
$x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$ it holds that

$[x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})]=0.$

Then one can show that $({\mathcal{M}}_{Q_{0},\kappa},T,\Omega)$ is a Borchers
triple as in the proof of Theorem 4.2. The formula for the S-matrix can be
proved analogously as in Section 3.2. ∎

We now proceed to completely determine the intersection property of
${\mathcal{M}}_{Q_{0},\kappa}$. As a preliminary, we describe the elements in
${\mathcal{M}}_{Q_{0},\kappa}$ in terms of the original algebra
${\mathcal{M}}$ componentwise. On
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
there acts the group $S^{1}\otimes S^{1}$ by the tensor product action:
$(s,t)\mapsto\alpha_{s,t}:=\alpha_{0,s}\otimes\alpha_{0,t}={\hbox{\rm
Ad\,}}(e^{isQ_{0}}\otimes e^{itQ_{0}})$. According to this action, we have a
decomposition of an element $z\in{\mathcal{M}}$ into Fourier components as in
Section 4.4.1:

$z_{m,n}:=\int_{S^{1}\times S^{1}}\alpha_{s,t}(z)e^{-i(ms+nt)}\,ds\,dt,$

which is still an element of ${\mathcal{M}}$, and with
$E(l,l^{\prime}):=E_{0}(l)\otimes E_{0}(l^{\prime})$, these components can be
obtained by

$z_{m,n}=\underset{l^{\prime}-k^{\prime}=n}{\sum_{l-k=m}}E(l,l^{\prime})zE(k,k^{\prime}).$

One sees that ${\hbox{\rm Ad\,}}(e^{isQ_{0}}\otimes e^{itQ_{0}})$ acts also on
${\mathcal{M}}_{Q_{0},\kappa}$ since it commutes with ${\hbox{\rm
Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}$. We still write this action by $\alpha$.
We can take their Fourier components by the same formula and the formula with
spectral projections still holds.

######  Lemma 4.14.

An element $z_{\kappa}\in{\mathcal{M}}_{Q_{0},\kappa}$ has the components of
the form

$(z_{\kappa})_{m,n}=z_{m,n}(e^{in\kappa Q_{0}}\otimes{\mathbbm{1}}),$

where $z=(z_{m,n})$ is some element in ${\mathcal{M}}$. Similarly, an element
$z^{\prime}_{\kappa}\in{\mathcal{M}}_{Q,\kappa}^{\prime}$ has the components
of the form

$(z^{\prime}_{\kappa})_{m,n}=z_{m,n}({\mathbbm{1}}\otimes e^{im\kappa
Q_{0}}),$

where $z^{\prime}=(z_{m,n})$ is some element in ${\mathcal{M}}^{\prime}$.

###### Proof.

We will show only the former statement since the latter is analogous. First we
consider an element of a simple form
$(x_{m}\otimes{\mathbbm{1}})S({\mathbbm{1}}\otimes y_{n})S^{*}$, where
$x_{m}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})_{m}$ and
$y_{n}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})_{n}$. We saw in Proposition 4.10
that this is equal to $(x_{m}\otimes y_{n})(e^{i\kappa
nQ_{0}}\otimes{\mathbbm{1}})$, thus this has the asserted form. Note that the
linear space spanned by these elements for different $m,n$ is closed even
under product. For a finite product and sum, the thesis is linear with respect
to $x$ and $y$, hence we obtain the desired decomposition. The von Neumann
algebra ${\mathcal{M}}_{Q_{0},\kappa}$ is linearly generated by these
elements. Recalling that $z_{m,n}$ is a matrix element with respect to the
decomposition ${\mathbbm{1}}=\sum_{l,l^{\prime}}E(l,l^{\prime})$, we obtain
the Lemma. ∎

Now we are going to determine the intersection of wedge algebras. At this
point, we need to use unexpectedly strong additivity and conformal covariance
(see Section 2). The fixed point subnet ${\mathcal{A}}_{0}^{\alpha_{0}}$ of a
strongly additive net ${\mathcal{A}}_{0}$ on $S^{1}$ with respect to the
action $\alpha_{0}$ of a compact group $S^{1}$ of inner symmetry is again
strongly additive [35].

######  Example 4.15.

The loop group nets ${\mathcal{A}}_{SU(N),k}$ are completely rational [17,
34], hence in particular they are strongly additive. Moreover, they are
conformal [28].

If ${\mathcal{A}}_{0}$ is diffeomorphism covariant, the strong additivity
follows from the split property and the finiteness of $\mu$-index [27]. We
have plenty of examples of nets which satisfy strong additivity and conformal
covariance since it is known that complete rationality passes to finite index
extensions and finite index subnets [24].

######  Theorem 4.16.

Let ${\mathcal{A}}_{0}$ be strongly additive and conformal and $e^{isQ_{0}}$
implement a periodic family of inner symmetries with the generator $Q_{0}$. We
write, with a little abuse of notation, $T(t_{+},t_{-}):=T_{0}(t_{+})\otimes
T_{0}(t_{-})$. For $t_{+}<0$ and $t_{-}>0$ we have

${\mathcal{M}}_{Q_{0},\kappa}\cap\left({\hbox{\rm
Ad\,}}T(t_{+},t_{-})({\mathcal{M}}_{Q_{0},\kappa}^{\prime})\right)={\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-})),$

where $G$ is the group of automorphisms generated by ${\hbox{\rm
Ad\,}}e^{i\kappa Q_{0}}$.

###### Proof.

Let us consider an element from the intersection. From Lemma 4.14, we have two
descriptions of such an element, namely,

$\displaystyle(z_{\kappa})_{m,n}$ $\displaystyle=$ $\displaystyle
z_{m,n}(e^{in\kappa
Q_{0}}\otimes{\mathbbm{1}}),\,\,\,z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+}),$
$\displaystyle(z^{\prime}_{\kappa})_{m,n}$ $\displaystyle=$ $\displaystyle
z^{\prime}_{m,n}({\mathbbm{1}}\otimes e^{im\kappa
Q_{0}}),\,\,\,z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{+})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-}).$

If these elements have to coincide, each $(m,n)$ component has to coincide. Or
equivalently, it should happen that $z_{m,n}(e^{in\kappa Q_{0}}\otimes
e^{-im\kappa Q_{0}})=z^{\prime}_{m,n}$.

Recall that an inner symmetry commutes with diffeomorphisms [11]. This implies
that the fixed point subalgebra contains the representatives of
diffeomorphisms. Furthermore, the fixed point subalgebra by a compact group is
again strongly additive [35]. This means that

$\displaystyle{\mathcal{A}}_{0}^{\alpha_{0}}((-\infty,t_{+}))\vee{\mathcal{A}}_{0}^{\alpha_{0}}((0,\infty))={\mathcal{A}}_{0}^{\alpha_{0}}((t_{+},0)^{\prime}),$
$\displaystyle{\mathcal{A}}_{0}^{\alpha_{0}}((-\infty,0))\vee{\mathcal{A}}_{0}^{\alpha_{0}}((t_{-},\infty))={\mathcal{A}}_{0}^{\alpha_{0}}((0,t_{-})^{\prime}),$

where $I^{\prime}$ means the complementary interval in $S^{1}$.

We claim that if for
$z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
and
$z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{+})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-})$
there holds $z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}$, then
$z=z^{\prime}\in\left({\mathcal{A}}_{0}({\mathbb{R}}_{-})\cap{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{0})\right)\otimes\left({\mathcal{A}}_{0}({\mathbb{R}}_{+})\cap{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-})\right)$.
Indeed, since
$z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
it commutes with $U(g_{+}\times g_{-})$ with ${\rm
supp}(g_{+})\subset{\mathbb{R}}_{+}$ and ${\rm
supp}(g_{-})\subset{\mathbb{R}}_{-}$. Similarly,
$z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{0})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{1})$
commutes with $U(g_{+}\times g_{-})$ with ${\rm
supp}(g_{+})\subset{\mathbb{R}}_{-}+t_{+}$ and ${\rm
supp}(g_{-})\subset{\mathbb{R}}_{+}+t_{-}$. Furthermore, the unitary
$e^{im\kappa Q}\otimes e^{-in\kappa Q}$ which implements an inner symmetry
commutes with any action of diffeomorphism [11]. Recall that the fixed point
subalgebra is strongly additive, hence by the assumed equality
$z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}$, this element
commutes with
${\mathcal{A}}_{0}^{\alpha_{0}}((t_{+},0)^{\prime})\otimes{\mathcal{A}}_{0}^{\alpha_{0}}((0,t_{-})^{\prime})$.
In particular, it commutes with any diffeomorphism of $S^{1}\times S^{1}$
supported in $(t_{+},0)^{\prime}\times(0,t_{-})^{\prime}$. There is a sequence
of diffeomorphisms $g_{i}$ which take ${\mathbb{R}}_{-}\times{\mathbb{R}}_{+}$
to $(t_{+}-\varepsilon_{i},0)\times(0,t_{-}+\varepsilon_{i})$ with support
disjoint from $(t_{+},0)\times(0,t_{-})$ for arbitrary small
$\varepsilon_{i}>0$. This fact and the diffeomorphism covariance imply that
$z$ is indeed contained in
${\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$. By a
similar reasoning, one sees that
$z^{\prime}\in{\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$
as well. Now by Reeh-Schlieder property for
${\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$ we have
$z=z^{\prime}$ since $z\Omega=z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa
Q})\Omega=z^{\prime}\Omega$.

Thus, if $z_{m,n}(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}_{m,n}$,
then
$z_{m,n}=z^{\prime}_{m,n}\in{\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$.
Furthermore, by Corollary 4.12, there is a separating vector $v\in
E(l,l^{\prime}){\mathcal{H}}$. Now it holds that $e^{inl\kappa-
iml^{\prime}\kappa}z_{m,n}v=z^{\prime}_{m,n}v$, hence from the separating
property of $v$ it follows that $e^{inl\kappa-
iml^{\prime}\kappa}z_{m,n}=z^{\prime}_{m,n}$ for each pair
$(l,l^{\prime})\in{\mathbb{Z}}\times{\mathbb{Z}}$. This is possible only if
both $n\kappa$ and $m\kappa$ are $2\pi$ multiple of an integer or
$z_{m,n}=z^{\prime}_{m,n}=0$. This is equivalent to that ${\hbox{\rm
Ad\,}}e^{i\kappa mQ_{0}}\otimes e^{i\kappa nQ_{0}}(z)=z$, namely, $z$ is an
element of the fixed point algebra
${\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-}))$ by
the action ${\hbox{\rm Ad\,}}e^{i\kappa mQ_{0}}\otimes e^{i\kappa nQ_{0}}$ of
$G\times G$. ∎

Note that the size of the intersection is very sensitive to the parameter
$\kappa$: If $\kappa$ is $2\pi$-multiple of a rational number, then the
inclusion $[{\mathcal{A}}_{0},{\mathcal{A}}_{0}^{G}]$ has finite index.
Otherwise, it has infinite index.

Finally, we comment on the net generated by the intersection. The intersection
takes a form of chiral net ${\mathcal{A}}_{0}^{G}\otimes{\mathcal{A}}_{0}^{G}$
where $G$ is generated by ${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}}$, hence the
S-matrix is trivial [15]. This result is expected also from [31], where Möbius
covariant net has always trivial S-matrix. Our construction is based on inner
symmetries which commute with Möbius symmetry, hence the net of strictly local
elements is necessarily Möbius covariant, then it should have trivial
S-matrix. But from this simple argument one cannot infer that the intersection
should be asymptotically complete, or equivalently chiral. This exact form of
the intersection can be found only by the present argument.

####  4.4.3 Construction through cyclic group actions

Here we briefly comment on the actions by the cyclic group ${\mathbb{Z}}_{k}$.
In previous Sections, we have constructed Borchers triples for the action of
$S^{1}$. It is not difficult to replace $S^{1}$ by a finite group
${\mathbb{Z}}_{k}$. Indeed, the main ingredient was the existence of the
Fourier components. For ${\mathbb{Z}}_{k}$-actions, the discrete Fourier
transform is available and all the arguments work parallelly (or even more
simply). For the later use, we state only the result without repeating the
obvious modification of definitions and proofs.

######  Theorem 4.17.

Let ${\mathcal{A}}_{0}$ be a strongly additive conformal net on $S^{1}$ and
$\alpha_{0,n}={\hbox{\rm Ad\,}}e^{i\frac{2\pi n}{k}Q_{0}}$ be an action of
${\mathbb{Z}}_{k}$ as inner symmetries. Then, for $n\in{\mathbb{Z}}_{k}$, the
triple

$\displaystyle{\mathcal{M}}_{Q_{0},n}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\frac{2\pi
n}{k}Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}({\mathbb{R}}_{+})\\}^{\prime\prime}$
$\displaystyle T$ $\displaystyle:=$ $\displaystyle T_{0}\otimes T_{0}$
$\displaystyle\Omega$ $\displaystyle:=$
$\displaystyle\Omega_{0}\otimes\Omega_{0}$

is an asymptotically complete Borchers triple with S-matrix $e^{i\frac{2\pi
n}{k}Q_{0}\otimes Q_{0}}$. As for strictly local elements, we have

${\mathcal{M}}_{Q_{0},n}\cap\left({\hbox{\rm
Ad\,}}T(t_{+},t_{-})({\mathcal{M}}_{Q,n}^{\prime})\right)={\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-})),$

where $G$ is the group of automorphisms of ${\mathcal{A}}_{0}$ generated by
${\hbox{\rm Ad\,}}e^{i\frac{2\pi n}{k}Q_{0}}$.

Note that, although the generator $Q_{0}$ of inner symmetries of the cyclic
group $Z_{k}$ is not unique, we used it always in the form $e^{i\kappa Q_{0}}$
or $e^{i\kappa Q_{0}\otimes Q_{0}}$ and these operators are determined by the
automorphisms. Spectral measures can be defined in terms of these
exponentiated operators uniquely on (the dual of) the cyclic group
${\mathbb{Z}}_{k}$. In this way, the choice of $Q_{0}$ does not appear in the
results and proofs.

##  5 Construction through endomorphisms on the $U(1)$-current net

###  5.1 The $U(1)$-current net and Longo-Witten endomorphisms

In this Section we will construct a family of Borchers triples for a specific
net on $S^{1}$. Since we need explicit formulae for the relevant operators, we
briefly summarize here some facts about the net called the $U(1)$-current net,
or the (chiral part of) free massless bosonic field. On this model, there has
been found a family of Longo-Witten endomorphisms [26]. We will construct a
Borchers triple for each of these endomorphisms. This model has been studied
with the algebraic approach since the fundamental paper [9]. We refer to [25]
for the notations and the facts in the following.

A fundamental ingredient is the irreducible unitary representation of the
Möbius group with the lowest weight $1$: Namely, we take the irreducible
representation of ${\rm PSL}(2,{\mathbb{R}})$ of which the smallest eigenvalue
of the rotation subgroup is $1$. We call the Hilbert space
${\mathcal{H}}^{1}$. We take a specific realization of this representation.
Namely, let $C^{\infty}(S^{1},{\mathbb{R}})$ be the space of real-valued
smooth functions on $S^{1}$. This space admits a seminorm

$\|f\|:=\sum_{k\geq 0}2k|\hat{f}_{k}|^{2},$

where $\hat{f}_{k}$ is the $k$-th Fourier component of $f$, and a complex
structure

$(\widehat{{\mathcal{I}}f})_{k}=-i\mathrm{sign}(k)\hat{f}_{k}.$

Then, by taking the quotient space by the null space with respect to the
seminorm, we obtain the complex Hilbert space ${\mathcal{H}}^{1}$. We say
$C^{\infty}(S^{1},{\mathbb{R}})\subset{\mathcal{H}}^{1}$. On this space, there
acts ${\rm PSL}(2,{\mathbb{R}})$ by naturally extending the action on
$C^{\infty}(S^{1},{\mathbb{R}})$.

Let us denote ${\mathcal{H}}^{n}:={\mathcal{H}}^{\otimes n}$ for a nonnegative
integer $n$. On this space, there acts the symmetric group $\mathrm{Sym}(n)$.
Let $Q_{n}$ be the projection onto the invariant subspace with respect to this
action. We put ${\mathcal{H}}^{n}_{s}:=Q_{n}{\mathcal{H}}^{n}$ and the
symmetric Fock space

${\mathcal{H}}^{\Sigma}_{s}:=\bigoplus_{n}{\mathcal{H}}^{n}_{s},$

and this will be the Hilbert space of the $U(1)$-current net on $S^{1}$. For
$\xi\in{\mathcal{H}}^{1}$, we denote by $e^{\xi}$ a vector of the form
$\sum_{n}\frac{1}{n!}\xi^{\otimes
n}=1\oplus\xi\oplus\left(\frac{1}{2}\xi\otimes\xi\right)\oplus\cdots$. Such
vectors form a total set in ${\mathcal{H}}^{\Sigma}_{s}$. The Weyl operator of
$\xi$ is defined by
$W(\xi)e^{\eta}=e^{-\frac{1}{2}\langle\xi,\xi\rangle-\langle\xi,\eta\rangle}e^{\xi+\eta}$.

The Hilbert space ${\mathcal{H}}^{\Sigma}_{s}$ is naturally included in the
unsymmetrized Fock space:

${\mathcal{H}}^{\Sigma}:=\bigoplus_{n}({\mathcal{H}}^{1})^{\otimes
n}={\mathbb{C}}\oplus{\mathcal{H}}^{1}\oplus\left({\mathcal{H}}^{1}\otimes{\mathcal{H}}^{1}\right)\oplus\cdots$

We denote by $Q_{\Sigma}$ the projection onto ${\mathcal{H}}^{\Sigma}_{s}$.
For an operator $X_{1}$ on the one particle space ${\mathcal{H}}^{1}$, we
define the second quantization of $X_{1}$ on ${\mathcal{H}}^{\Sigma}_{s}$ by

$\Gamma(X_{1}):=\bigoplus_{n}(X_{1})^{\otimes n}=1\oplus
X_{1}\oplus\left(X_{1}\otimes X_{1}\right)\oplus\cdots$

Obviously, $\Gamma(X_{1})$ restricts to the symmetric Fock space
${\mathcal{H}}^{\Sigma}_{s}$. We still write this restriction by
$\Gamma(X_{1})$ if no confusion arises. For a unitary operator $V_{1}\in
B({\mathcal{H}}^{1})$ and $\xi\in{\mathcal{H}}^{1}$, it holds that
$\Gamma(V_{1})e^{\xi}=e^{V_{1}\xi}$ and ${\hbox{\rm
Ad\,}}\Gamma(V_{1})(W(\xi))=W(V_{1}\xi)$. On the one particle space
${\mathcal{H}}^{1}$, there acts the Möbius group ${\rm PSL}(2,{\mathbb{R}})$
irreducibly by $U_{1}$. Then ${\rm PSL}(2,{\mathbb{R}})$ acts on
${\mathcal{H}}^{\Sigma}$ and on ${\mathcal{H}}^{\Sigma}_{s}$ and by
$\Gamma(U_{1}(g))$, $g\in{\rm PSL}(2,{\mathbb{R}})$. The representation of the
translation subgroup in ${\mathcal{H}}^{1}$ is denoted by
$T_{1}(t)=e^{itP_{1}}$ with the generator $P_{1}$.

The $U(1)$-current net ${{\mathcal{A}}^{(0)}}$ is defined as follows:

${{\mathcal{A}}^{(0)}}(I):=\\{W(f):f\in
C^{\infty}(S^{1},{\mathbb{R}})\subset{\mathcal{H}}^{1},{\rm supp}(f)\subset
I\\}^{\prime\prime}.$

The vector
$1\in{\mathbb{C}}={\mathcal{H}}^{0}\subset{\mathcal{H}}^{\Sigma}_{s}$ serves
as the vacuum vector $\Omega_{0}$ and $\Gamma(U_{1}(\cdot))$ implements the
Möbius symmetry. We denote by $T^{\Sigma}_{s}$ the representation of one-
dimensional translation of ${{\mathcal{A}}^{(0)}}$.

For this model, a large family of endomorphisms has been found by Longo and
Witten.

######  Theorem 5.1 ([26], Theorem 3.6).

Let $\varphi$ be an inner symmetric function on the upper-half plane
$\SS_{\infty}\subset{\mathbb{C}}$: Namely, $\varphi$ is a bounded analytic
function of $\SS_{\infty}$ with the boundary value $|\varphi(p)|=1$ and
$\varphi(-p)=\overline{\varphi(p)}$ for $p\in{\mathbb{R}}$. Then
$\Gamma(\varphi(P_{1}))$ commutes with $T^{\Sigma}_{s}$ (in particular
$\Gamma(\varphi(P_{1}))\Omega_{0}=\Omega_{0}$) and ${\hbox{\rm
Ad\,}}\Gamma(\varphi(P_{1}))$ preserves
${{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$. In other words,
$\Gamma(\varphi(P_{1}))$ implements a Longo-Witten endomorphism of
${{\mathcal{A}}^{(0)}}$.

###  5.2 Construction of Borchers triples

In this Section, we construct a Borchers triple for a fixed $\varphi$, the
boundary value of an inner symmetric function (see Section 5.1). Many
operators are naturally defined on the unsymmetrized Fock space, hence we
always keep in mind the inclusion
${\mathcal{H}}^{\Sigma}_{s}\subset{\mathcal{H}}^{\Sigma}$. The full Hilbert
space for the two-dimensional Borchers triples will be
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$.

On ${\mathcal{H}}^{m}$, there act $m$ commuting operators

$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
i\leq m\\}.$

We construct a unitary operator by the functional calculus on the
corresponding spectral measure. We set

  * •

$P_{i,j}^{m,n}:=({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})\otimes({\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})$,
which acts on ${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$, $1\leq i\leq m$ and
$1\leq j\leq n$.

  * •

$S^{m,n}_{\varphi}:=\prod_{i,j}\varphi(P_{i,j}^{m,n})$, where
$\varphi(P_{i,j}^{m,n})$ is the functional calculus on
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$.

  * •

$S_{\varphi}:=\bigoplus_{m,n}S_{\varphi}^{m,n}=\bigoplus_{m,n}\prod_{i,j}\varphi(P^{m,n}_{i,j})$

By construction, the operator $S_{\varphi}$ acts on
${\mathcal{H}}^{\Sigma}\otimes{\mathcal{H}}^{\Sigma}$. Furthermore, it is easy
to see that $S_{\varphi}$ commutes with both $Q_{\Sigma}\otimes{\mathbbm{1}}$
and ${\mathbbm{1}}\otimes Q_{\Sigma}$: In other words, $S_{\varphi}$ naturally
restricts to partially symmetrized subspaces
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ and
${\mathcal{H}}^{\Sigma}\otimes{\mathcal{H}}^{\Sigma}_{s}$ and to the totally
symmetrized space
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$. Note that
$S^{m,n}_{\varphi}$ is a unitary operator on the Hilbert spaces
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$ and $S_{\varphi}$ is the direct
sum of them.

Let $E_{1}\otimes E_{1}\otimes\cdots\otimes E_{1}$ be the joint spectral
measure of operators
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
j\leq n\\}$. The operators $\\{\varphi(P_{i,j}^{m,n}):1\leq i\leq m,1\leq
j\leq n\\}$ and $S_{\varphi}^{m,n}$ are compatible with the spectral measure
$\left(\overset{m\mbox{-}\mathrm{times}}{\overbrace{E_{1}\otimes
E_{1}\otimes\cdots\otimes
E_{1}}}\right)\otimes\left(\overset{n\mbox{-}\mathrm{times}}{\overbrace{E_{1}\otimes
E_{1}\otimes\cdots\otimes E_{1}}}\right)$ and one has

$\varphi(P_{i,j}^{m,n})=\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes\left({\mathbbm{1}}\otimes\cdots\underset{j\mbox{-th}}{dE_{1}(p_{j})}\otimes\cdots{\mathbbm{1}}\right).$

For $m=0$ or $n=0$ we set $\varphi_{i,j}^{m,n}={\mathbbm{1}}$ as a convention.

According to this spectral decomposition, we decompose $S_{\varphi}$ with
respect only to the right component as in the commutativity Lemma 4.1:

$\displaystyle S_{\varphi}$ $\displaystyle=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}\varphi(P_{i,j}^{m,n})$
$\displaystyle=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes
dE_{0}(p_{1})\otimes\cdots\otimes dE(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{m,n}\int\prod_{i,j}\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes
dE_{0}(p_{1})\otimes\cdots\otimes dE(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\bigoplus_{m}\prod_{j}(\varphi(p_{j}P_{1}))^{\otimes
m}\otimes dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\bigoplus_{m}(\varphi(p_{j}P_{1}))^{\otimes
m}\otimes dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\Gamma(\varphi(p_{j}P_{1}))\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n}),$

where the integral and the product commute in the third equality since the
spectral measure is disjoint for different values of $p$’s, and the sum and
the product commute in the fifth equality since the operators in the integrand
act on mutually disjoint spaces, namely on
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{\Sigma}$ for different $m$. Since all
operators appearing in the integrand in the last expression are the second
quantization operators, this formula naturally restricts to the partially
symmetrized space ${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$.

######  Lemma 5.2.

It holds for $x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$ that

$[x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0,$

on the Hilbert space
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$.

###### Proof.

The operator $S_{\varphi}$ is disintegrated into second quantization operators
as we saw above. If $\varphi$ is an inner symmetric function, then so is
$\varphi(p_{j}\cdot)$, $p_{j}\geq 0$, thus each $\Gamma(\varphi(p_{j}P_{1}))$
implements a Longo-Witten endomorphism.

Note that $S_{\varphi}$ restricts naturally to
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ by construction and
$x\otimes{\mathbbm{1}}$ and $x^{\prime}\otimes{\mathbbm{1}}$ extend naturally
to ${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ since the right-
components of them are just the identity operator ${\mathbbm{1}}$. Then we
calculate the commutation relation on
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$. This is done in the
same way as Lemma 4.1: Namely, we have

${\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})=\bigoplus_{n}\int{\hbox{\rm
Ad\,}}\left(\prod_{j}\Gamma(\varphi(p_{j}P_{1}))\right)(x^{\prime})\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n}).$

And this commutes with $x\otimes{\mathbbm{1}}$. Indeed, since
$x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$, hence ${\hbox{\rm
Ad\,}}\Gamma(\varphi(p_{j}))(x^{\prime})\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$
for any $p_{j}\geq 0$ by Theorem 5.1 of Longo-Witten, and by the fact that the
spectral support of $E_{1}$ is positive. Precisely, we have
$[x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0$ on
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$.

Now all operators $S_{\varphi}$, $x\otimes{\mathbbm{1}}$ and
$x^{\prime}\otimes{\mathbbm{1}}$ commute with ${\mathbbm{1}}\otimes
Q_{\Sigma}$, we obtain the thesis just by restriction. ∎

Finally we construct a Borchers triple by following the prescription at the
end of Section 3.1.

######  Theorem 5.3.

The triple

  * •

${\mathcal{M}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes
y):x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-}),y\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})\\}^{\prime\prime}$

  * •

$T=T_{0}\otimes T_{0}$ of ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$

  * •

$\Omega=\Omega_{0}\otimes\Omega_{0}$ of
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$

is an asymptotically complete Borchers triple with S-matrix $S_{\varphi}$.

###### Proof.

This is almost a repetition of the proof of Theorem 4.2. Namely, the
conditions on $T$ and $\Omega$ are readily satisfied since they are same as
the chiral triple. The operators $S_{\varphi}$ and $T$ commute since both are
the functional calculus of the same spectral measure, hence $T(t_{0},t_{1})$
sends ${\mathcal{M}}_{\varphi}$ into itself for $(t_{0},t_{1})\in
W_{\mathrm{R}}$. The vector $\Omega$ is cyclic for ${\mathcal{M}}_{\varphi}$
since ${\mathcal{M}}_{\varphi}\Omega\supset\\{x\otimes{\mathbbm{1}}\cdot
S_{\varphi}\cdot{\mathbbm{1}}\otimes
y\cdot\Omega\\}=\\{x\otimes{\mathbbm{1}}\cdot{\mathbbm{1}}\otimes
y\cdot\Omega\\}$ and the latter is dense by the Reeh-Schlieder property of the
chiral net. The separating property of $\Omega$ is shown through Lemma 5.2. ∎

######  Remark 5.4.

In this approach, the function $\varphi$ itself appears in two-particle
scattering, not the square as in [22]. Thus, although the formulae look
similar, the present construction contains much more examples.

#### Intersection property for constant functions $\varphi$

For the simplest cases $\varphi(p)=1$ or $\varphi(p)=-1$, we can easily
determine the strictly local elements. Indeed, for $\varphi(p)=1$,
$S_{\varphi}={\mathbbm{1}}$ and the Borchers triple coincides with the one
from the original chiral net. For $\varphi(p)=-1$,
$S_{\varphi}^{m,n}=(-1)^{mn}\cdot{\mathbbm{1}}$ and it is not difficult to see
that if one defines an operator $Q_{0}:=2P_{e}-{\mathbbm{1}}$, where $P_{e}$
is a projection onto the “even” subspace $\bigoplus_{n}{\mathcal{H}}^{2n}_{s}$
of ${\mathcal{H}}^{\Sigma}_{s}$, then $e^{i\pi Q_{0}}$ implements a
${\mathbb{Z}}_{2}$-action of inner symmetries on ${{\mathcal{A}}^{(0)}}$ and
$S_{\varphi}=e^{i\pi Q_{0}\otimes Q_{0}}$. Then Theorem 4.17 applies to find
that the strictly local elements are of the form
${{\mathcal{A}}^{(0)}}^{{\mathbb{Z}}_{2}}\otimes{{\mathcal{A}}^{(0)}}^{{\mathbb{Z}}_{2}}$
where the action of ${\mathbb{Z}}_{2}$ is realized by ${\hbox{\rm
Ad\,}}e^{i\pi nQ_{0}}$.

###  5.3 Free fermionic case

As explained in [26], one can construct a family of endomorphisms on the
Virasoro net ${\rm Vir}_{c}$ with the central charge $c=\frac{1}{2}$ by
considering the free fermionic field. With a similar construction using the
one-particle space on which the Möbius group acts irreducibly and projectively
with the lowest weight $\frac{1}{2}$, one considers the free fermionic
(nonlocal) net on $S^{1}$, which contains ${\rm Vir}_{\frac{1}{2}}$ with index
$2$.

The endomorphisms are implemented again by the second quantization operators.
By “knitting up” such operators as is done for bosonic $U(1)$-current case,
then by restricting to the observable part ${\rm Vir}_{\frac{1}{2}}$, we
obtain a family of Borchers triples with the asymptotic algebra ${\rm
Vir}_{\frac{1}{2}}\otimes{\rm Vir}_{\frac{1}{2}}$ with nontrivial S-matrix. In
the present article we omit the detail, and hope to return to this subject
with further investigations.

##  6 Conclusion and outlook

We showed that any two-dimensional massless asymptotically complete model is
characterized by its asymptotic algebra which is automatically a chiral Möbius
net, and the S-matrix. Then we reinterpreted the Buchholz-Lechner-Summers
deformation applied to chiral conformal net in this framework: It corresponds
to the S-matrix $e^{i\kappa P_{0}\otimes P_{0}}$. Furthermore we obtained
wedge-local nets through periodic inner symmetries which have S-matrix
$e^{i\kappa Q_{0}\otimes Q_{0}}$. We completely determined the strictly local
contents in terms of the fixed point algebra when the chiral component is
strongly additive and conformal. Unfortunately, the S-matrix restricted to the
strictly local part is trivial. For the $U(1)$-current net and the Virasoro
net ${\rm Vir}_{c}$ with $c=\frac{1}{2}$, we obtained families of wedge-local
nets parametrized by inner symmetric functions $\varphi$.

One important lesson is that construction of wedge-local nets should be
considered as an intermediate step to construct strictly local nets: Indeed,
any Möbius covariant net has trivial S-matrix [31], hence the triviality of
S-matrix in the construction through inner symmetries is interpreted as a
natural consequence. Although the S-matrix as a Borchers triple is nontrivial,
this should be treated as a false-positive. The true nontriviality should be
inferred by examining the strictly local part. On the other hand, we believe
that the techniques developed in this paper will be of importance in the
further explorations in strictly local nets. The sensitivity of the strictly
local part to the parameter $\kappa$ in the case of the construction with
respect to inner symmetries gives another insight.

Apart from the problem of strict locality, a more systematic study of the
necessary or sufficient conditions for S-matrix is desired. Such a
consideration could lead to a classification result of certain classes of
massless asymptotically complete models. For the moment, a more realistic
problem would be to construct S-matrix with the asymptotic algebra
${\mathcal{A}}_{N}\otimes{\mathcal{A}}_{N}$, where ${\mathcal{A}}_{N}$ is a
local extension of the $U(1)$-current net [9, 26]. A family of Longo-Witten
endomorphisms has been constructed also for ${\mathcal{A}}_{N}$, hence a
corresponding family of wedge-local net is expected and recently a similar
kind of endomorphisms has been found for a more general family of nets on
$S^{1}$ [3]. Or a general scheme of deforming a given Wightman-field theoretic
net has been established [22]. The family of S-matrices constructed in the
present paper seems rather small, since there is always a pair of spectral
measures and their tensor product diagonalizes the S-matrix. This could mean
in physical terms that the interaction between two waves is not very strong.
We hope to address these issues in future publications.

#### Acknowledgment.

I am grateful to Roberto Longo for his constant support. I wish to thank
Marcel Bischoff, Kenny De Commer, Wojciech Dybalski and Daniele Guido for
useful discussions.

##  Appendix Appendix A A remark on BLS deformation

In the proof of Theorem 4.5 we used the fact that
$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}\xi\otimes\Omega_{0}=x\xi\otimes\Omega_{0}$.
The equation (2.2) from [7] translates in our notation to

$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}=\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B}{\hbox{\rm
Ad\,}}U(\kappa t_{1},\kappa t_{0})(x\otimes{\mathbbm{1}})FdE(t_{0},t_{1}),$

where $B$ is a bounded subset in ${\mathbb{R}}^{2}$ and $F$ is a finite
dimensional subspace in ${\mathcal{H}}$. Now it is easy to see that
$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}(\xi\otimes\Omega_{0})=x\xi\otimes\Omega_{0}$.
Indeed, we have $\xi\otimes\Omega_{0}\in E({\mathrm{L}}_{+})$, where
${\mathrm{L}}_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}^{2}:p_{0}+p_{1}=0\\}$, hence
the integral above is concentrated in ${\mathrm{L}}_{+}$, and for
$(t,t)\in{\mathrm{L}}_{+}$ it holds that ${\hbox{\rm Ad\,}}U(\kappa t,\kappa
t)(x\otimes{\mathbbm{1}})={\hbox{\rm Ad\,}}{\mathbbm{1}}\otimes
U_{0}(\sqrt{2}\kappa t)(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$. Then
the integral simplifies as follows:

$\displaystyle(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}(\xi\otimes\Omega_{0})$
$\displaystyle=$
$\displaystyle\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B\cap{\mathrm{L}}_{+}}{\hbox{\rm
Ad\,}}U(\kappa t,\kappa t)(x\otimes{\mathbbm{1}})\cdot F\cdot
dE(t,t)(\xi\otimes\Omega_{0})$ $\displaystyle=$
$\displaystyle\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B\cap{\mathrm{L}}_{+}}x\otimes{\mathbbm{1}}\cdot
dE(t,t)(\xi\otimes\Omega_{0})$ $\displaystyle=$ $\displaystyle
x\xi\otimes\Omega_{0}.$

This is what we had to prove.

## References

  * [1] Huzihiro Araki and László Zsidó.  Extension of the structure theorem of Borchers and its application to half-sided modular inclusions.  Rev. Math. Phys., 17(5):491–543, 2005.
  * [2] Hellmut Baumgärtel.  Operator algebraic methods in quantum field theory.  Akademie Verlag, Berlin, 1995.
  * [3] Marcel Bischoff.  Models in boundary quantum field theory associated with lattices and loop group models.  Comm. Math. Phys., to appear.
  * [4] H.-J. Borchers.  The CPT-theorem in two-dimensional theories of local observables.  Comm. Math. Phys., 143(2):315–332, 1992.
  * [5] R. Brunetti, D. Guido, and R. Longo.  Modular structure and duality in conformal quantum field theory.  Comm. Math. Phys., 156(1):201–219, 1993.
  * [6] D. Buchholz.  Collision theory for waves in two dimensions and a characterization of models with trivial $S$-matrix.  Comm. Math. Phys., 45(1):1–8, 1975.
  * [7] D. Buchholz, G. Lechner, and S.J. Summers.  Warped convolutions, Rieffel deformations and the construction of quantum field theories.  Commun. Math. Phys., 304:95–123, 2011.
  * [8] Detlev Buchholz, Claudio D’Antoni, and Roberto Longo.  Nuclear maps and modular structures. I. General properties.  J. Funct. Anal., 88(2):233–250, 1990.
  * [9] Detlev Buchholz, Gerhard Mack, and Ivan Todorov.  The current algebra on the circle as a germ of local field theories.  Nuclear Phys. B Proc. Suppl., 5B:20–56, 1988.  Conformal field theories and related topics (Annecy-le-Vieux, 1988).
  * [10] Detlev Buchholz and Stephen J. Summers.  Warped convolutions: a novel tool in the construction of quantum field theories.  In Quantum field theory and beyond, pages 107–121. World Sci. Publ., Hackensack, NJ, 2008.
  * [11] Sebastiano Carpi and Mihály Weiner.  On the uniqueness of diffeomorphism symmetry in conformal field theory.  Comm. Math. Phys., 258(1):203–221, 2005.
  * [12] Jacques Dixmier.  von Neumann algebras, volume 27 of North-Holland Mathematical Library.  North-Holland Publishing Co., Amsterdam, 1981.
  * [13] S. Doplicher and R. Longo.  Standard and split inclusions of von Neumann algebras.  Invent. Math., 75(3):493–536, 1984.
  * [14] Wojciech Dybalski and Yoh Tanimoto.  Asymptotic completeness for infraparticles in two-dimensional conformal field theory.  arXiv:1112.4102.
  * [15] Wojciech Dybalski and Yoh Tanimoto.  Asymptotic completeness in a class of massless relativistic quantum field theories.  Comm. Math. Phys., 305(2):427–440, 2011.
  * [16] Martin Florig.  On Borchers’ theorem.  Lett. Math. Phys., 46(4):289–293, 1998.
  * [17] Fabrizio Gabbiani and Jürg Fröhlich.  Operator algebras and conformal field theory.  Comm. Math. Phys., 155(3):569–640, 1993.
  * [18] Harald Grosse and Gandalf Lechner.  Wedge-local quantum fields and noncommutative Minkowski space.  J. High Energy Phys., (11):012, 26, 2007.
  * [19] Harald Grosse and Gandalf Lechner.  Noncommutative deformations of Wightman quantum field theories.  J. High Energy Phys., (9):131, 29, 2008.
  * [20] Rudolf Haag.  Local quantum physics.  Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1996.  Fields, particles, algebras.
  * [21] Yasuyuki Kawahigashi and Roberto Longo.  Classification of local conformal nets. Case $c<1$.  Ann. of Math. (2), 160(2):493–522, 2004.
  * [22] Gandalf Lechner.  Deformations of quantum field theories and integrable models.  Comm. Math. Phys., 212:265–302, 2012.
  * [23] Gandalf Lechner.  Construction of quantum field theories with factorizing $S$-matrices.  Comm. Math. Phys., 277(3):821–860, 2008.
  * [24] Roberto Longo.  Conformal subnets and intermediate subfactors.  Comm. Math. Phys., 237(1-2):7–30, 2003.  Dedicated to Rudolf Haag.
  * [25] Roberto Longo.  Real Hilbert subspaces, modular theory, ${\rm SL}(2,{\bf R})$ and CFT.  In Von Neumann algebas in Sibiu: Conference Proceedings, pages 33–91. Theta, Bucharest, 2008.
  * [26] Roberto Longo and Edward Witten.  An algebraic construction of boundary quantum field theory.  Commun. Math. Phys., 303:213–232, 2011.
  * [27] Roberto Longo and Feng Xu.  Topological sectors and a dichotomy in conformal field theory.  Comm. Math. Phys., 251(2):321–364, 2004.
  * [28] Andrew Pressley and Graeme Segal.  Loop groups.  Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1986.  Oxford Science Publications.
  * [29] K.-H. Rehren.  Chiral observables and modular invariants.  Comm. Math. Phys., 208(3):689–712, 2000.
  * [30] M. Takesaki.  Theory of operator algebras. II, volume 125 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2003.  Operator Algebras and Non-commutative Geometry, 6.
  * [31] Y. Tanimoto.  Noninteraction of waves in two-dimensional conformal field theory.  Comm. Math. Phys., 314(2):419–441, 2012.
  * [32] Antony Wassermann.  Operator algebras and conformal field theory. III. Fusion of positive energy representations of ${\rm LSU}(N)$ using bounded operators.  Invent. Math., 133(3):467–538, 1998.
  * [33] Hans-Werner Wiesbrock.  Half-sided modular inclusions of von-Neumann-algebras.  Comm. Math. Phys., 157(1):83–92, 1993.
  * [34] Feng Xu.  Jones-Wassermann subfactors for disconnected intervals.  Commun. Contemp. Math., 2(3):307–347, 2000.
  * [35] Feng Xu.  Strong additivity and conformal nets.  Pacific J. Math., 221(1):167–199, 2005.