# Ground state representations of some non-rational conformal nets

Yoh Tanimoto111Supported by Programma per giovani ricercatori, anno 2014 “Rita
Levi Montalcini” of the Italian Ministry of Education, University and
Research.
Dipartimento di Matematica, Universitá di Roma Tor Vergata
Via della Ricerca Scientifica, 1, I-00133 Roma, Italy
email<EMAIL_ADDRESS>

###### Abstract

We construct families of ground state representations of the
$\mathrm{U}(1)$-current net and of the Virasoro nets ${\rm Vir}_{c}$ with
central charge $c\geq 1$. We show that these representations are not covariant
with respect to the original dilations, and those on the ${\rm U(1)}$-current
net are not solitonic. Furthermore, by going to the dual net with respect to
the ground state representations of ${\rm Vir}_{c}$, we obtain a possibly new
family of Möbius covariant nets on $S^{1}$.

##  1 Introduction

A model of quantum field theory may be in various states: in quantum theory, a
system is defined by the algebra of observables and a physical state is
realized as a normalized positive linear functional on it [20]. Among them,
the most important state on a quantum field theory is the vacuum state, and
indeed, notable axiomatizations of quantum field theory are formulated in
terms of the vacuum state or the vacuum correlation functions. Yet, some other
states are of particular interest. For example, DHR representations correspond
to charged states. KMS states represent thermally equilibrium states. Limiting
cases of KMS states, where the temperature is zero, are called ground states
and they enjoy particular properties. In this paper, we construct ground
states on chiral components of certain two-dimensional conformal field
theories.

Ground states can be characterized in various ways, see e.g. [4]. In one of
them, ground states are invariant under the dynamics and the generator of the
dynamics has positive generator. In the operator-algebraic setting, this is
particularly interesting, because if one has a ground state on a one-
dimensional Haag-Kastler net, one can take the dual net in the GNS
representation, which can possibly give a new Haag-Kastler net, hence a
different model.

In this work, we focus on the simplest class of quantum field theory: chiral
components of two-dimensional conformal field theory. Previously we studied
KMS states on them [7, 8]. Differently from KMS representations, it appears to
be difficult to classify ground states for a given net: the techniques to
classify KMS states do not directly apply to ground states, because we do not
have a direct connection between ground states and modular automorphisms,
which was crucial in [7, Section 4.2.1, Appendix B]222 The arguments for the
case of maximal nets [7, Theorem 4.7] might still work for ground states., nor
extension results of KMS states to larger net (cf. [1][8, Appendix A]) are
available for ground states. On the other hand, we classified ground states on
loop algebras [29], yet it is still unclear whether the results passes to
ground states on the corresponding Haag-Kastler nets. Therefore, rather than
classifying them, we try to construct various examples.

Contrary to the case of loop algebras where ground state is unique [29], we
find that the $\mathrm{U}(1)$-current net and the Virasoro nets with $c\geq 1$
admit continuously many ground states. It turns out that the
$\mathrm{U}(1)$-current net admits a one-parameter family of automorphisms
commuting with translations, and by composing them with the vacuum state, we
obtain a family of ground states. As the Virasoro nets with $c\geq 1$
restricted to ${\mathbb{R}}$ can be embedded in the $\mathrm{U}(1)$-current
net [6], they also admit a family of ground states. In the GNS representations
of these ground states, one can take the dual net and they must be different
from the original net for the case of the Virasoro nets with $c>1$, since the
original nets are not strongly additive,

We also obtain explicit expressions of the current and the stress-energy
tensor in the GNS representation. These expressions in turn serve to show that
the GNS representations of the ${\rm U(1)}$-current net are not normal on
half-lines. This shows that the implication “positivity of energy
$\Longrightarrow$ solitonic representation”, conjectured in [21, Conjecture
34] for loop groups, does not hold for the $\mathrm{U}(1)$-current net.

This paper is organized as follows. In Section 2 we recall Möbius covariant
nets, operator-algebraic setting of chiral components of conformal field
theory and collect general facts on ground states. In Section 3 we introduce
examples of Möbius covariant nets, the ${\rm U(1)}$-current net and the
Virasoro nets. In Section 4 we construct ground states on these nets and study
their property. We conclude with some open problems in Section 5.

##  2 Preliminaries

###  2.1 Möbius covariant net on circle

The following is a mathematical setting for chiral components of two-
dimensional conformal field theory. These chiral components are essentially
quantum field theories on the real line ${\mathbb{R}}$, and by conformal
covariance they extend to $S^{1}$. In the operator-algebraic setting, they are
realized as nets (precosheaves) of von Neumann algebras on $S^{1}$. More
precisely, let ${\mathcal{I}}$ be the set of open, non-dense, non empty
intervals in $S^{1}$. A Möbius covariant net $({\mathcal{A}},U,\Omega)$ is a
triple of the map ${\mathcal{A}}:{\mathcal{I}}\ni
I\mapsto{\mathcal{A}}(I)\subset{\mathcal{B}}({\mathcal{H}})$, where
${\mathcal{A}}(I)$ is a von Neumann algebra on a Hilbert space
${\mathcal{H}}$, $U$ a strongly continuous representation of the group ${\rm
PSL}(2,{\mathbb{R}})$ on ${\mathcal{H}}$ and a unit vector
$\Omega\in{\mathcal{H}}$ satisfying the following conditions:

  1. (MN1)

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

  2. (MN2)

Locality: If $I_{1}\cap I_{2}=\emptyset$, then ${\mathcal{A}}(I_{1})$ and
${\mathcal{A}}(I_{2})$ commute.

  3. (MN3)

Möbius covariance: for $\gamma\in{\rm PSL}(2,{\mathbb{R}})$, ${\hbox{\rm
Ad\,}}U(\gamma)({\mathcal{A}}(I))={\mathcal{A}}(\gamma I)$.

  4. (MN4)

Positive energy: the generator $L_{0}$ of rotations $U(\rho(t))=e^{itL_{0}}$
is positive.

  5. (MN5)

Vacuum: $\Omega$ is the unique (up to a phase) invariant vector for
$U(\gamma),\gamma\in{\rm PSL}(2,{\mathbb{R}})$ and
$\overline{{\mathcal{A}}(I)\Omega}={\mathcal{H}}$ (the Reeh-Schlieder
property).

From (MN1)–(MN5), one can prove the following [17, 16]:

  1. (MN6)

Haag duality: ${\mathcal{A}}(I^{\prime})={\mathcal{A}}(I)^{\prime}$, where
$I^{\prime}$ is the interior of the complement of $I$.

  2. (MN7)

Additivity: If $I=\bigcup I_{j}$, then
${\mathcal{A}}(I)=\bigvee_{j}{\mathcal{A}}(I_{j})$.

We consider also the following additional properties:

  * •

Strong additivity: If $I_{1}$ and $I_{2}$ are the intervals obtained from $I$
by removing one point, then
${\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2})={\mathcal{A}}(I)$.

  * •

Conformal covariance: $U$ extends to a projective unitary representation of
the group ${\rm Diff}_{+}(S^{1})$ of orientation preserving diffeomorphisms
and ${\hbox{\rm Ad\,}}U(\gamma){\mathcal{A}}(I)={\mathcal{A}}(\gamma I)$, and
${\hbox{\rm Ad\,}}U(\gamma)(x)=x$ if $I\cap{\rm supp\,}\gamma=\emptyset$.

Strong additivity does not follow from (MN1)–(MN5), and indeed the Virasoro
nets with $c>1$ fail to have strong additivity [6, Section 4]. If conformal
covariance holds, $({\mathcal{A}},U,\Omega)$ is called a conformal net.

Concrete examples relevant to this work will be presented in Section 3.

###  2.2 Ground state representations

Let us identify ${\mathbb{R}}$ with a dense interval in $S^{1}$ by the
stereographic projection. By convention, the point of infinity $\infty$
corresponds to $-1\in S^{1}\subset{\mathbb{C}}$. We denote by
${\mathcal{A}}|_{\mathbb{R}}$ the net of algebras
$\\{{\mathcal{A}}(I):I\Subset{\mathbb{R}}\\}$ restricted to finite open
intervals in the real line ${\mathbb{R}}$. Translations $\tau(t)$ and
dilations $\delta(s)$ acts on ${\mathcal{A}}|_{\mathbb{R}}$, and they are
elements of ${\rm PSL}(2,{\mathbb{R}})$.

${\hbox{\rm Ad\,}}U(\tau(t))$ acts on
$\overline{\bigcup_{I\Subset{\mathbb{R}}}{\mathcal{A}}(I)}^{\|\cdot\|}$ as an
automorphism. A state $\varphi$ on ${\mathcal{A}}|_{\mathbb{R}}$ is by
definition a state on the $C^{*}$-algebra
$\overline{\bigcup_{I\Subset{\mathbb{R}}}{\mathcal{A}}(I)}^{\|\cdot\|}$. Let
$\varphi$ be a state on ${\mathcal{A}}|_{\mathbb{R}}$ and let $\pi_{\varphi}$
be the GNS representation with respect to $\varphi$. If $\pi_{\varphi}$
extends to ${\mathcal{A}}({\mathbb{R}}_{\pm})$ in the $\sigma$-weak topology,
then we say that $\pi_{\varphi}$ is solitonic. If there is a representation
$U^{\varphi}$ of the translation group ${\mathbb{R}}$ such that ${\hbox{\rm
Ad\,}}U^{\varphi}(\tau(t))(\pi_{\varphi}(x))=\pi_{\varphi}({\hbox{\rm
Ad\,}}U(\tau(t))(x))$, then we say that $\pi_{\varphi}$ is translation-
covariant. Furthermore, if the generator of $U^{\varphi}$ can be taken
positive, then we say $\pi_{\varphi}$ is a positive-energy representation.

If a state $\varphi$ on ${\mathcal{A}}|_{\mathbb{R}}$ is invariant under
${\hbox{\rm Ad\,}}U(\tau(t))$, then $\pi_{\varphi}$ is automatically
translation-covariant, because one can define
$U^{\varphi}(\tau(t))\pi_{\varphi}(x)\Omega_{\varphi}=\pi_{\varphi}({\hbox{\rm
Ad\,}}U(\tau(t))(x))\Omega_{\varphi}$, where $\Omega^{\varphi}$ is the GNS
vector for $\varphi$. If furthermore $U^{\varphi}$ has positive energy, then
we say that it is a ground state representation, and the state $\varphi$ is
called the ground state. By the Reeh-Schlieder argument (see e.g. [2, Theorem
1.3.2], [24, Theorem 3.2.1]), the GNS vector $\Omega_{\varphi}$ is cyclic for
$\pi_{\varphi}({\mathcal{A}}(I))$ for any $I\Subset{\mathbb{R}}$.

If $\pi_{\varphi}$ is a ground state representation with the ground state
vector $\Omega_{\varphi}$, then one can consider the translation-covariant net
$\\{\pi_{\varphi}({\mathcal{A}}(I))\\}$ on intervals in ${\mathbb{R}}$ on the
GNS representation space ${\mathcal{H}}_{\varphi}$. Its dual net is defined by
$\hat{\mathcal{A}}_{\varphi}({\mathbb{R}}_{+}+a):=\left(\bigvee_{I\Subset{\mathbb{R}}_{+}+a}\pi_{\varphi}({\mathcal{A}}(I))\right)^{\prime\prime}$
and
$\hat{\mathcal{A}}_{\varphi}((a,b))=\hat{\mathcal{A}}_{\varphi}({\mathbb{R}}_{+}+b)\cap\hat{\mathcal{A}}_{\varphi}({\mathbb{R}}_{+}+a)^{\prime}$.
By [19, Corollary 1.9], $\hat{\mathcal{A}}$ extends uniquely to a strongly
additive Möbius covariant net on $S^{1}$:

######  Theorem 2.1 (Guido-Longo-Wiesbrock).

There is a one-to-one correspondence between

  * •

Isomorphism classes of strongly additive Möbius covariant nets

  * •

Isomorphism classes of Borchers triples $({\mathcal{M}},U,\Omega)$,
(${\mathcal{M}}$ is a von Neumann algebra, $U$ is a representation of
${\mathbb{R}}$ with positive generator, $\Omega$ is cyclic and separating for
${\mathcal{M}}$ and $U(t)\Omega=\Omega$ such that ${\hbox{\rm
Ad\,}}U(t)({\mathcal{M}})\subset{\mathcal{M}}$ for $t\geq 0$) with the
property that $\Omega$ is cyclic for ${\hbox{\rm
Ad\,}}U(t)({\mathcal{M}})^{\prime}\cap{\mathcal{M}}$

Even if the given net ${\mathcal{A}}$ is conformally covariant, it is not
known whether the net $\hat{\mathcal{A}}_{\varphi}$ is conformally covariant.
Although $\hat{\mathcal{A}}_{\varphi}$ is ${\rm
Diff}_{\mathrm{c}}({\mathbb{R}})$-covariant in the natural sense, where ${\rm
Diff}_{\mathrm{c}}({\mathbb{R}})$ is the group of diffeomorphisms of
${\mathbb{R}}$ with compact support, we do not have the uniqueness results of
${\rm Diff}_{\mathrm{c}}({\mathbb{R}})$-action (cf. [10, Theorem 5.5] for a
uniqueness theorem for ${\rm Diff}_{+}(S^{1})$-action). See [31, Chapter 4]
for some attempts to construct ${\rm Diff}_{+}(S^{1})$-covariance), and [25,
Section 4.2] for some examples in $(1+1)$-dimensions where Möbius covariance
gets lost by passing to the dual net.

The purpose of this paper is to construct ground state representations of
certain nets. One of the ideas to construct ground state representations is
the following: an automorphism $\alpha$ of the net
${\mathcal{A}}|_{\mathbb{R}}$ is a family
$\\{\alpha_{I}\\}_{I\Subset{\mathbb{R}}}$ of automorphisms of
$\\{{\mathcal{A}}(I)\\}$ such that
$\alpha_{\tilde{I}}|_{{\mathcal{A}}(I)}=\alpha_{I}$ for $I\subset\tilde{I}$.
$\alpha$ extends naturally to
$\overline{\bigcup_{I\Subset{\mathbb{R}}}{\mathcal{A}}(I)}^{\|\cdot\|}$.

######  Lemma 2.2.

If $\alpha$ commutes with ${\hbox{\rm Ad\,}}U(\tau(t))$, then
$\omega\circ\alpha$ is a ground state.

###### Proof.

The GNS representation of $\omega\circ\alpha$ is given by
$\pi_{\omega}\circ\alpha=\alpha$, where $\pi_{\omega}$ is the vacuum
representation, the identity map. As $\alpha$ commutes with ${\hbox{\rm
Ad\,}}U(\tau(t))$, the same representation $U(\tau(t))$ with positive energy
implements translations. ∎

Furthermore, combining [7, Lemma 4.4, Lemma 4.5], we obtain the following
dichotomy: if ${\mathcal{A}}|_{\mathbb{R}}$ admits one nontrivial ground
state, then there must be continuously many ground states.

######  Proposition 2.3.

If there is an automorphism $\alpha$ of ${\mathcal{A}}|_{\mathbb{R}}$
commuting with translations and if $\omega\circ\alpha\neq\alpha$, then
$\\{\omega\circ\alpha\circ{\hbox{\rm Ad\,}}U(\delta(s))\\}_{s\in{\mathbb{R}}}$
are mutually different and their GNS representations $\\{{\hbox{\rm
Ad\,}}U(\delta(-s))\circ\alpha\circ{\hbox{\rm
Ad\,}}U(\delta(s))\\}_{s\in{\mathbb{R}}}$ are mutually unitarily inequivalent.

Note that $\alpha$ as in Proposition 2.3 cannot be implemented by a unitary
operator, because $\Omega$ is unique up to a phase. Hence, the second
statement implies that the dilation automorphisms $\\{{\hbox{\rm
Ad\,}}U(\delta(s))\\}$ are not unitarily implemented in the GNS
representations (namely, there are no $U^{\alpha}(s)$ such that ${\hbox{\rm
Ad\,}}U^{\alpha}(s)\circ\alpha=\alpha\circ{\hbox{\rm Ad\,}}U(\delta(s))$): if
they were implemented, then it would imply that $\alpha$ and ${\hbox{\rm
Ad\,}}U(\delta(-s))\circ\alpha\circ{\hbox{\rm Ad\,}}U(\delta(s))$ are
unitarily equivalent, which is a contradiction.

##  3 Main examples

Here we introduce examples of conformal nets. Although our emphasis is on
their restriction to ${\mathbb{R}}$, they are most conveniently defined on
$S^{1}$. Our treatment of these models follows that of [8, Section 4.1].

###  3.1 The $\mathrm{U}(1)$-current net

Buchholz, Mack and Todorov obtained some fundamental results in the operator-
algebraic treatment of the ${\rm U(1)}$-current algebra [5]. The current
$J(f)$ can be defined by the Fourier modes $\\{J_{n}\\}_{n\in{\mathbb{Z}}}$,
where $J_{n}$ satisfy the following commutation relations
$[J_{m},J_{n}]=m\delta_{m+n,0}$ and hence the current $J$ as a quantum field
on $S^{1}$ satisfies the following relation:
$[J(f),J(g)]=i\int_{S^{1}}f(\theta)g^{\prime}(\theta)d\theta$, for $f,g\in
C^{\infty}(S^{1},{\mathbb{R}})$. It admits a vacuum state
$\langle\Omega,\cdot\,\Omega\rangle$ such that $J_{n}\Omega=0$ for $n\geq 0$,
and the representation is uniquely determined by this property, unitarity
$J_{n}^{*}=J_{-n}$ and the commutation relation. The smeared current
$J(f)=\sum_{n}\hat{f}_{n}J_{n}$, where $f\in C^{\infty}(S^{1},{\mathbb{R}})$
and $\hat{f}_{n}=\frac{1}{2\pi}\int f(\theta)e^{-in\theta}$ is the Fourier-
coefficient, is essentially self-adjoint on the subspace
${\mathcal{H}}^{\mathrm{fin}}$ spanned by vectors $J_{-n_{1}}\cdots
J_{-n_{k}}\Omega$, $n_{j}\in{\mathbb{N}}$.

The vacuum state is invariant under ${\rm
PSL}(2,{\mathbb{R}})$-transformations $J(f)\mapsto
J(f\circ\gamma^{-1}),\gamma\in{\rm PSL}(2,{\mathbb{R}})$, hence this can be
implemented by a unitary operator $U(\gamma)$, and in this way we obtain a
unitary representation $U$ of ${\rm PSL}(2,{\mathbb{R}})$ with positive
energy.

The exponential $W(f)=e^{iJ(f)}$ is called the Weyl operator. They satisfy
$W(f)W(g)=e^{-\frac{i}{2}\sigma(f,g)}W(f+g)$, where
$\sigma(f,g)=\int_{S^{1}}f(\theta)g^{\prime}(\theta)d\theta$. If we define von
Neumann algebras by ${\mathcal{A}}_{\rm U(1)}(I)=\\{W(f):{\rm supp\,}f\subset
I\\}^{\prime\prime}$, then $({\mathcal{A}}_{\rm U(1)},U,\Omega)$ is a Möbius
covariant net. This is called the ${\rm U(1)}$-current net. It is known that
it satisfies strong additivity [6, (4.21) and below] and conformal covariance
[27, Secction 5.3].

We can also define $J$ in the real line picture. To do this, note that the
above commutation relation is invariant under diffeomorphisms. Therefore, if
we introduce $J(f)$ for $f\in
C^{\infty}_{\mathrm{c}}({\mathbb{R}},{\mathbb{R}})$ by $J(f\circ C^{-1})$
where $C(t)=-\frac{t-i}{t+i}$ is the Cayley transform mapping ${\mathbb{R}}\to
S^{1}$ (with a slight abuse of notation: the definition of $J(f)$ depends on
whether $f\in C^{\infty}_{\mathrm{c}}({\mathbb{R}})$ or $f\in
C^{\infty}(S^{1})$), we obtain the same commutation relation

$\displaystyle[J(f),J(g)]=i\int_{\mathbb{R}}f(t)g^{\prime}(t)dt,\qquad f,g\in
C^{\infty}({\mathbb{R}},{\mathbb{R}}).$ (1)

We call it $J$ in the real line picture. Again by the diffeomorphism
covariance, translations and dilations acts on $J$ in the natural way: if
$\gamma$ is such a transformation, then ${\hbox{\rm
Ad\,}}U(\gamma)(J(f))=J(f\circ\gamma^{-1})$.

######  Lemma 3.1.

For a function $f$ on $S^{1}$ such that $\sum_{k\geq
0}k|\hat{f}_{k}|^{2}<\infty$, $J(f)$ and $W(f)$ can be defined by continuity.

###### Proof.

Note first that if $J(f_{n})\Omega\to J(f)\Omega$, then $W(f_{n})\to W(f)$ in
the strong operator topology. Indeed, recall that
$\langle\Omega,W(g)\Omega\rangle=e^{-\frac{1}{2}\|J(g)\Omega\|^{2}}$ [24,
Section 6.5] and $\sigma(f,g)=2\mathrm{Im}\,\langle
J(f)\Omega,J(g)\Omega\rangle$. Therefore, it is straightforward to see that,
if $J(f_{n})\Omega\to J(f)\Omega$, then $W(f_{n})\Omega\to W(f)\Omega$ and
$\sigma(f_{n},g)\to\sigma(f,g)$. It then follows that for a fixed $g$

$\displaystyle\|(W(f_{n})-W(f))W(g)\Omega\|\to 0,$

and since $W(g)\Omega$ is total and $\\{W(f_{n})\\}$ are unitary, this implies
the claimed strong convergence.

For a smooth function $f$ on $S^{1}$, $\|J(f)\Omega\|^{2}=\sum_{k\geq
0}k|\hat{f}_{k}|^{2}$, and hence $J$ can be extended by continuity to any
$L^{2}$-function $f$ on $S^{1}$ such that $\sum_{k\geq
0}k|\hat{f}_{k}|^{2}<\infty$. ∎

If $f$ is piecewise smooth and continuous, then $\sum_{k\geq
0}k|\hat{f}_{k}|^{2}<\infty$. Indeed, as an operator on $L^{2}(S^{1})$,
$i\frac{d}{d\theta}$ is self-adjoint on the following domain [28, X.1 Example
1]

$\\{f\in C(S^{1}):f\text{ is absolutely continuous},f^{\prime}\text{ is in
}L^{2}(S^{1})\\}$

and
$\sum_{k}|\hat{f}^{\prime}_{k}|^{2}=\sum_{k}k^{2}|\hat{f}_{k}|^{2}<\infty$,
therefore, $\sum_{k\geq 0}k|\hat{f}_{k}|^{2}<\infty$. These facts can be seen
as an easier version of [10, Proposition 4.5][32, Lemma 2.2].

###  3.2 The Virasoro nets

On the Hilbert space of the ${\rm U(1)}$-current net, one can construct
another quantum field. Let $L_{n}=\frac{1}{2}\sum_{m}:J_{-m}J_{n+m}:$ where
the normal ordering means $J_{k}$ with negative $k$ comes to the left [22,
(2.9)]. Then they satisfy the Virasoro algebra
$[L_{m},L_{n}]=(m-n)L_{m+n}+c\frac{m(m^{2}-1)}{12}$ with the central charge
$c=1$. The stress-energy tensor $T(f)=\sum\hat{f}_{n}L_{n}$ is essentially
self-adjoint on ${\mathcal{H}}^{\mathrm{fin}}$ and satisfies the commutation
relations $[T(f),J(g)]=iJ(fg^{\prime})$. Especially, if $f$ and $g$ have
disjoint support, they commute. By the linear energy bound [10, Proposition
4.5][3, Lemma 3.2] and the Driessler-Fröhlich theorem [13, Theorem 3.1](or its
adaptation to the case of operators [3, Theorem 3.4]), they actually strongly
commute. For $\gamma\in{\rm Diff}(S^{1})$, it holds that ${\hbox{\rm
Ad\,}}U(\gamma)(T(f))=T(\gamma_{*}f)+\beta(\gamma,f)$ with a certain
$\beta(\gamma,f)\in{\mathbb{R}}$, where $\gamma_{*}f$ is the pullback of $f$
as a vector field by $\gamma$ [15, Proposition 3.1]. Moreover, $T(f)$
integrates to the representation $U$ in the sense that $U({\rm
Exp}(tf))=e^{itT(f)}$ up to a phase [18, Section 4], see also [15, Proposition
5.1].

Let us denote ${\mathcal{B}}_{1}(I)=\\{e^{iT(f)}:{\rm supp\,}f\subset
I\\}^{\prime\prime}$ and call it the Virasoro subnet. By Haag duality, it
holds that ${\mathcal{B}}_{1}(I)\subset{\mathcal{A}}_{\rm U(1)}(I)$.
Furthermore, by the Reeh-Schlieder argument, ${\mathcal{H}}_{{\rm
Vir}_{1}}:=\overline{{\mathcal{B}}_{1}(I)\Omega}$ does not depend on $I$. The
restriction of $\\{{\mathcal{B}}_{1}(I)\\}$ and $U$ to ${\mathcal{H}}_{{\rm
Vir}_{1}}$ together with $\Omega\in{\mathcal{H}}_{{\rm Vir}_{1}}$ is called
the Virasoro net with $c=1$ and we denote it by $({\rm
Vir}_{1},U_{1},\Omega_{1})$.

We can also consider the real line picture for $T$. For a vector field $f$ on
${\mathbb{R}}$, or more precisely $f(t)\frac{d}{dt}$, we define
$T(f):=T(C^{-1}_{*}f)$. Conversely, a vector field $f$ on $S^{1}$ corresponds
to $C_{*}f(t)=\frac{t^{2}+1}{2}f(C^{-1}(t))$.

In [8, Proof of Theorem 4.7], we also computed the exponentiation of the
relation between $J(g)$ and $T(f)$:

$\displaystyle{\hbox{\rm
Ad\,}}W(g)(T(f))=T(f)-J(fg^{\prime})+\frac{1}{2}\sigma(fg^{\prime},g),$

where this equality holds on ${\mathcal{H}}^{\mathrm{fin}}$.

In order to obtain the Virasoro nets with $c>1$, we need to perturb $T$ as in
[6, (4.6)]. We have computed its real-line picture [8, Section 5.3]:

$\displaystyle T^{\kappa}(f)=T(f)+\kappa J(f^{\prime}),$

then $T^{\kappa}$ satisfies the commutation relation

$[T^{\kappa}(f),T^{\kappa}(g)]=iT^{\kappa}(fg^{\prime}-f^{\prime}g)+i\frac{1+\kappa^{2}}{12}\int_{\mathbb{R}}f^{\prime\prime\prime}(t)g(t)dt,$

hence the central charge is $c=1+\kappa^{2}$. Define
${\mathcal{B}}_{c}(I):=\\{e^{iT^{\kappa}(f)}:{\rm supp\,}f\subset
I\\}^{\prime\prime}$. We have seen that
${\mathcal{B}}_{c}(I)\subset{\mathcal{A}}_{{\rm U(1)}}(I)$, but
${\mathcal{B}}_{c}(I)$ is covariant only with respect to $U$ restricted to
translations and dilations. Yet, ${\mathcal{H}}_{{\rm
Vir}_{c}}:=\overline{{\mathcal{B}}_{c}(I)\Omega}$ does not depend on $I$ again
by the Reeh-Schlieder argument. It was shown in [6, Section 4, (4.8)] that by
exploiting the Möbius invariance of $n$-points functions of $T^{\kappa}$, the
restriction of ${\mathcal{B}}_{c}$ to ${\mathcal{H}}_{{\rm Vir}_{c}}$ can be
extended to a conformal net, the Virasoro net ${\rm Vir}_{c}$ with central
charge $c=1+\kappa^{2}$. ${\rm Vir}_{c},c>1$ does not satisfy strong
additivity [6, Section 4, (4.13)]. It is an open problem whether the dual net
of ${\rm Vir}_{c}$ is conformal (${\rm Diff}(S^{1})$-covariant).

##  4 Ground states

###  4.1 On the ${\rm U(1)}$-current net

Following Lemma 2.2, we construct automorphisms of ${\mathcal{A}}_{{\rm
U(1)}}|_{\mathbb{R}}$ commuting with translations. This has been done in [8,
Proposition 4.1]. Let us recall it in our present notations.

######  Lemma 4.1.

For $q\in{\mathbb{R}}$, there is an automorphism $\alpha_{q}$ of
${\mathcal{A}}_{{\rm U(1)}}|_{\mathbb{R}}$ commuting with translations such
that $\alpha_{q}(W(f))=e^{iq\int_{\mathbb{R}}f(t)dt}W(f)$.

###### Proof.

Let $I\Subset{\mathbb{R}}$. There is a smooth function $s_{I}$ with compact
support such that $s_{I}(t)=t$. By commutation relation (1), for $f$ with
${\rm supp\,}f\subset I$ we have

${\hbox{\rm
Ad\,}}W(qs_{I})(W(f))=e^{iq\int_{\mathbb{R}}s_{I}^{\prime}(t)f(f)dt}W(f)=e^{iq\int_{\mathbb{R}}f(f)dt}W(f).$

We define $\alpha_{q,I}={\hbox{\rm Ad\,}}W(qs_{I})$. This does not depend on
the choice of $s_{I}$ within the restriction above, because
${\mathcal{A}}_{{\rm U(1)}}(I)$ is generated by $W(f)$ with ${\rm
supp\,}f\subset I$. $\alpha_{q}$ commutes with translations because
$\int_{\mathbb{R}}f(t)dt$ is invariant under translations. ∎

From here, it is immediate to construct ground states by Lemma 2.2. We collect
some properties of the resulting GNS representations.

######  Theorem 4.2.

We have the following.

  * •

The states $\\{\omega\circ\alpha_{q}\\}$ are mutually different ground states
of ${\mathcal{A}}_{{\rm U(1)}}|_{\mathbb{R}}$, and they are connected with
each other by dilations. Dilations are not implemented unitarily.

  * •

The GNS vector $\Omega_{q}$ of $\omega\circ\alpha_{q}$ is in the domain of
$J_{q}(f)$, where $\alpha_{q}(W(f))=e^{iJ_{q}(f)}$ and
$J_{q}(f)=J(f)+q\int_{\mathbb{R}}f(t)dt$. Especially, it holds that
$\omega\circ\alpha_{q}(J(f))=q\int_{\mathbb{R}}f(t)dt$

  * •

The GNS representations $\\{\alpha_{q}\\}$ do not extend to
${\mathcal{A}}({\mathbb{R}}_{\pm})$ in the weak operator topology, hence are
not solitonic.

  * •

The dual nets $\\{\hat{\alpha}_{q}({\mathcal{A}}_{{\rm U(1)}}(I))\\}$ are
equal to the original net ${\mathcal{A}}_{{\rm U(1)}}$.

###### Proof.

As $\\{\alpha_{q}\\}$ are automorphisms commuting with translations, the
states $\\{\omega\circ\alpha_{q}\\}$ are ground states by Lemma 2.2, and their
GNS representations are $\\{\alpha_{q}\\}$. They are different states because
they give the scalar $e^{iq\int_{\mathbb{R}}f(t)dt}$ to $W(f)$. It follows
from Proposition 2.3 that they are connected by dilations and dilations are
not implemented unitarily.

Let us recall the action
$\alpha_{q}(W(sf))=e^{iqs\int_{\mathbb{R}}f(t)dt}W(sf)$. From this it is
immediate that

$\frac{d}{ds}\alpha_{q}(W(sf))\Omega|_{s=0}=iJ(f)\Omega+iq\int_{\mathbb{R}}f(t)dt.$

We also showed [8, Lemma 4.6] that
${\mathcal{H}}^{\infty}=\bigcap_{n}{\mathrm{Dom}}(L_{0}^{n})$ is invariant
under Weyl operators $W(g)$, hence we have, the following equation of
operators on ${\mathcal{H}}^{\infty}$:

${\hbox{\rm Ad\,}}W(q\sigma_{I})(J(f))=J(f)+q\int_{\mathbb{R}}f(t)dt.$

${\mathcal{H}}^{\infty}$ is a core of $L_{0}$, and hence of $J(f)$ by the
commutator theorem [28, Theorem X.37] and the estimate $\|J(f)\xi\|\leq
c_{f}\|(L_{0}+1)^{\frac{1}{2}}\xi\|$, see the computations in [23, Proposition
1] (this estimate is known since [6, below (2.23)]. Since
${\mathcal{H}}^{\infty}$ is invariant under $J(f)$, to apply the commutator
theorem with slightly different assumptions [28, Theorem X.36], it is enough
to have a linear bound $\|J(f)\xi\|\leq c_{f}\|(L_{0}+1)\xi\|$).

To show the non-normality on the half line ${\mathbb{R}}_{+}$
(${\mathbb{R}}_{-}$ is analogous), we take a sequence of smooth functions
$g_{n}(-e^{i\theta})$ on $S^{1}$:

$g_{n}(-e^{i\theta})=\left\\{\begin{array}[]{ll}0&\text{ for
}0\leq\theta<\pi\\\ \theta-\pi&\text{ for }\pi\leq\theta<\frac{3\pi}{2}\\\
-\theta+2\pi&\text{ for }\frac{3\pi}{2}\leq\theta<2\pi-\frac{1}{2n}\\\
-2(\theta-2\pi+\frac{1}{2n})&\text{ for
}2\pi-\frac{1}{n}\leq\theta<2\pi-\frac{1}{2n}\\\ 0&\text{ for
}2\pi-\frac{1}{2n}\leq\theta\leq 2\pi\end{array}\right.$

$\theta$$g(-e^{i\theta})$$0$$2\pi$$\pi$$\frac{\pi}{2}$ Figure 1: The functions
$g_{n}(-e^{i\theta})$ (the thick line) and $g(-e^{i\theta})$ (the thin line
above), restricted to $0\leq\theta\leq 2\pi$.

It is clear that each $g_{n}$ is piecewise smooth (linear) and continuous, and
both $g_{n},g_{n}^{\prime}$ converges in the $L^{2}$-norm to

$g(-e^{i\theta})=\left\\{\begin{array}[]{ll}0&\text{ for }0\leq\theta<\pi\\\
\theta-\pi&\text{ for }\pi\leq\theta<\frac{3\pi}{2}\\\ -\theta+2\pi&\text{ for
}\frac{3\pi}{2}\leq\theta<2\pi\\\ \end{array}\right..$

Therefore, $J(g_{n})\Omega\to J(g)$ and hence $W(g_{n})\to W(g)$ in the strong
operator topology, and in the weak operator topology, too. On the other hand,
$\alpha_{q}(W(g_{n}))=e^{iq_{n}}W(g_{n})$, where

$q_{n}=q\int_{\mathbb{R}}g_{n}(C(t))dt.$

Note that, $C(t)=-\frac{t-i}{t+i}$, and $g(C(t))$ behaves as $\frac{1}{t}$ as
$t\to\infty$, hence $q_{n}\to\infty$. Accordingly,
$\alpha_{q}(W(g_{n}))=e^{iq_{n}}W(g_{n})$ is not convergent.

The last claim follows immediately because $\alpha_{q}$ are automorphisms,
hence $\alpha_{q}({\mathcal{A}}_{\rm U(1)}(I))={\mathcal{A}}_{\rm U(1)}(I)$
and by strong additivity of ${\mathcal{A}}_{\rm U(1)}$. ∎

In this case it is trivial that the dual net coincides with the original net.
In contrast, we have a markedly different result for ${\rm Vir}_{c}$.

The fact that $\omega\circ\alpha_{q}(J(f))=q\int_{\mathbb{R}}f(t)dt$ suggests
that $q$ is an analogue of the lowest weight $g$ of [5, Section 1C], cf. [29,
Lemmas 5.3, 5.6] and the state $\Omega_{q}$ has uniform charge density $q$.

##### The ${\rm U(1)}$-current as a multiplier representation of the loop
group $SLS^{1}$.

Let $SLS^{1}$ be the group of smooth maps from $S^{1}$ into $S^{1}$ with
winding number $0$ with pointwise multiplication [27, Section 5.1]. These
elements can be identified with smooth functions on $S^{1}$ in ${\mathbb{R}}$
by logarithm, and the group operation becomes addition. Recall the Weyl
relation $W(f)W(g)=e^{-\frac{i}{2}\sigma(f,g)}W(f+g)$: this can be seen as a
multiplier representation of $SLS^{1}$ with the cocycle
$e^{-\frac{i}{2}\sigma(f,g)}$. From this perspective, one can also consider
the subgroup $S\Omega S^{1}$ of functions $f$ on $S^{1}$ such that
$f^{(n)}(-1)=0$ for all $n\in{\mathbb{N}}$ [21, Section 2.1.2].

Let us observe that $f\mapsto\alpha_{q}(W(f))$ is a multiplier representation
of functions $f$ whose support does not contain $-1$, because for such $f$,
$\alpha_{q}(W(f))$ is defined and respects product. We claim that this extends
to $f\in S\Omega S^{1}$. Indeed, for such $f$, $f(C(t))$ is a rapidly
decreasing smooth function, hence we can find compactly supported test
functions $f_{n}\to f$ in the topology of Schwartz-class functions, and then
$\alpha_{q}(W(f_{n}))=e^{iq\int_{\mathbb{R}}f_{n}(t)dt}W(f_{n})\to
e^{iq\int_{\mathbb{R}}f(t)dt}W(f)$. On the other hand, we have seen in Theorem
4.2 that the representation $\alpha_{q}$ is not normal (continuous in the
$\sigma$-weak topology, which coincides with the weak operator topology on
norm-bounded sets) on any half-line.

Henriques conjectured that any positive-energy representation of $\Omega G$
should be normal on half-lines [21, Conjecture 34], where $G$ is a simple and
simply connected compact group. The above observation shows that an analogous
conjecture for $S\Omega S^{1}$ does not hold. On the other hand, we have also
shown that for such $G$, the only ground state of the algebra
$\mathscr{S}\mathfrak{g}_{\mathbb{C}}$ of Schwartz-class maps in
$\mathfrak{g}_{\mathbb{C}}$ is the vacuum state [29, Corollary 5.8]. This
supports to some extent the conjecture for $\Omega G$. Yet, there are many
other positive-energy representations of $\Omega G$ without ground state, and
the situation is not conclusive.

###  4.2 On Virasoro nets

We can simply restrict the states $\\{\omega\circ\alpha_{q}\\}$ to the
Virasoro nets ${\rm Vir}_{c}$. Let us denote
${\mathcal{H}}^{\kappa}_{q}=\overline{\alpha_{q}({\rm Vir}_{c}(I))\Omega}$,
where $c=1+\kappa^{2}$ and this does not depend on $I$ by the Reeh-Schlieder
argument. Let us denote the GNS representation
$P_{{\mathcal{H}}^{\kappa}_{q}}\alpha_{q}$ by $\rho_{\kappa,q}$, where
$P_{{\mathcal{H}}^{\kappa}_{q}}$ is the projection onto
${\mathcal{H}}^{\kappa}_{q}$.

######  Theorem 4.3.

We have the following.

  * •

The states $\\{\omega\circ\alpha_{q}\\}$ are mutually different ground states
of ${\rm Vir}_{c}|_{\mathbb{R}}$ for different $\frac{q^{2}}{2}$, and they are
connected with each other by dilations. Dilations are not implemented
unitarily in $\rho_{\kappa,q}$.

  * •

The GNS vector $\Omega_{q}$ of $\omega\circ\alpha_{q}$ is in the domain of
$T^{\kappa}_{q}(f)$, where
$\alpha_{q}(e^{isT^{\kappa}(f)})=e^{iT^{\kappa}_{q}(f)}$ and
$T^{\kappa}_{q}(f)=T(f)+qJ(f)+\kappa
J(f^{\prime})+\frac{q^{2}}{2}\int_{\mathbb{R}}f(t)dt$ on
${\mathcal{H}}^{\infty}$.

  * •

If $c>1$, the dual nets $\\{\hat{\rho}_{q}({\rm Vir}_{c}(I))\\}$ are not
unitarily equivalent to any of ${\rm Vir}_{c^{\prime}}$, $c^{\prime}>1$.

###### Proof.

Let us first prove the second statement. We have seen in [8, Lemma 4.6] that
$W(g)$ preserves ${\mathcal{H}}^{\infty}$ and for $I\supset{\rm supp\,}f$ it
holds on ${\mathcal{H}}^{\infty}$ that

${\hbox{\rm
Ad\,}}W(qs_{I})(T(f))=T(f)+qJ(f)+\frac{q^{2}}{2}\int_{\mathbb{R}}f(t)dt.$

Furthermore, we have ${\hbox{\rm
Ad\,}}W(qs_{I})(J(f))=J(f)+q\int_{\mathbb{R}}f(t)dt$, and
$T^{\kappa}(f)=T(f)+\kappa J(f^{\prime})$. Applying ${\hbox{\rm
Ad\,}}W(qs_{I})$ term by term, we obtain on ${\mathcal{H}}^{\infty}$

${\hbox{\rm Ad\,}}W(qs_{I})(T^{\kappa}(f))=T(f)+qJ(f)+\kappa
J(f^{\prime})+\frac{q^{2}}{2}\int_{\mathbb{R}}f(t)dt,$

since $\int_{\mathbb{R}}f^{\prime}(t)dt=0$, and this operator has $\Omega$ in
its domain.

To show that they are different states for different $\frac{q^{2}}{2}$, it is
enough to observe that

$\frac{d}{ds}\omega\circ\alpha_{q}(e^{isT^{\kappa}(f)})=\langle\Omega,{\hbox{\rm
Ad\,}}W(qs_{I})(T^{\kappa}(f))\Omega\rangle=\frac{q^{2}}{2}\int_{\mathbb{R}}f(t)dt,$

giving different values.

Let us also check that these states depend only on $c=1+\kappa^{2}$ and not on
$\kappa$. To see this, note that $J(f)\mapsto J(-f)$ is a vacuum-preserving
automorphism of ${\mathcal{A}}|_{\mathbb{R}}$ commuting with translations.
This automorphism $\alpha_{-}$clearly intertwines $\alpha_{q}$ and
$\alpha_{-q}$. Under $\alpha_{-}$, $T(f)$ is mapped to $T(f)$ itself because
it is quadratic in $J$, therefore, $T^{\kappa}(f)$ is mapped to
$T^{-\kappa}(f)$. Since $\omega\circ\alpha_{q}$ and
$\omega\circ\alpha_{q}\circ\alpha_{-}=\omega\circ\alpha_{-}\circ\alpha_{-q}=\omega\circ\alpha_{-q}$
give the same state on ${\rm Vir}_{c}$, they depend on only on
$c=1+\kappa^{2}$.

They are connected by dilation $\delta(s)$ on the larger algebra
${\mathcal{A}}_{\rm U(1)}|_{\mathbb{R}}$ and from the action333Formally, $J$
has scaling dimension $1$: ${\hbox{\rm
Ad\,}}(U(\delta(s))(J(t))=e^{s}J(e^{s}t)$. By integrating this equation again
$f$, we obtain ${\hbox{\rm Ad\,}}U(\delta(s))(J(f))=J(f\circ\delta(-s))$,
where $f\circ\delta(-s)(t)=f(e^{-s}t)$. Similarly, the stress-energy tensor
$T$ has scaling dimension $2$: ${\hbox{\rm
Ad\,}}(U(\delta(s))(T(t))=e^{2s}T(e^{s}t)$ and ${\hbox{\rm
Ad\,}}U(\delta(s))(T(f))=e^{s}T(f\circ\delta(-s))$. of dilations ${\hbox{\rm
Ad\,}}U(\delta(s))(J(f))=J(f\circ\delta(-s))$ that

$\displaystyle{\hbox{\rm Ad\,}}[W(qs_{I})U(\delta(s))](J(f))$
$\displaystyle=J(f\circ\delta(-s))+q\int_{\mathbb{R}}f\circ\delta(-s)(t)dt$
$\displaystyle=J(f\circ\delta(-s))+e^{s}q\int_{\mathbb{R}}f(t)dt$
$\displaystyle={\hbox{\rm Ad\,}}U(\delta(s))(J_{e^{s}q}(f))$

Hence $\omega\circ\alpha_{q}\circ{\hbox{\rm
Ad\,}}U(\delta(s))=\omega\circ\alpha_{e^{s}q}$,

Since $\Omega_{q}$ is the unique translation-invariant vector, it follows that
$\rho_{\kappa,q}$ are irreducible (see the arguments of [2, Corollary 1.2.3]).
If dilations were implemented by unitary operators, they would bring a vector
state to another vector state which are again translation-invariant, which
contradicts the uniqueness of invariant vector.

As ${\rm Vir}_{c},c>1$ are not strongly additive while dual nets are strong
additive, they cannot be unitarily equivalent. ∎

We expect that the GNS representations $\\{\rho_{\kappa,q}\\}$ are not normal
on $\bigcup_{I\Subset{\mathbb{R}}_{\pm}}{\rm Vir}_{c}(I)$. A candidate for a
convergent sequence which is not convergent in $\\{\rho_{k,q}\\}$ is
$e^{T(f_{n})}$, where $f_{n}$ are smooth functions on ${\mathbb{R}}$,
supported in ${\mathbb{R}}_{+}$ and converging to a function $f$ which decays
as $\frac{1}{t}$. We expect that $T(f)+\kappa J(f^{\prime})+qJ(f)$ is self-
adjoint, while $q^{2}\int_{\mathbb{R}}f(t)dt$ is divergent. To do this, the
commutator theorem would not suffice (because formally
$[L_{0},J(f)]=J(f^{\prime})$ would not be defined for this $f$) but one would
need an analysis similar to [10, Theorem 4.4]

Here the value $\frac{q^{2}}{2}$ can be viewed as the energy density of the
state $\Omega_{q}$, hence the representation as a whole has infinite energy
with respect the original vacuum, cf. [14] where it was shown that, under
certain conditions, translation-invariant states cannot be created by finite
energy. This also supports that $\rho_{\kappa,q}$ are not normal even on half-
lines.

Even for the case $c=1$, the dual nets $\\{\hat{\rho}_{q}({\rm
Vir}_{1}(I))\\}$ cannot be easily identified with any known conformal net. We
remark that strong additivity of the original net ${\rm Vir}_{1}$ does not
appear to imply that $\\{\rho_{q}({\rm Vir}_{c}(I))\\}$ is already dual, cf.
[7, Lemma 3.2].

##  5 Outlook

##### Classification.

As we have seen, the ground states we constructed in this paper are
characterized by a number $q$ (for the ${\rm U(1)}$-current net) or
$\frac{q^{2}}{2}$ (for the Virasoro nets). This is a structure very similar to
the invariant $\psi$ of [29, Lemma 5.3]. If one studies corresponding Lie
algebras, it should be possible to classify ground state representations as in
[29, Theorem 5.6].

More precisely, to the net ${\mathcal{A}}_{\rm U(1)}|_{\mathbb{R}}$ there
corresponds the central extension of the Lie algebra of compactly supported
smooth functions on ${\mathbb{R}}$ with the relation $[f,g]=\sigma(f,g)$. To
the Virasoro nets ${\rm Vir}_{c}|_{\mathbb{R}}$ one considers the central
extension of smooth vector fields on ${\mathbb{R}}$ with
$[f,g]=fg^{\prime}-f^{\prime}g+\frac{c}{12}\int
f(t)g^{\prime\prime\prime}(f)dt$. By positivity of energy, one should be able
to determine the representation in terms of $q$ and $\frac{q^{2}}{2}$,
respectively.

Actually, for the ${\rm U(1)}$-current net, the direct classification of
ground states might be possible. Some techniques from [8, Section 4.2] used to
classify KMS states should be useful, although the boundary condition ($t\to
t+i\beta$ for $\beta$-KMS states) is missing for ground states. As for the
Virasoro nets, it is still not clear whether such classification results pass
to the Virasoro nets. For that implication, one needs that the ground state
vector is infinitely differentiable. We are currently not able to do this,
because the corresponding Lie group ${\rm Diff}_{\mathrm{c}}({\mathbb{R}})$ of
compactly supported diffeomorphisms of ${\mathbb{R}}$ do not possess any
compact subgroup (in contrast to ${\rm Diff}(S^{1})$ which contains finite
covers of ${\rm PSL}(2,{\mathbb{R}})$, which is the key to differentiate
representations [9, Appendix], cf. [33]).

##### Dual nets.

The most fundamental properties of the dual nets
$\\{\hat{\rho}_{\kappa,q}({\rm Vir}_{c}(I))\\}$ remain open. Among them is
conformal covariance. Conformal covariance implies the split property [26],
and even the split property is unknown to hold in these dual nets. The split
property may fail in the dual net in two-dimensional Haag-Kastler net [25,
Section 4.2], therefore, it may be worthwhile to try to (dis)prove the split
property in these nets.

##### More positive-energy representations.

Ground states consist only a particular class of positive-energy
representations. It was shown in [11, 12] that there is a huge class of
locally normal positive-energy representations of the free massless fermion
field in $(3+1)$-dimensions. A similar construction should be possible in one
dimension. Furthermore, important conformal nets, including some loop group
nets, can be realized as subnets of (the tensor product of ) the free fermion
field nets (see [30, Examples 4.13-16]). Among them, there might be
counterexamples to [21, Conjecture 34].

#### Acknowledgements.

We acknowledge the MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006.

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# TelecomRAG: Taming Telecom Standards with Retrieval Augmented Generation and
LLMs

Girma M. Yilma, Jose A. Ayala-Romero, Andres Garcia-Saavedra, and Xavier
Costa-Perez

###### Abstract

Large Language Models (LLMs) have immense potential to transform the
telecommunications industry. They could help professionals understand complex
standards, generate code, and accelerate development. However, traditional
LLMs struggle with the precision and source verification essential for telecom
work. To address this, specialized LLM-based solutions tailored to
telecommunication standards are needed. Retrieval-augmented generation (RAG)
offers a way to create precise, fact-based answers. This paper proposes
TelecomRAG, a framework for a Telecommunication Standards Assistant that
provides accurate, detailed, and verifiable responses. Our implementation,
using a knowledge base built from 3GPP Release 16 and Release 18 specification
documents, demonstrates how this assistant surpasses generic LLMs, offering
superior accuracy, technical depth, and verifiability, and thus significant
value to the telecommunications field.

###### Index Terms:

Telecommunications, LLM, Standards, 3GPP, O-RAN, ETSI

##  I Introduction

Large Language Models (LLMs), built upon the popular Transformer architecture
and massive training datasets, are revolutionizing the way we interact with
language. Their ability to “understand” complex text, generate responses, and
translate between languages has far-reaching implications across industries.
Among these, the telecommunications domain stands to benefit significantly
from the advancements in LLMs.

Telecommunications professionals are tasked with navigating the intricate and
evolving landscape of standard specification documents established by bodies
such as 3GPP, ETSI, and O-RAN. These standards define the fundamental
technologies that enable global connectivity and interoperability. AI-powered
assistants based on LLMs have the potential to revolutionize how professionals
interact with these standards. Such assistants could rapidly analyze dense
specifications, generate code snippets, assist in debugging, and even offer
insights that bridge the gap between different standards. This holds the
promise of increased efficiency, innovation, and the streamlined development
of robust telecom solutions.

However, conventional LLM-based assistants often struggle with knowledge-
intensive tasks, a point discussed in detail in Section II of this paper.
While LLMs can store factual information implicitly within their vast
parameter sets, this knowledge is not easily accessible, updated, or
traceable. In the context of telecommunications, where precision and reliance
on standard-specified sources are critical, this limitation creates hurdles
for widespread adoption in the industry.

To overcome these challenges, there is a growing need for specialized LLM-
based solutions tailored to telecommunication standards. Section II explores
various domains where LLMs have shown success, highlighting the need for such
adaptation within telecommunications. Retrieval-Augmented Generation (RAG), a
technique that combines LLM capabilities with a non-parametric knowledge base,
offers a promising avenue for addressing the need for precise and factual
answers. Accordingly, we discuss in Section III the challenges of applying
this approach specifically to telecommunication standards, followed by the
design of an architecture that tackles these challenges to build a
Telecommunication Standards Assistant that is accurate, technically deep, and
provides verifiable answers.

In Section IV, we first present an implementation of this framework. Our work
leverages state-of-the-art APIs and libraries to ingest telecom standards
documents, create knowledge embeddings, and seamlessly integrate them within a
user-friendly interface. We have evaluated our assistant with a vast pool of
technical questions, one of which is presented in Section IV as an example,
rendering accurate, thorough, technically rich, and verifiable answers in
marked contrast with the responses provided by general-purpose LLM-based
assistants such as ChatGPT-4 or Gemini Ultra.

##  II LLMs and Language Generation

This section provides some background on LLMs, their abilities (Sec. II-A),
and LLMs for specific application domains (Sec. II-B). Then, we review the
related work retrieval-augmented methods (Sec. II-C), which are the base of
our solution.

###  II-A Background on LLMs

Large language models (LLMs) are Transformer-based language models with
hundreds of billions or more parameters trained on an enormous corpus of text.
Some examples include Gemini, LLaMa, and GPT-4. These models demonstrate
significant capabilities in comprehending natural language and tackling
intricate tasks through text generation. Although existing LLMs adopt the same
Transformer architecture and pre-training objectives as other smaller language
models, the significant increase in the model size, data size, and computing
capacity largely improves the performance of LLMs. In fact, some works propose
different scaling laws of LLMs, showing that model performance has strong
dependencies with these tree dimensions [1].

One of the most prominent features of LLMs is the emergent abilities, formally
defined as the abilities that are not present in smaller models but arise in
larger ones. The authors in [2] show several examples of emergent abilities,
where the performance in a task via few-shot prompting is random until a
particular scale. When the model size surpasses a certain scale the
performance increases well above random. This ability, also referred to as in-
context learning, is formally introduced by GPT-3 [3] but is not shown by
previous models such as GPT-1 and GPT-2. Another ability that emerged in large
models is the step-by-step reasoning also called chain of thoughts [4]. This
strategy is very effective in solving complex tasks that involve multiple
steps such as math problems.

As emergent abilities cannot be predicted through the scaling laws of LLMs,
this raises the question of whether additional scaling could potentially
further expand the range of capabilities of language models [2].

###  II-B LLMs for specific application domains

LLMs have been applied to different specific domain areas. In healthcare, Med-
PaLM [5] achieves expert-level performance on the United States Medical
Licensing Examination (USMLE). It relies on the pre-trained PaLM model
enhanced with several techniques, namely, few-shot prompting, chain-of-thought
and self-consistency prompting, and prompt tuning. However, the use of LLMs in
healthcare comes with the risk of fabricating medical misinformation (e.g.,
suggesting advice inconsistent with medical guidelines) [6].

Finance is another important field where the application of LLMs is promising.
In this field, BloombergGPT [7] is a proprietary model based on BLOOM
architecture trained with a mix of general purpose and specialized finance
data. This results in a model that achieves SoTA results in financial
benchmarks, while keeping competitive performance on general-purpose tasks. As
an alternative, FinGPT is an open-source framework comprising several layers
for data gathering and engineering, pre-trained LLM models (e.g., LLaMA), and
different fine-tuning techniques such as Fine-tuning via Low-rank Adaptation
(LoRA) and Reinforcement Learning from Human Feedback (RLHF).

In the domain of scientific research, LLMs specially pre-trained on
scientific-related corpora have been proposed (e.g. Galactica [8], Minerva
[9]). These models aim to assist in different stages of the scientific
research pipeline.

However, all these solutions have the potential risk of hallucination, i.e.,
the generation of content that is not grounded in factual information but is
instead a product of the model’s creative extrapolation. This phenomenon
occurs even when the models are trained with a high-quality and highly curated
corpus of data, as in the case of Galactica [8]. Moreover, hallucination may
have serious implications in critical domains such as healthcare [6]. In order
to reduce the hallucination phenomenon and provide reliable factual
information, other LLM-based approaches have been proposed as explained in the
next section.

###  II-C Retrieval-Augmented Methods

LLMs have demonstrated the ability to retain factual information within their
parameters and excel in achieving state-of-the-art performance when fine-tuned
for natural language processing tasks. However, the acquired world knowledge
is implicitly encoded within the parameters of the underlying neural network,
which poses a challenge in discerning what knowledge is stored in the network
and where. Consequently, LLMs have serious limitations to $i$) access and
precisely manipulate knowledge; $ii$) add new information or include updates
to the current knowledge. Thus, task-specific architectures can outperform
language models in knowledge-intensive tasks111This term refers to tasks that
humans could not reasonably be expected to accomplish without relying on an
external knowledge source. [10, 11].

To improve the performance in knowledge-intensive tasks, some works in the
literature propose hybrid models that combine a non-parametric memory (e.g., a
textual knowledge corpus such as Wikipedia) with a pre-trained parametric
memory (e.g., a LLM) [10, 11]. This architecture retrieves relevant
information from the non-parametric memory, which is later processed and
interpreted by the parametric memory (usually a language model), resulting in
more factual and specific answers. Moreover, this architecture allows us to
directly modify or expand the knowledge stored in the parametric memory
without having to modify the parametric memory.

REALM [10] augments a masked language model with a differentiable retriever,
which gathers knowledge from a textual knowledge corpus. Both the language
model and the retriever are trained together end-to-end according to the
language model objectives. This strategy poses a significant computational
challenge since the backpropagation through the retriever must consider
millions of documents in the textual knowledge corpus. Moreover, REALM only
explores open-domain extractive question answering, i.e., its objective is to
predict masked tokens in a sentence.

The authors in [11] expand this idea by proposing the retrieval-augmented
generation (RAG) methodology for sequence-to-sequence (seq2seq) language
generation. RAG considers a retriever based on Dense Passage Retrieval (DPR)
[12] and a seq2seq model (e.g., BART architecture) for language generation.
The DPR retriever follows a bi-encoder architecture, one for the queries and
the other for the documents in the knowledge corpus. This method can be fine-
tuned on any seq2seq task, whereby both the generator and retriever are
jointly learned. In the fine-tuning phase, only the query encoder is trained.
This reduces considerably the computational burden compared to REALM, which
periodically updates the document index during pre-training [10]. The results
in [11] show that RAG responses are more factual, specific, and diverse than
other baselines based on language models. For these reasons and given the
nature of the task involved in telecommunication standards, we rely on RAG to
design TelecomRAG as detailed in the next section.

##  III TelecomRAG Design

Figure 1: TelecomRAG: Architecture of the RAG-based Telecommunication
Standards Assistant.

This section details the conceptual design of our solution, which is mainly
based on RAG with some practical modifications. This solution combines
information retrieval techniques and language models. As detailed in the
previous section, this combination outperforms language models alone in
knowledge-intensive tasks by providing more factual answers. Moreover, there
are several advantages of generating answers instead of just extracting them
directly from the knowledge base. For example, we may need to combine and
elaborate the information included in several documents to obtain the answer
to a specific query. Also, sometimes documents can contain clues about an
answer but not the literal response.

TelecomRAG comprises two stages. In the first stage, we generate a vector
database from the Telecom knowledge base. This stage is executed offline
before launching the system. The vector database can be updated whenever new
files need to be considered (e.g., new 3GPP releases or document updates). The
second stage is executed for every user query and establishes the
conversational question-answering pipeline.

###  III-A Generation of Telecom vector database (Offline stage).

The vector database creation involves the generation of embedding
representations of the files in the Telecom knowledge base. Embeddings are
dense vector representations in a latent space and they aim to capture the
semantic meaning of the content. Thus, the embedding of two different
documents should be close in the vectorial space if the content of these
documents is related (even if they use completely different words). Based on
this, documents with content related to a query can be retrieved and the
system’s output can be grounded on factual information contained in these
documents.

As an initial step, the files in the Telecom Knowledge Base need to be cleaned
and chunked. The configuration of the chunking is crucial. Overly short chunks
provide insufficient information, while excessively long passages hinder the
creation of a rich semantic space and may exceed the LLM’s context window
limit. The documents are divided into passages (referred to as documents
henceforth) of a fixed number of words with an overlap between consecutive
chunks. For each document, one embedding is computed and stored in the vector
database. A common practice to compute the embedding is to use a language
model based on the BERT architecture [11, 13].

###  III-B Document Retrieval and Presentation (Online stage).

The pipeline of the online stage is depicted in Fig. 1, comprising several
building blocks: $i$) the retriever, which processes the input query, gathers
related documents from the knowledge base, and outputs the LLM input; $ii$)
the history module, which stores past queries and responses; iii) the response
generation and optimization module, which receives the LLM input and generates
the final output. In the following, we detail the steps of the online
pipeline.

First, the retriever generates the standalone query based on the user’s new
query and the history of past queries and responses. It is important the
standalone query to be self-contained for the retrieval step. If we only use
the new query, relevant context content may be missing. Conversely, if we
input the full past conversation, irrelevant information may hinder an
accurate retrieval. Therefore, the standalone query is generated by an LLM
taking as input the new user’s query and the history provided by the history
module.

Second, the embedding of the standalone query is computed using an encoder. We
use the same encoder as in the offline stage allowing the documents and the
queries to be encoded in the same latent semantic space.

Third, we perform a semantic search in the vector database. We look for
document embeddings that are similar to the encoded query based on a
similarity metric. As mentioned before, the embeddings capture the “gist” of
the text regardless of its length or the specific wording. Given that the
dimensionality of the embeddings is usually high and the vector database can
contain a large number of embeddings, an efficient search algorithm is needed
to reduce the computational burden. We retrieve $K$ documents from the
knowledge base whose embeddings obtained the highest similarity metric value.

Fourth, we compose the LLM input with the standalone query, the $K$ retrieved
documents, and the history of queries and outputs, as shown in Fig. 1. The LLM
input is passed to the Base LLM in the response generation & optimization
module.

Fifth, the Base LLM generates a tentative output based on the LLM input. To
adapt the answer to different contexts, we use role-playing prompting
techniques. In some cases, we may need a very detailed and comprehensive
answer. To this end, we can assign the LLM the role of a “standard expert” and
the user the role of a “non-experienced trainee”. Based on this, we can
generate a prompt as follows: “Assume you are a 3GPP standard expert and need
to provide a very comprehensive answer to a non-experienced trainee.” Thus,
the tone and verbosity of the output can be easily adjusted.

Sixth, the tentative output generated by the Base LLM is checked by the
verification & optimization module. The goal of this module is to check the
appropriateness of the output. For example, if no documents are retrieved, a
predefined message is sent to the user informing that there are no documents
in the knowledge base related to the input query. Also, improper outputs can
be filtered based on keywords. In other cases, the user can actively
participate in the verification & optimization loop by, for example, improving
the query or discarding irrelevant parts of the retrieved documents or history
manually. Once the verification & optimization is passed, the system outputs
the final response.

Figure 2: Graphical User Interface of TelecomRAG, an Assistant for Telecommunication Standards. TABLE I: ECN Failure Indication data from the correct 3GPP specification document [14, Section 8.30]. Procedure | Initiated | Information element name |  | Information
---
element
required
Information element description
ECN Failure Indication | IMS-AGW | Context | Mandatory |  This information element indicates the context for the bearer termination.
Bearer Termination | Mandatory |  This information element indicates the bearer termination for which the ECN failure is reported.
ECN Error Indication | Mandatory |  This information element indicates an ECN failure event.
ECN Failure Indication Ack | IMS-ALG | Context | Mandatory |  This information element indicates the context where the command was executed.
Bearer Termination | Mandatory |  This information element indicates the bearer termination where the command was executed.
TABLE II: What are the information elements included in the “ECN Failure Indication”, and how are they defined? TelecomRAG’s response |  The ECN Failure Indication procedure, as described in the context of 3GPP standards, is used to report ECN (Explicit Congestion Notification) related failures. This procedure is part of the communication between IMS-ALG (IP Multimedia Subsystem - Application Layer Gateway) and IMS-AGW (IP Multimedia Subsystem - Access Gateway). The specific information elements involved in the ECN Failure Indication procedure are as follows: 1. Context: This mandatory information element indicates the context for the bearer termination. The context provides the necessary background or situation associated with the bearer that is being terminated. 2. Bearer Termination: This mandatory information element indicates the bearer termination for which the ECN failure is being reported. The bearer termination refers to the point where a particular data session or communication is being terminated. 3. ECN Error Indication: This mandatory information element indicates an ECN failure event. An ECN failure event is a situation where the ECN mechanism has detected congestion in the network, but due to some error, the congestion notification could not be conveyed or processed as expected. In the ECN Failure Indication procedure, when the IMS-AGW receives any IP headers indicating ECN Congestion Experienced (ECN-CE), it should trigger rate adaptation in accordance with the specifications given in 3GPP TS 26.114. If there is an ECN failure, the IMS-AGW reports this failure to the IMS-ALG using the ECN Failure Indication message containing the aforementioned information elements. Additionally, the ECN Failure Indication Acknowledgment (Ack) is used to acknowledge the receipt of the ECN Failure Indication. It also contains the Context and Bearer Termination information elements, which indicate the context and bearer termination where the command was executed.
References: “IP Multimedia Subsystem (IMS) Application Level Gateway (IMS-ALG)
- IMS Access Gateway (IMS-AGW) interface” 3GPP Technical Specification 23.334
---|---

##  IV TelecomRAG: Implementation and Evaluation

In the following, we present our implementation of the above design (Section
IV-A) and then, using an example query, we compare our implementation against
vanilla chat assistants commonly used today (Section IV-B).

###  IV-A Implementation

Our implementation of the RAG-based Telecommunication Standard assistant
design presented above is mostly founded on Langchain [15], a framework that
provides an array of built-in wrappers and utilities designed to extract
reasoning capabilities from language models.

To build a knowledge base for telecommunication standards specialization, we
gathered the complete set of 3GPP release-16 and release-18 standard
specification documents. These documents are in various formats (PDF, TXT,
DOCX, DOC), which can be readily processed by the DirectoryLoader method from
the Langchain.document_loaders library, along with custom functions for
progress tracking, multi-threading, and error handling (e.g., corrupted file
formats).

We divided the data into documents of 4000 characters with a 100-characters
overlap between consecutive chunks using the CharacterTextSplitter method from
Langchain.text_splitter. For every document, we then generated an embedding
using the OpenAI’s text-embedding-ada-002 embedding model via the
Langchain.embeddings.openai library. This embedding model is also user to
encode the standalone query at the retriever. After all the text documents are
encoded, we store them in a _vector database_ using Chroma, an AI-native
vector database from Langchain.vectorstores.

We implement the pipeline of the online stage using the
ConversationalRetrievalChain method from Langchain.chains library. We use the
LLM gpt-4-1106-preview from OpenAI for the query condensation (at the
retriever) and for the answer elaboration (at the response generation &
optimization module).

We use _cosine similarity_ as the similarity metric in our system. Cosine
similarity is a computationally cheap solution that has proven good
performance in this type of application. The semantic search is performed
using an approximate nearest neighbor (ANN) algorithm called Hierarchical
Navigable Small World (HNSW) implemented in Chroma. We consider the $K=4$
documents with the highest cosine similarity obtained by HNSW algorithm to be
included in the LLM input. The system’s pipeline is conveniently wrapped up
into a user-friendly web interface using Gradio that resembles that of
conventional AI chatbots such as Gemini or ChatGPT, as shown in Fig. 2.

###  IV-B Evaluation

We evaluated TelecomRAG on a large set of technical questions derived from
3GPP specifications. The assistant consistently provided thorough, accurate,
and technically detailed responses useful for telecom engineers, including
references to specific 3GPP documents for verification.

For example, consider the query “What are the information elements included in
the ’ECN Failure Indication’, and how are they defined?”, which is described
in [14] (Section 8.30) and summarized in Table I for validation. Table II
depicts the response provided by our solution, including a reference to the
right 3GPP document, demonstrating accuraccy and verifiability for this
example. Notably, our solution maintained this depth and accuracy across all
queries in our evaluation.

In marked contrast, when issuing the same queries to general-purpose
assistants such as ChatGPT-4, Gemini Ultra or TelecomGPT (a third-party
application built on ChatGPT to provide specialization on the
telecommunication sector222Available in https://chat.openai.com/g/g-eSwyzwwDV-
telecom-gpt.), the responses provided are vague, inaccurate and not
verifiable, as illustrated in Table III for the query “What are the
information elements included in the ’ECN Failure Indication’, and how are
they defined?” presented earlier.

TABLE III: What are the information elements included in the “ECN Failure Indication”, and how are they defined? ChatGPT 4’s response |  The “ECN Failure Indication” is a term that is typically associated with network protocols, specifically with mechanisms that deal with congestion control. ECN stands for Explicit Congestion Notification. This feature enables network nodes to notify each other about network congestion without dropping packets. However, the specific term “ECN Failure Indication” isn’t standard, and you may be referring to situations where ECN signaling fails or is not properly communicated between network entities. Here’s a detailed look at the typical elements related to ECN, which might be relevant to a failure scenario:  1. ECN-Capable Transport (ECT) \- This is a codepoint in the IP header that indicates that the end-points of the TCP connection are ECN-capable, meaning they can both send and receive ECN signals. There are two ECT codepoints, ECT(0) and ECT(1), which serve slightly different purposes but essentially signal ECN capabilities. 2. Congestion Experienced (CE) \- This is another codepoint in the IP header used by routers experiencing congestion to indicate this status to the end-points of the transmission. When a packet marked with an ECT codepoint encounters congestion, routers that support ECN can mark these packets as CE instead of dropping them. 3. ECN Echo (ECE) \- This is a flag in the TCP header used to indicate that a segment with the CE codepoint has been received. The receiving end uses this flag to notify the sender about the congestion. 4. Congestion Window Reduced (CWR) \- This flag is set by the sending end to signal to the receiving end that it has received the ECE flag and has reduced its congestion window size accordingly. This is an acknowledgment that the sender is responding to the congestion. When discussing “ECN Failure Indication”, it might relate to any anomalies or failures in the signaling process among these elements. Here are a few scenarios where ECN could “fail”:  • Unsupported ECN: If any part of the path between sender and receiver does not support ECN, then the ECN signals (ECT, CE) may not be handled or propagated correctly. • Incorrect ECN Implementation: Bugs or incorrect implementations in routers or end hosts can lead to incorrect ECN behavior, such as failing to set CE flags despite congestion or not reacting to ECE flags. • ECN Negotiation Failure: During the TCP handshake, both sides must agree to use ECN. If there is a failure in this negotiation, ECN will not be used, even if both endpoints support it. Understanding specific ECN failure modes would depend on the context in which ECN is being used (like in specific network architectures or for certain types of traffic) and the specific tools and methods used for diagnosing such failures.
---|---
Gemini Ultra’s response |  Unfortunately, there isn’t a universally defined “ECN Failure Indication” with a set list of information elements. “ECN” can refer to different things depending on the context, and the specific details of a failure indication would depend on the particular system or protocol involved. Here’s a breakdown of the possibilities:  • IP Networking: In the context of Internet Protocol (IP) networking, ECN (Explicit Congestion Notification) is a mechanism for routers to signal congestion to senders. It’s unlikely there would be a separate “ECN Failure Indication”. Issues with ECN functionality might be reported through other mechanisms within the specific routing protocol. • Other Fields: ECN can also stand for Engineering Change Notice or Entry Change Notice in various fields. A failure related to an ECN would likely be specific to the system or process that uses those terms and wouldn’t have a standardized format. Here are some ways to get more information:  • Context Clues: If you encountered the term “ECN Failure Indication” somewhere specific, look for surrounding information that might indicate the context (e.g., manuals, software documentation). • System Identification: If you know the system or protocol where you came across this term, searching for its documentation might reveal details about error reporting mechanisms. If you can provide more context about where you encountered “ECN Failure Indication”, I might be able to offer more specific guidance.
TelecomGPT response |  The “ECN Failure Indication” message in the context of telecommunications and protocols isn’t explicitly defined in the documents I have available, such as the SMPP specifications and various telecom-related white papers and presentations. It appears you might be referring to a specific protocol message or error indication that might be part of a broader telecommunications standard or proprietary system not covered in these documents. If the “ECN Failure Indication” you are asking about is related to a specific application, protocol, or vendor-specific implementation, it would be helpful to have more context or details about the system in which this term is used. If it is part of a standard not covered in the documents or discussions I have access to, you may need to consult the specific documentation or standards applicable to that protocol or system. If you can provide more details or clarify the specific protocol or context in which you encountered “ECN Failure Indication”, I can assist you better. Alternatively, if this is related to a standard or system, the relevant documentation or technical support resources for that system may be the best source for detailed information.

##  V Conclusions

This paper presented a specialized LLM solution tailored to telecommunication
standards. This solution is based on RAG and combines information retrieval
techniques and LLMs, providing precise, fact-based, and verifiable answers. We
evaluate our solution using a knowledge base built from 3GPP Release 18
specification documents. Our solution surpasses generic LLMs, offering
superior accuracy, technical depth, and verifiability. In the next phase, we
plan to extend the knowledge base with other SDO documentation. We believe
that this work can bring significant value to the telecommunications field.

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Girma M. Yilma is a Senior Research Engineer at NEC Laboratories Europe in
Heidelberg, Germany. He received his M.Sc. in Telecommunications Engineering
from the University of Trento, Italy. His current research interests are
focused on GenAI for Telecom Services, Trusted Execution Environments for NFV
MANO, and Cloud Native Networking.
---
Jose A. Ayala-Romero received his Ph.D. degree from the Technical University
of Cartagena, Spain, in 2019. Then, he joined Trinity College Dublin, Ireland,
as a post-doctoral researcher. From January 2021 to September 2022, he was a
Senior Data Scientist at Huawei Ireland Research Center, Ireland. Currently,
he is a Senior Researcher at NEC Laboratories Europe, Germany. His research
interests include the application of diverse machine learning techniques to
mobile networks problems.
---
Andres Garcia-Saavedra received his PhD degree from the University Carlos III
of Madrid (UC3M) in 2013. He then joined Trinity College Dublin (TCD),
Ireland, as a research fellow until 2015. Currently, he is a Principal
Researcher at NEC Laboratories Europe. His research interests lie in the
application of fundamental mathematics to real-life wireless communication
systems.
---
Xavier Costa-Perez is a Research Professor at ICREA, Scientific Director at
i2Cat and Head of 5G/6G R&D at NEC Laboratories Europe. His team generates
research results which are regularly published at top scientific venues,
produces innovations which have received several awards for successful
technology transfers, participates in major European Commission R&D
collaborative projects and contributes to standardization bodies. He received
both his M.Sc. and Ph.D. degrees in Telecommunications from the Polytechnic
University of Catalonia (UPC).
---