# Loop groups and QNEC

Lorenzo Panebianco*

*Dipartimento di Matematica, Università di Roma “La Sapienza”
Piazzale Aldo Moro 5, 00185–Roma, Italy
E-mail<EMAIL_ADDRESS>
(November 2020)

###### Abstract

We investigate some analytical properties of loop group models, showing that a
Positive Energy Representation (PER) of a loop group $LG$ can be extended to a
PER of $H^{3/2}(S^{1},G)$ for any compact, simple and simply connected Lie
group $G$. We then explicitly compute the adjoint action of $H^{5/2}(S^{1},G)$
on the stress energy tensor and we use these results to prove the Quantum Null
Energy Condition (QNEC) and the Bekenstein Bound for states obtained by
applying a Sobolev loop to the vacuum. We also give a simpler proof of these
last results in the case $G=SU(n)$. Finally, we construct and study solitonic
representations of the loop group conformal nets induced by the conjugation by
a loop with a discontinuity in $-1$.

Keywords: Loop Group, Positive Energy Representation, Relative Entropy

MSC2020: primary 81T05, 81T40

##  1 Introduction

Recently, much attention has been focused on quantum information aspects of
Quantum Field Theory, which naturally takes place in the framework of quantum
black holes thermodynamics. However, more unexpected and interesting
connections between the relative entropy and the stress energy tensor have
arisen, and in particular it is of interest to provide and prove an axiomatic
formulation of the Quantum Null Energy Condition (QNEC). A general proof of
the QNEC is given in [4]. While in [11] we study the QNEC on the Virasoro net,
in this work we focus on loop group models.

We mainly follow [14]. Let $G$ be a compact, simple and simply connected Lie
group. A Positive Energy Representation (PER) of the loop group
$LG=C^{\infty}(S^{1},G)$ on a separable Hilbert space $\mathcal{H}$ is a
projective strongly continuous unitary representation $\pi$ of
${LG}\rtimes\mathbb{T}$ with a commutative diagram

$\textstyle{{LG}\rtimes\mathbb{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{PU(\mathcal{H})}$$\textstyle{\mathbb{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{U(\mathcal{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the torus $\mathbb{T}\cong\text{Rot}$ acts on $LG$ by rotations and $R$
is a strongly continuous unitary representation which induces an isotypical
decomposition $\mathcal{H}=\bigoplus_{n\geq 0}\mathcal{H}(n)$. In the
following we will consider only PERs of finite type, namely we require
$\dim\mathcal{H}(n)<+\infty$ for every $n$. We also require, without loss of
generality, that $\mathcal{H}(0)$ is not zero-dimensional. Irreducible PERs
are of finite type.

We denote by $\mathfrak{g}_{0}$ the Lie algebra of $G$ and by $\mathfrak{g}$
the complexification of $\mathfrak{g}_{0}$. Recall that $\mathfrak{g}_{0}$ is
a compact Lie algebra, that is its Killing form is negative definite. In
particular, there is an antilinear involution $x\mapsto x^{*}$ of
$\mathfrak{g}$ such that

$\mathfrak{g}_{0}=\\{x\in\mathfrak{g}\colon x^{*}=-x\\}\,.$

We define an involution of $L^{\text{pol}}\mathfrak{g}$ by
$X(n)^{*}=X^{*}(-n)$, where $X(n)=Xe^{in\theta}$ for $X$ in $\mathfrak{g}$. We
denote by $\mathcal{H}^{\text{fin}}$ the subspace of finite energy vectors,
namely the algebraic sum of the subspaces $\mathcal{H}(n)$, and by
$\mathcal{H}^{\infty}\subseteq\mathcal{H}$ the Fréchet space of smooth
vectors. For $X$ in $L\mathfrak{g}$ and $\gamma$ in $LG$, on
$\mathcal{H}^{\infty}$ we have that $[d,\pi(X)]=i\pi(\dot{X})$ and that
$\pi(X)^{*}=\pi(X^{*})$. Moreover, we have the commutation relations

$[\pi(X),\pi(Y)]=\pi([X,Y])+i\ell B(X,Y)\,,\qquad
B(X,Y)=\int_{0}^{2\pi}\braket{X,\dot{Y}}\frac{d\theta}{2\pi}\,.$

The adjoint action of $LG$ is given by

$\begin{split}\pi(\gamma)\pi(X)\pi(\gamma)^{*}&=\pi(\gamma
X\gamma^{-1})+ic(\gamma,X)\,,\\\
\pi(\gamma)d\pi(\gamma)^{*}&=d-i\pi(\dot{\gamma}\gamma^{-1})+c(\gamma,d)\,,\end{split}$
(1)

where $d$ is the generator of rotations, namely $\pi(R_{\theta})=e^{i\theta
d}$. The real constants $c(\gamma,X)$ and $c(\gamma,d)$ are explicitly given
by

$c(\gamma,X)=-\ell\int_{0}^{2\pi}\braket{\gamma^{-1}\dot{\gamma},\dot{X}}\frac{d\theta}{2\pi}\,,\qquad
c(\gamma,d)=-\frac{\ell}{2}\int_{0}^{2\pi}\braket{\gamma^{-1}\dot{\gamma},\gamma^{-1}\dot{\gamma}}\frac{d\theta}{2\pi}\,.$

Here $\braket{\cdot,\cdot}$ denotes the basic inner product, namely the
Killing form normalized on the highest root $\theta$ in such a way that
$\braket{\theta,\theta}=2$. Notice that the map $X\mapsto\pi(X)$ is actually a
representation if restricted on $\mathfrak{g}$. Indeed, the projective
representation of $G$ lifts to a unitary representation and the subspaces
$\mathcal{H}(n)$ are $G$-invariant. We will use the following notation:

$x=\pi(X)\,,\quad x(n)=\pi(X(n))\,,\quad\braket{x,y}=\braket{X,Y}\,.$

We can define a representation of the Virasoro algebra Vir

$[L_{n},L_{m}]=(n-m)L_{n+m}+\delta_{n+m,0}\frac{n(n^{2}-1)}{12}c\,,$

by Sugawara construction, that is such a representation is given by defining

$L_{n}=\frac{1}{2(\ell+g)}\sum_{m}\colon x_{i}(-m)x^{i}(m+n)\colon\,,$

where we used the Einstein convention on summations. Here $\\{x_{i}\\}$ and
$\\{x^{i}\\}$ are dual basis with respect to the basic inner product, namely
$\braket{x^{i},x_{j}}=\delta_{ij}$, and $g$ is the dual Coxeter number, that
is half the Casimir of the adjoint representation:

$[X_{i},[X^{i},Y]]=2gY\,,\quad Y\in\mathfrak{g}\,.$

Notice that if $X_{i}$ belongs to $\mathfrak{g}_{0}$ then
$x_{i}(n)^{*}=-x_{i}(-n)$. If the PER is irreducible, then the central charge
$c$ and the trace anomaly $h$ are given by

$c=\frac{\ell\dim\mathfrak{g}}{\ell+g}\,,\qquad
h=\frac{C_{\lambda}}{2(\ell+g)}\,,$

where $C_{\lambda}$ is the Casimir associated to the basic inner product
$\braket{\cdot,\cdot}$ and to the null energy space
$\mathcal{H}(0)=\mathcal{H}_{\lambda}$, which is the irreducible highest
weight representation of $\mathfrak{g}$ associated to the dominant integral
weight $\lambda$ satisfying

$\braket{\lambda,\theta}\leq\ell\,.$ (2)

The set of dominant integral weights $\lambda$ satisfying condition (2) is
called the level $\ell$ alcove. We will say that $\pi$ is a vacuum positive
energy representation, or simply a vacuum representation, if
$\mathcal{H}(0)=\mathbb{C}\Omega$ is one-dimensional. The state $\omega$
associated to $\Omega$ is called the vacuum state.

######  Lemma 1.

If $\mathfrak{g}$ is simple and $\ell\geq 1$, then all the unitary irreducible
representations $V(c,h)$ of the Virasoro algebra appearing from an irreducible
level $\ell$ positive energy representation of $LG$ satisfy

$c\geq 1\,,\quad 0\leq
h\leq\frac{\ell^{2}}{\delta(\ell+g)}=\frac{c\ell}{\delta\dim\mathfrak{g}}\,,$

for some $\delta>0$.

###### Proof.

First we recall that since $\mathfrak{g}$ is a complex semisimple Lie algebra
then $C_{\lambda}=\braket{\lambda,\lambda}$. In particular, since
$\mathfrak{g}$ is simple then the dual Coxeter number is a strictly positive
number. The estimate on the central charge $c$ follows by
$g+1\leq\dim\mathfrak{g}$, which can be noticed by studying the following
table:

Dynkin diagram | $A_{n}$ | $B_{n}$ | $C_{n}$ | $D_{n}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---|---
Complex simple Lie algebra | $\mathfrak{sl}_{n+1}$ | $\mathfrak{so}_{2n+1}$ | $\mathfrak{sp}_{2n}$ | $\mathfrak{so}_{2n}$ | $\mathfrak{e}_{6}$ | $\mathfrak{e}_{7}$ | $\mathfrak{e}_{8}$ | $\mathfrak{f}_{4}$ | $\mathfrak{g}_{2}$
Complex dimension | $n^{2}+2n$ | $2n^{2}+n$ | $2n^{2}+n$ | $2n^{2}-n$ | $78$ | $133$ | $248$ | $52$ | $14$
Dual Coxeter number | $n+1$ | $2n-1$ | $n+1$ | $2n-2$ | $12$ | $18$ | $30$ | $9$ | $4$

Now we use the fact that $m=\min_{i}\cos\alpha_{\theta,\lambda^{i}}$ is
strictly positive. Here the minimum is above all the fundamental dominant
integral weights $\lambda^{i}$ and $\alpha_{\theta,\lambda^{i}}$ is the angle
between the weights $\theta$ and $\lambda^{i}$. Therefore if
$\braket{\lambda,\theta}\leq\ell$ then

$C_{\lambda}\leq\ell^{2}/2\cos^{2}\alpha_{\theta,\lambda}\leq\ell^{2}/2m^{2}\,,$

from which we have $h\leq\ell^{2}/4m^{2}(\ell+g)$, and the proof is concluded.
∎

######  Lemma 2.

[7] $[L_{n},x(k)]=-kx(n+k)$ on $\mathcal{H}^{\text{\em fin}}$.

###### Proof.

Let $\\{x_{i}\\}$ be a orthonormal basis in $\mathfrak{g}_{0}$, so that its
dual basis is given by $x^{i}=-x_{i}$. We have the following relations:

$\begin{split}[xy,z]&=x[y,z]+[x,z]y\,,\\\
[x(a),[y,z](b)]&=[[z,x],y](a+b)+[[x,y](a),z(b)]\,,\\\
[x_{i}(a),[x^{i},z](b)]&=[x_{i},[x^{i},z]](a+b)=2gz(a+b)\,,\end{split}$

for any integers $a$ and $b$. Notice also that on $\mathcal{H}^{\text{fin}}$
we have

$\begin{split}L_{n}&=\frac{1}{2(\ell+g)}\sum_{m\geq-n/2}(2-\delta_{-m,n/2})x_{i}(-m)x^{i}(m+n)\,.\end{split}$

Therefore we can compute

$\begin{split}[x_{i}(-m)x^{i}(m+n),x_{j}(k)]&=x_{i}(-m)[x^{i}(m+n),x_{j}(k)]+[x_{i}(-m),x_{j}(k)]x^{i}(m+n)\\\
&=-\ell k\delta_{i,j}(\delta_{k,-m-n}+\delta_{k,m})x_{j}(n+k)\\\
&\qquad+x^{i}(-m)[x_{i},x_{j}](m+n+k)+[x_{i},x_{j}](-m+k)x^{i}(m+n)\,.\end{split}$

The structure constants $\\{c^{h}_{ij}\\}$ relative to $\\{x_{i}\\}$, that is
the constants determined by $[x_{i},x_{j}]=c^{h}_{ij}x_{h}$, verify the
relations $c^{h}_{ij}=c^{i}_{jh}=-c^{i}_{hj}$. It follows that

$\begin{split}[x_{i},x_{j}](a)x^{i}(b)&=-x^{i}(a)[x_{i},x_{j}](b)\,,\\\
x^{i}(a)[x_{i},x_{j}](b)+x^{i}(b)[x_{i},x_{j}](a)&=2gx_{j}(a+b)\,,\\\
x^{i}(a)[x_{i},x_{j}](a)&=gx_{j}(2a)\,.\end{split}$

In particular, if we set $X_{m}=x^{i}(-m)[x_{i},x_{j}](m+n+k)$ then
$X_{m}+X_{-m-n-k}=2gx_{j}(n+k)$. Thus, by explicit computation one can prove
that

$\begin{split}[L_{n},x_{j}(k)]&=-\frac{k\ell}{(\ell+g)}x_{j}(n+k)+\frac{1}{2(\ell+g)}\sum_{m\geq-n/2}(2-\delta_{-m,n/2})(X_{m}-X_{m-k})\\\
&=-kx_{j}(n+k)\,,\end{split}$

and therefore we have $[L_{n},x(k)]=-kx(n+k)$ on $\mathcal{H}^{\text{fin}}$
for every $x$ in $\mathfrak{g}$ and $k,n$ in $\mathbb{Z}$. ∎

As a corollary, the representation of $\text{Vir}=\mathbb{C}\cdot
c\oplus\partial$, with $\partial$ the Witt algebra, extends to a
representation of the semidirect product
$\mathfrak{g}[t,t^{-1}]\rtimes\text{Vir}\cong\tilde{\mathfrak{g}}\rtimes\partial$,
with $\tilde{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}\cdot c$.
Indeed, if we set $L_{n}=\pi(\ell_{n})$, where
$\ell_{n}(\theta)=e^{in\theta}\frac{d}{d\theta}$, then we can define the
stress energy tensor $\pi(h)=\sum_{n}\hat{h}_{n}L_{n}$ for any polynomial
vector field $h$ on the circle, namely a vector field which is a finite linear
combination of the fields $\ell_{n}$. Therefore, by Lemma 2 we have
$[\pi(h),\pi(X)]=\pi(h.X)$ on $\mathcal{H}^{\text{fin}}$ for every $X$ in
$L^{\text{pol}}\mathfrak{g}$, where
$h.X(\theta)=h(\theta)\frac{d}{d\theta}X(\theta)$.

The space $L^{\text{pol}}\mathfrak{g}$ can be completed to a Banach Lie
algebra $L\mathfrak{g}_{t}$, with $t\geq 0$. Indeed, given
$X=\sum_{k}a_{k}e^{ik\theta}$ in $L^{\text{pol}}\mathfrak{g}$, we define
$L\mathfrak{g}_{t}$ as the completion of $L^{\text{pol}}\mathfrak{g}$ with
respect to the norm

$|X|_{t}=\sum_{k}(1+|k|)^{t}\|a_{k}\|\,.$

We have norm continuous embeddings with dense range $C^{\left\lceil
t\right\rceil+1}(S^{1},\mathfrak{g})\hookrightarrow
L\mathfrak{g}_{t}\hookrightarrow C^{\lfloor t\rfloor}(S^{1},\mathfrak{g})$,
and for any $t\geq n$ we have $\|X^{(n)}\|_{\infty}\leq|X|_{t}$. Notice that,
in general, we can similarly define the space $L\mathfrak{g}_{s,p}$ by

$|X|_{s,p}=\bigg{(}\sum_{k}(1+|k|)^{sp}\|a_{k}\|^{p}\bigg{)}^{1/p}\,.$

We set $\mathcal{S}_{t}=L\mathbb{C}_{t}$, namely the space of continuous
complex functions $h$ on $S^{1}$ satisfying

$|h|_{t}=\sum_{k}(1+|k|)^{t}\|\hat{h}_{k}\|<+\infty\,.$

Notice that by Fourier expansion we can naturally identify $\mathcal{S}_{t}$
with a space of Sobolev vector fields on the circle. Finally, we define two
actions of $\mathcal{S}_{t}$ by
$h.X(\theta)=h(\theta)\frac{d}{d\theta}X(\theta)$ and
$hX(\theta)=h(\theta)X(\theta)$. Indeed, by noticing that
$(1+|n+k|)^{t}\leq(1+|n|)^{t}(1+|k|)^{t}$ we have
$|h.X|_{s,p}\leq|h|_{s}|X|_{s+1,p}$ and $|hX|_{s,p}\leq|h|_{s}|X|_{s,p}$.

##  2 Sobolev loop groups

We know that $LG=C^{\infty}(S^{1},G)$ is a Fréchet Lie group if endowed with
the Whitney smooth topology. Its topology is induced by the norms defined on
the Banach Lie groups $L^{k}G=C^{k}(S^{1},G)$. The exponential map
$\exp_{LG}\colon L\mathfrak{g}_{0}\to LG$ is naturally defined by
$\exp_{LG}(X)=\exp_{G}\cdot X$ and is a local diffeomorphism (see [12]). Here
we define and describe some properties of Sobolev loop groups.

Let $M$ be a riemannian manifold. Suppose $M$ to be isometrically embedded in
some $\mathbb{R}^{\nu}$. Define, for $1\leq p<\infty$ and $0\leq s<\infty$,

$W^{s,p}(S^{1},M)=\\{f\in W^{s,p}(S^{1},\mathbb{R}^{\nu})\colon f(\theta)\in
M\,\text{\em a.e.}\\}\,.$

Here $W^{s,p}(S^{1},\mathbb{R}^{\nu})$ is the completion of
$C^{\infty}(S^{1},\mathbb{R}^{\nu})$ with respect to the norm
$\|f\|_{s,p}=\|\Delta^{s/2}f\|_{p}+\|f\|_{p}$, where $\Delta\geq 0$ is the
smallest closure on $L^{p}(S^{1},\mathbb{R}^{\nu})$ of the laplacian seen as
an operator on $C^{\infty}(S^{1},\mathbb{R}^{\nu})$.

In the following, every compact Lie group $G$ will be considered as a
riemannian Lie group with respect to the unique riemannian structure extending
$-\braket{\cdot,\cdot}$, that is the opposite of the basic inner product, and
such that left and right translations are smooth isometries. We show that if
$G$ is compact and simple then every faithful unitary representation
$\rho\colon G\to U(n)$ induces an isometrical embedding of $G$ in some real
euclidean space. By continuity of the representation we have that $G$ is
represented as a compact embedded Lie subgroup of $U(n)$. Moreover, by
simplicity of $\mathfrak{g}_{0}$ we have that
$\lambda\text{tr}(\rho(x)^{*}\rho(y))=-\braket{x,y}$ for some $\lambda>0$.
Therefore if we consider $M_{n}(\mathbb{C})$ as a real vector space with inner
product $\lambda\text{Re}\,\text{tr}(X^{*}Y)$ then we have an isometric
embedding $G\hookrightarrow M_{n}(\mathbb{C})$.

######  Theorem 3.

If $G$ is a compact, simple and simply connected Lie group faithfully
represented in some space of matrices, then $W^{s,p}(S^{1},G)$ is an analytic
Banach Lie group for $1<p,sp<\infty$ whose Lie algebra is
$W^{s,p}(S^{1},\mathfrak{g}_{0})$. Moreover, $C^{\infty}(S^{1},G)$ is dense in
$W^{s,p}(S^{1},G)$ and thus $W^{s,p}(S^{1},G)$ is connected.

###### Proof.

First we show that $W^{s,p}(S^{1},G)$ is a topological group. This can be
proved by using the fact that any two functions $f,g$ in
$W^{s,p}(S^{1},\mathbb{R}^{\nu})$ verify, for $p$ and $sp$ in $(1,\infty)$,
the estimate [5]

$\|fg\|_{s,p}\leq C_{s,p}\|f\|_{s,p}\|g\|_{s,p}\,.$

By this estimate and by the identity $f^{-1}-g^{-1}=f^{-1}(g-f)g^{-1}$ it
follows that $W^{s,p}(S^{1},G)$ is a topological group for $p$ and $sp$ in
$(1,\infty)$, since it is clearly a Hausdorff space. Now we define the map

$\exp_{s,p}\colon W^{s,p}(S^{1},\mathfrak{g}_{0})\to
W^{s,p}(S^{1},G)\,,\quad\exp_{s,p}(X)(z)=\exp_{G}(X(z))\,.$

This map is a local homeomorphism since $\exp_{G}\cdot X$ is an absolutely
convergent series for $\|X\|_{s,p}<1/C_{s,p}$. We check that
$W^{s,p}(S^{1},G)$ is connected. By the density of $C^{\infty}(S^{1},G)$ in
$W^{s,p}(S^{1},G)$ (see Theorem 1.1. of [1]), it suffices to prove that
$C^{\infty}(S^{1},G)$ is path connected and then connected. But a smooth
homotopy between two loops in $G$ is a path in $C^{\infty}(S^{1},G)$ and the
connectedness follows. Finally, we conclude if we prove that the group
operations of inversion and multiplication are analytic. By connectedness we
can reduce to prove this in an open neighborhood of the identity (see [14],
Lemma 2.2.1.). The inversion $X\mapsto-X$ is clearly analytic. The analyticity
of left and right multiplication follows from the Baker-Campbell-Hausdorff-
Dynkin formula, and the theorem is proved. ∎

######  Corollary 4.

Every loop $\gamma$ in $W^{s,p}(S^{1},G)$ is a finite product of exponentials.

We have formally defined our Sobolev loop group $W^{s,p}(S^{1},G)$ and we have
checked that such a space has good topological and analytical properties. Now
we are finally ready to extend our positive energy representation of $LG$.

######  Lemma 5.

Let $X$ be a topological space. Consider a net $(x^{i})_{i\in I}$ in $X$ such
that $x^{i}\to x$ for some $x$ in $X$. Suppose that for every $i\in I$ there
is a directed set $A_{i}$ and a net $(x^{i}_{\alpha})_{\alpha\in A_{i}}$
convergent to $x^{i}$. Then there is a net with values in
$D=\\{x^{i}_{\alpha}\\}$ which is convergent to $x$.

###### Proof.

Fix a neighborhood $U$ of $x$ and an element ${\alpha}_{i}$ in $A_{i}$ for
some $i\in I$. By convergence one can find an element ${\alpha}^{\prime}_{j}$
in $A_{j}$ for some $j\geq i$, with eventually
${\alpha}_{i}\leq{\alpha}^{\prime}_{j}$ if $i=j$, which belongs to $U$. This
means that $x$ is an accumulation point of the net

$A=\coprod_{i\in I}A_{i}\to D\,,\quad\alpha_{i}\mapsto y^{i}_{\alpha_{i}}\,.$

Here $A$ is the disjoint union of the sets $A_{i}$ considered with the
lexicographical order: $\alpha_{i}\leq\alpha^{\prime}_{j}$ if and only if
$i\leq j$ in $I$ and $\alpha_{i}\leq\alpha^{\prime}_{i}$ in $A_{i}$ in the
case $i=j$. It follows the existence of a subnet $B\to D$ of $A$ convergent to
$x$. ∎

######  Example 1.

Let $G$ be a topological group. Let $(x_{\alpha})_{\alpha\in A}$ and
$(y_{\beta})_{\beta\in B}$ be nets in $G$, with $x_{\alpha}\to x$ and
$y_{\beta}\to y$. Then the net $(x_{\alpha}y_{\beta})_{(\alpha,\beta)\in
A\times B}$, where $A\times B$ is the product directed set, is convergent to
$xy$. Indeed, given a neighborhood $W$ of the identity, there is a
neighborhood $V$ of the identity such that $V^{2}\subseteq W$. By definition
there exist $\alpha_{0}$ in $A$ and $\beta_{0}$ in $B$ such that $x_{\alpha}$
is in $xV$ and $y_{\beta}$ is in $Vy$ for every $\alpha\geq\alpha_{0}$ and
$\beta\geq\beta_{0}$, and by $xV^{2}y\subseteq xWy$ the assertion follows.

######  Proposition 6.

Let $i\colon G\to H$ and $\pi\colon G\to U$ be two homomorphisms of
topological groups. We suppose $H$ to be connected and $i(G)$ to be dense in
$H$. We suppose the existence of a neighborhood $V$ in $H$ of the identity
such that

$p(v)=\lim_{\alpha}\pi(g_{\alpha})\,,\quad i(g_{\alpha})\to v\,,$

is well defined and does not depend on the the choice of the net
$(g_{\alpha})_{\alpha\in A}$. Then, we can define a homomorphism $p\colon H\to
U$ such that $\pi=p\cdot i$.

###### Proof.

By the connectedness of $H$ we have that $H=\cup_{n}V^{n}$. We show by
induction that $p$ is well defined on $V^{n}$ for every $n$. Suppose the
thesis true for $V^{n}$, and consider elements $w$ in $V^{n}$ and $v$ in $V$.
Pick a net $h_{\beta}$ such that $i(h_{\beta})\to v$. By inductive hypothesis
and Lemma 5 we have that

$\lim_{\alpha}\pi(i(g_{\alpha}))=\lim_{\beta}\lim_{\alpha}\pi(i(g_{\alpha}h^{-1}_{\beta}))\pi(v)$
(3)

is well defined and does not depend on the net $g_{\alpha}$ such that
$i(g_{\alpha})\to wv$. Hence $p$ is well defined and clearly $\pi=p\cdot i$.
The continuity follows by Lemma 5, and Example 1 shows that $p$ is actually a
group homomorphism. ∎

We now consider the Fréchet Lie group $\textit{\em Diff}_{+}(S^{1})$. The
action of $\textit{\em Diff}_{+}(S^{1})$ on $LG$ is smooth and any irreducible
positive energy representation of $LG$ is $\textit{\em
Diff}_{+}(S^{1})$-covariant. Let $h(\theta)\frac{d}{d\theta}$ be a smooth
vector field on the circle, or more generally in $(L\mathfrak{g}_{0})_{3/2}$.
We recall that the stress energy tensor

$T(h)=\sum_{n}\hat{h}_{n}L_{n}\,,$

is an essentially self-adjoint operator with $\mathcal{H}^{\text{fin}}$ as a
dense core. For any $X+i\alpha h$ in $L\mathfrak{g}\rtimes i\mathbb{R}h$ we
define $\pi(X+i\alpha h)=\pi(X)+i\alpha T(h)$. We notice that by the bound

$\|(1+L_{0})^{k}L_{n}\xi\|\leq\sqrt{c/2}(1+|n|)^{k+3/2}\|(1+L_{0})^{k+1}\xi\|\,,\quad
k\in\mathbb{N}\,,$

we have

$\|\pi(X+ih)\xi\|_{k}\leq\sqrt{2\kappa}|X|_{k+1/2}\|\xi\|_{k+1/2}+\sqrt{c/2}|{h}|_{k+3/2}\|\xi\|_{k+1}\,.$

######  Proposition 7.

If $X$ is in $W^{s,p}(S^{1},\mathfrak{g}_{0})$ for $1\leq p\leq 2$ and
$s>3/2+1/p$, then $\pi(X)$ is a closable operator which is essentially skew-
adjoint on any core of $d$.

###### Proof.

We know that for every vector $\xi$ in $\mathcal{H}^{\text{fin}}$ and $X$ in
$L^{\text{pol}}\mathfrak{g}$ we have [14]

$\begin{split}\|\pi(X)\xi\|_{t}&\leq\sqrt{2(\ell+g)}|X|_{|t|+1/2}\|\xi\|_{t+1/2}\,,\\\
\|[1+d,\pi(X)]\xi\|_{t}&\leq\sqrt{2(\ell+g)}|X|_{|t|+3/2}\|\xi\|_{t+1/2}\,,\end{split}$
(4)

for any $t$ in $\mathbb{R}$. By density one extends $\pi$ to
$(L\mathfrak{g})_{|t|+3/2}$ in such a way to still verify the same estimates
for $\xi$ in $\mathcal{H}^{t+1/2}$. It follows that if $X$ is in
$L\mathfrak{g}_{3/2}$ then both $\pi(X)$ and $[1+d,\pi(X)]$ are bounded
operators from $\mathcal{H}^{1/2}$ to
$\mathcal{H}^{0}\subseteq\mathcal{H}^{-1/2}$. Therefore, by the Nelson
commutator theorem (Thm. X.36 in [13]) we have that if $X$ is in
$(L\mathfrak{g}_{0})_{3/2}$ then the restriction of $\pi(X)$ on

$\mathcal{D}=\\{\psi\in\mathcal{H}\cap\mathcal{H}^{1/2}\colon\pi(X)\in\mathcal{H}\\}$

is a closable operator on $\mathcal{H}$ which is essentially skew-adjoint on
any core of $d$ such as $\mathcal{H}^{\text{fin}}$. Notice now that, by
standard arguments, there is a norm continuous embedding
$W^{s,p}(S^{1},\mathfrak{g})\hookrightarrow L\mathfrak{g}_{3/2}$. Indeed, if
$X(\theta)=\sum_{k}a_{k}e^{ik\theta}$ then by the Hölder inequality

$|X|_{3/2}=\sum_{k}(1+|k|)^{3/2}\|a_{k}\|=\sum_{k}(1+|k|)^{3/2-s}(1+|k|)^{s}\|a_{k}\|\leq
A_{s,p}|{X}|_{s,p^{\prime}}\leq B_{s,p}\|{X}\|_{s,p}\,,$ (5)

where $A_{s,p}$ exists by construction and $B_{s,p}$ exists by Riesz-Thorin.
Therefore, by the arguments given above we have that if $X$ is in
$W^{s,p}(S^{1},\mathfrak{g}_{0})$ then $\pi(X)$ is a symmetric operator on
$\mathcal{H}^{\textit{\em fin}}$ which is essentially self-adjoint on any core
of $d$. ∎

Propositions 6 and 7 can be used to extend a positive energy representation of
$LG$ to a strongly continuous projective representation of $W^{s,p}(S^{1},G)$.
However, for simplicity and convenience in the following we will focus on
$H^{s}(S^{1},G)=W^{s,2}(S^{1},G)$.

######  Proposition 8.

The representation $\pi$ of $L\mathfrak{g}$ can be extended to
$H^{3/2}(S^{1},\mathfrak{g})$, with $\pi(X)$ closable and such that

$\|\pi(X)\xi\|\leq
C\|X\|_{H^{3/2}}\|\xi\|_{1/2}\,,\quad\xi\in\mathcal{H}^{1/2}\,.$ (6)

Moreover, $\pi(X)^{*}=\overline{\pi(X^{*})}$, and in particular $\pi(X)$ is
essentially skew-adjoint if $X$ is in $H^{3/2}(S^{1},\mathfrak{g}_{0})$.

###### Proof.

We use some techniques shown in [3]. Consider $X=\sum_{k}a_{k}e^{ik\theta}$ in
$H^{3/2}(S^{1},\mathfrak{g})$. By

$\begin{split}|X|_{1/2}&=\sum_{k}(1+|k|)^{1/2}\|a_{k}\|\leq\bigg{(}\sum_{k}(1+|k|)^{-2}\bigg{)}^{1/2}\bigg{(}\sum_{k}(1+|k|)^{3}\|a_{k}\|^{2}\bigg{)}^{1/2},\end{split}$
(7)

we have that $\pi(X)$ is well defined and (6) follows by the previous
estimates. It is also closable since $\pi(X^{*})\subseteq\pi(X)^{*}$. Notice
that since $\mathcal{H}^{\text{fin}}$ is a core for $(1+d)^{1/2}$ then
$\pi(X^{*})$ is the formal adjoint of $\pi(X)$ on $\mathcal{H}^{1/2}$ for any
$X$ in $H^{3/2}(S^{1},\mathfrak{g})$. Now we define on $\mathcal{H}^{1/2}$ the
operator

$R_{X,\epsilon}=[\pi(X),e^{-\epsilon d}]\,,$

which is well defined since $e^{-\epsilon
d}\colon\mathcal{H}\to\mathcal{H}^{\infty}\subseteq\mathcal{H}^{1/2}$. Since
$-R_{X^{*},\epsilon}\subseteq R^{*}_{X,\epsilon}$ then $R_{X,\epsilon}$ is
closable. Notice that if $dv_{k}=kv_{k}$ then

$R_{x(n),\epsilon}v_{k}=f_{n,k}(\epsilon)x(n)v_{k}\,,\quad
f_{n,k}(\epsilon)=e^{-\epsilon k}-e^{-\epsilon(k-n)}\,.$

By simple analysis techniques one can prove that

$|f_{n,k+n}(\epsilon)|^{2}(1+k+n)\leq 2(1+|n|)^{2}\,,$

for any $\epsilon\geq 0$, $k\geq 0$ and $n+k\geq 0$. Therefore if
$v=\sum_{k\geq 0}v_{k}$ is in $\mathcal{H}^{\text{fin}}$ then for every
$X=\sum_{j}x_{j}(n_{j})$ in $L^{\text{pol}}\mathfrak{g}$ we have

$\begin{split}\|R_{X,\epsilon}v\|^{2}&=\sum_{k}\|(R_{X,\epsilon}v)_{k}\|^{2}=\sum_{j,k}|f_{n_{j},k+n_{j}}(\epsilon)|^{2}\|x_{j}(n_{j})v_{k+n_{j}}\|^{2}\\\
&\leq
2(\ell+g)\sum_{j,k}|f_{n_{j},k+n_{j}}(\epsilon)|^{2}(1+|n_{j}|)(1+k+n_{j})\|x_{j}\|^{2}\|v_{k+n_{j}}\|^{2}\\\
&\leq 4(\ell+g)\sum_{j,k}(1+|n_{j}|)^{3}\|x_{j}\|^{2}\|v_{k+n_{j}}\|^{2}\\\
&\leq 4(\ell+g)\|X\|_{H^{3/2}}^{2}\|v\|^{2}\,.\end{split}$ (8)

In particular, $R_{X,\epsilon}$ can be defined as a bounded operator for every
$X$ in $H^{3/2}(S^{1},\mathfrak{g})$ and $R_{X,\epsilon}\to 0$ strongly as
$\epsilon\to 0$. Moreover, by the identity
$R^{*}_{X,\epsilon}=-\overline{R_{X^{*},\epsilon}}$ we have that
$R^{*}_{X,\epsilon}\to 0$ strongly as well. Now we arrive to the crucial
point: if $v$ is in $\mathcal{D}(\pi(X)^{*})$ then

$\pi(X^{*})e^{-\epsilon d}v=\pi(X)^{*}e^{-\epsilon d}v=e^{-\epsilon
d}\pi(X)^{*}v-R^{*}_{X,\epsilon}v\to\pi(X)^{*}v\,,\quad\epsilon\to 0\,,$

and this concludes the proof since $e^{-\epsilon d}v\to v$. ∎

######  Theorem 9.

Let $\pi\colon LG\to PU(\mathcal{H})$ be a positive energy representation of
$LG$. Then $\pi$ can be extended to a positive energy representation of
$H^{3/2}(S^{1},G)$.

###### Proof.

We consider an open neighborhood $U$ in $H^{3/2}(S^{1},\mathfrak{g}_{0})$ on
which the exponential map of $H^{3/2}(S^{1},G)$ is a homeomorphism and set
$V=\exp_{H^{3/2}}(U)$. For $\gamma=\exp_{H^{3/2}}(X)$ in $V$ we define in
$PU(\mathcal{H})$

$\pi(\gamma)=e^{\pi(X)}\,,\quad X\in U\,.$

The neighborhood $V$ verifies Proposition 6, since if
$\gamma_{\alpha}=\exp(X_{\alpha})$ converges to $\gamma=\exp(X)$ in $V$ then
the estimate (6) implies that $\pi(X_{\alpha})\xi$ is a Cauchy net for every
$\xi$ in $\mathcal{H}^{1/2}$. But the pointwise convergence of self-adjoint
operators on a common core implies the strong resolvent convergence of such
operators (Theorem VIII.25.(a) of [13]), thus $\pi$ can be continuously
extended. Finally, since the rotation group acts on $H^{3/2}(S^{1},G)$ by
continuous operators (see Lemma A.3 of [2]) and since $LG$ is dense in
$H^{3/2}(S^{1},G)$, we have that $\pi$ is actually a positive energy
representation since it is Rot-covariant. ∎

######  Proposition 10.

[14] Let $R_{s}=\exp_{\textit{\em Diff}_{+}(S^{1})}(sh)$ be a smooth
diffeomorphism of $S^{1}$, with $h$ a smooth vector field of the circle. Set
$R_{h}=\\{R_{s}\\}_{s\in\mathbb{R}}$. Then the exponential map
$L\mathfrak{g}_{0}\rtimes\mathbb{R}h\to LG\rtimes R_{h}$ is well defined and
continuous. Moreover, if $X_{\alpha}=R_{\alpha}.X$ then

$\exp_{LG\rtimes R_{h}}(X+\alpha h)=\exp_{LG}(X_{\alpha})R_{\alpha}\,.$ (9)

###### Proof.

To compute the exponential map, we fix $X+\alpha h$ in
$L\mathfrak{g}_{0}\rtimes\mathbb{R}h$ and look for $f\colon\mathbb{R}\to
LG\rtimes R_{h}$ which satisfies $(X+\alpha h)f=\dot{f}$ and $f(0)=1$. We
suppose $f$ to be of the form $f_{t}=\gamma^{t}R_{\phi(t)}$ with $\gamma$ in
$LG$. As a manifold, $LG\rtimes R_{h}$ is the product of $LG$ and $R_{h}$,
thus $s\mapsto\exp_{LG}(sX)R_{s\alpha}$ is the integral curve for $X+\alpha h$
at the identity. Therefore, with the notation
$\gamma_{s}(\theta)=\gamma(R_{s}^{-1}(\theta))$ we have

$\begin{split}(X+\alpha
h)f_{t}&=\frac{d}{ds}\bigg{|}_{s=0}\exp_{LG}(sX)R_{s\alpha}\gamma^{t}R_{\phi(t)}=\frac{d}{ds}\bigg{|}_{s=0}\exp_{LG}(sX)(\gamma^{t})_{s\alpha}R_{s\alpha+\phi(t)}\\\
&=X\gamma^{t}R_{\phi(t)}+\alpha\frac{d}{ds}\bigg{|}_{s=0}(\gamma^{t})_{s}R_{\phi(t)}+\alpha\gamma^{t}hR_{\phi(t)}\,,\\\
\dot{f}_{t}&=\bigg{(}\frac{d}{dt}{\gamma^{t}}\bigg{)}R_{\phi(t)}+\phi^{\prime}(t)\gamma^{t}hR_{\phi(t)}\,,\end{split}$

whence $\phi(t)=\alpha t$, and we must solve

$\frac{d}{dt}{\gamma^{t}}=X\gamma^{t}+\alpha\frac{d}{ds}\bigg{|}_{s=0}(\gamma^{t})_{s}\,,\quad\gamma^{0}=1\,.$

Since the solution of this equation is $\gamma^{t}=\exp_{LG}(tX)_{\alpha
t}=\exp_{LG}(tX_{\alpha t})$, the assertion follows. ∎

######  Corollary 11.

$LG\rtimes R_{h}$ is a smooth Fréchet Lie group, and in particular the map

$LG\times R_{h}\to LG\,,\quad(\gamma,R_{s})\mapsto\gamma_{s}\,,$

is smooth.

###### Proof.

We first point out that the map $L\mathfrak{g}_{0}\times\mathbb{R}h\to
L\mathfrak{g}_{0}$ given by $(X,sh)\mapsto X_{s}$ is smooth and it is also a
local homeomorphism. Hence it suffices to notice that $LG\times R_{h}$ is
connected, check that the map $\exp_{LG\rtimes R_{h}}$ is a local
homeomorphism thanks to Proposition 10 and apply Lemma 2.2.1. of [14]. ∎

######  Corollary 12.

The following holds in $PU(\mathcal{H})$:

$e^{\pi(X+i\alpha h)}=\pi(\exp_{LG\rtimes R_{h}}(X+\alpha h))\,.$

###### Proof.

By the Trotter product formula and Proposition 10 we have the following
identities in $PU(\mathcal{H})$:

$\begin{split}e^{\pi(X+i\alpha
h)}&=\lim_{n\to\infty}(e^{i\alpha\pi(h)/n}e^{\pi(X/n)})^{n}=\lim_{n\to\infty}\pi(R_{\alpha/n}\exp_{LG}(X/n))^{n}\\\
&=\lim_{n\to\infty}\pi(\exp_{LG}(X_{\alpha/n}/n)R_{\alpha/n})^{n}=\lim_{n\to\infty}\pi(\exp_{LG\rtimes
R_{h}}((X+\alpha h)/n))^{n}\\\ &=\pi(\exp_{LG\rtimes R_{h}}(X+\alpha
h))\,,\end{split}$ (10)

where we used the identity $e^{iT(h)}=\pi(e^{h})$ which holds in
$PU(\mathcal{H})$. ∎

######  Lemma 13.

Let $\pi\colon G\to PU(\mathcal{H})$ be a strongly continuous projective
representation of a topological group $G$. Then the map

$G\times U(\mathcal{H})\to
U(\mathcal{H})\,,\quad(g,u)\mapsto\pi(g)u\pi(g)^{*}\,,$

is well defined and strongly continuous.

###### Proof.

The map is clearly well defined, and if $g_{\alpha}$ converges to $g$ in $G$
then we can choose lifts $v_{\alpha}$ and $v$ of $\pi(g_{\alpha})$ and
$\pi(g)$ such that $v_{\alpha}$ converges to $v$ in $U(\mathcal{H})$. But in
the unitary group the strong topology and the $*$-strong topology coincide,
and multiplication is continuous on bounded sets by the uniform boundedness
principle, so the assertion follows. ∎

######  Remark 14.

A continuous projective representation $\pi\colon G\to PU(\mathcal{H})$ can be
naturally lifted to a continuous unitary representation $\tilde{\pi}$ of
$\widetilde{G}=\big{\\{}(g,u)\in G\times
U(\mathcal{H})\colon\pi(g)=[u]\big{\\}}$ given by $\tilde{\pi}(g,u)=u$.

######  Theorem 15.

If $\gamma$ is in $H^{3/2}(S^{1},G)$ and $X$ is in
$H^{3/2}(S^{1},\mathfrak{g}_{0})$, then

$\pi(\gamma)\pi(X)\pi(\gamma)^{*}=\pi(\text{\em Ad}(\gamma)X)+ic(\gamma,X)\,,$
(11)

for some continuous real function $c(\gamma,X)$. Moreover, if $\gamma$ is in
$H^{5/2}(S^{1},G)$ and $h$ is a real vector field with $|h|_{3/2}<+\infty$,
then

$\pi(\gamma)\pi(X+ih)\pi(\gamma)^{*}=\pi(\text{\em
Ad}(\gamma)X)+iT(h)+\pi(h\dot{\gamma}\gamma^{-1})+ic(\gamma,X)+ic(\gamma,h)$
(12)

for some continuous real function $c(\gamma,h)$.

###### Proof.

We first prove (12) in the smooth case. By the previous propositions, if
$\gamma$ is in $LG$ and $Y=X+ih$ is in $L\mathfrak{g}_{0}\rtimes
i\mathbb{R}h$, then the following identities hold in $PU(\mathcal{H})$:

$\begin{split}\pi(\gamma)e^{s\pi(Y)}\pi(\gamma)^{*}&=\pi(\gamma)\pi(\exp_{LG\rtimes
R_{h}}(sY))\pi(\gamma)^{*}\\\ &=\pi(\gamma\exp_{LG\rtimes
R_{h}}(sY)\gamma^{-1})\\\ &=\pi(\exp_{LG\rtimes
R_{h}}(s\text{Ad}(\gamma)Y))\\\ &=e^{s\pi(\text{Ad}(\gamma)Y)}\,,\end{split}$
(13)

and consequently
$\pi(\gamma)e^{s\pi(Y)}\pi(\gamma)^{*}=\lambda(s)e^{s\pi(\text{Ad}(\gamma)Y))}$
for some function $\lambda\colon\mathbb{R}\to\mathbb{T}$. But
$\lambda\colon\mathbb{R}\to\mathbb{T}$ is a continuous homomorphism and
therefore $\lambda(t)=e^{iat}$ for a unique real number $a=c(\gamma,Y)$. We
point out that $\text{Ad}(\gamma)$ has to be intended as the adjoint action
with respect to the semidirect product $LG\rtimes R_{h}$. Notice also that
$c(\gamma,Y)$ is linear in $Y$, so we can write

$c(\gamma,X+ih)=c(\gamma,X)+c(\gamma,h)\,,$

where we set $c(\gamma,h)=c(\gamma,ih)$ for simplicity. Therefore, the claimed
expression follows by the Stone theorem and by using the product rule for the
derivative on $1=\gamma_{s}\cdot\gamma_{s}^{-1}$.

Now we prove (11) in the Sobolev case. Consider $(\gamma_{\alpha},X_{\alpha})$
in $LG\times L\mathfrak{g}_{0}$ converging to $(\gamma,X)$ in
$H^{3/2}(S^{1},G)\times H^{3/2}(S^{1},\mathfrak{g}_{0})$. We have that both
$\pi(\gamma_{\alpha})e^{s\pi(X_{\alpha})}\pi(\gamma_{\alpha})^{*}$ and
$e^{s\pi(\text{Ad}(\gamma_{\alpha})X_{\alpha})}$ strongly converge to the
corresponding terms in $\gamma$ and $X$. By the argument used before we have
that $e^{ic(\gamma_{\alpha},X_{\alpha})}$ converges to $e^{ic(\gamma,X)}$,
that is $e^{ic(\gamma,X)}$ is continuous in $\gamma$ and $X$. But continuity
is a local property and the exponential map has local left inverses, thus
$c(\gamma,X)$ is continuous and the first part of the theorem is proved. Now
we prove (12) in the Sobolev case. Consider $\gamma$ in $H^{5/2}(S^{1},G)$ and
$h$ real with $|h|_{3/2}<+\infty$. Notice that $i\pi(h)$ and
$\pi(h\dot{\gamma}\gamma^{-1})$ are both essentially skew-adjoint. Consider
now smooth approximating nets $\gamma_{\alpha}\to\gamma$, $X_{\alpha}\to X$
and $h_{\alpha}\to h$ as before. By the previous propositions, the
approximating right hand side of (12) minus $c(\gamma_{\alpha},h_{\alpha})$
converges in the strong resolvent sense to the corresponding term in $\gamma$,
$X$ and $h$ since we have a net of skew-adjoint operators pointwise convergent
on a common core. Similarly, $\pi(X_{\alpha}+ih_{\alpha})$ converges in the
strong resolvent sense to $\pi(X+ih)$ and therefore

$\pi(\gamma_{\alpha})e^{s\pi(X_{\alpha}+ih_{\alpha})}\pi(\gamma_{\alpha})^{*}\to\pi(\gamma)e^{s\pi(X+ih)}\pi(\gamma)^{*}$

strongly for every $s$ in $\mathbb{R}$. By the argument used before we have
that $e^{ic(\gamma,h)}$ is continuous and thus $c(\gamma,h)$ is continuous.
The thesis is proved. ∎

######  Corollary 16.

The subspace $\mathcal{H}^{s}\subseteq\mathcal{H}$ is
$H^{5/2}(S^{1},G)$-invariant for $s\geq 0$. Moreover, for $s\geq{5/2}$ real
and $n\leq\lfloor s-1\rfloor$ integer, the corresponding map
$H^{s}(S^{1},G)\times\mathcal{H}^{n}\to\mathcal{H}^{n}/\mathbb{T}$ is jointly
continuous.

###### Proof.

Since $\mathcal{D}(u^{*}Au)=u^{*}\mathcal{D}(A)$ for every unitary $u$ and
every self-adjoint operator $A$, then

$\begin{split}\mathcal{D}((1+d)^{s})&=\pi(\gamma)^{*}\mathcal{D}((1+d-i\pi(\dot{\gamma}\gamma^{-1})+c(\gamma,d))^{s})\\\
&\subseteq\pi(\gamma)^{*}\mathcal{D}((1+d)^{s})\,.\end{split}$ (14)

Since $\mathcal{D}((1+d)^{s})=\mathcal{H}^{s}$ for $s\geq 0$, the
$\mathcal{H}^{s}$-invariance follows. Now we prove the second statement, where
we can suppose $n\geq 1$. By Proposition 1.5.3. of [14] we have
$\|\pi(\gamma)\xi\|_{n}\leq(1+M_{n-1})^{n}\|\xi\|_{n}$, where
$M_{p}=C|\gamma^{-1}\dot{\gamma}|_{p+1/2}+|c(\gamma^{-1},d)|$ for some $C>0$,
and the joint continuity can be proved as in [14]. ∎

######  Theorem 17.

With the hypotheses of Theorem 15, we have

$c(\gamma,X)=-\ell\int_{0}^{2\pi}\braket{\gamma^{-1}\dot{\gamma},X}\frac{d\theta}{2\pi}\,,\quad
c(\gamma,h)=-\frac{\ell}{2}\int_{0}^{2\pi}h\braket{\gamma^{-1}\dot{\gamma},\gamma^{-1}\dot{\gamma}}\frac{d\theta}{2\pi}\,.$

###### Proof.

We follow Theorem 1.6.3. of [14], skipping some computations for the sake of
brevity. Consider a smooth loop $\gamma$ in $LG$ and a smooth real vector
field $h$. For $Y$ in $L\mathfrak{g}_{0}\rtimes i\mathbb{R}h$ we have

$c(\gamma_{1}\gamma_{2},Y)=c(\gamma_{2},Y)+c(\gamma_{1},\text{Ad}(\gamma_{2})Y)\,.$
(15)

If $\gamma^{t}=\exp_{LG}(tX)$, then the map $t\mapsto c(\gamma^{t},Y)$ is
differentiable at $t=0$ since $LG\times R_{h}\to LG$ is smooth. In particular,
we have that

$\partial_{t}\big{|}_{t=0}c(\gamma^{t},Y)=\ell B(X,Y)\,,$

and so

$\partial_{t}\big{|}_{t=0}c(\gamma^{t},h)=0\,.$

By using (15) we have that $c(\gamma^{t},h)$ is differentiable everywhere,
with

$\partial_{t}c(\gamma^{t},Y)=\ell B(X,Y)+c(\gamma^{t},[X,Y])\,,$

or more compactly

$\dot{c}_{t}(Y)=i_{X}\ell B(Y)-(X.c_{t})(Y)\,.$ (16)

We naturally expect the solution of the ODE to be given by the Duhamel formula

$c(\gamma^{t},Y)=\ell
B(X,\text{Ad}(\gamma^{t})\int_{0}^{t}\text{Ad}(\gamma^{-\tau})Yd\tau)=\ell
B(X,\int_{0}^{t}\text{Ad}(\gamma^{s})Yds)\,.$ (17)

Using $\frac{d}{dt}\text{Ad}(\gamma_{t})Y=[X,\text{Ad}(\gamma_{t})Y]$, it is
easy to verify that (17) defines a $C^{1}(\mathbb{R},(L\mathfrak{g}\rtimes
i\mathbb{R}h)^{*})$ solution of (16) with initial condition $c_{0}=0$. The
solution is unique. Finally, one can use (17) and Corollary 1.6.2. of [14] to
obtain the claimed expressions in the smooth case. By the continuity of
$c(\gamma,Y)$ shown in Theorem 15 the thesis is proved. ∎

######  Corollary 18.

By repeating the proof of Theorem 15, one can show that if $\gamma$ is in
$H^{3/2}(S^{1},G)$ and $X$ is in $H^{3/2}(S^{1},\mathfrak{g}_{0})$, then

$\pi(\gamma)^{*}\pi(X)\pi(\gamma)=\pi(\text{\em
Ad}(\gamma^{-1})X)+ib(\gamma,X)\,,$ (18)

for some continuous real function $b(\gamma,X)$. Moreover, if $\gamma$ is in
$H^{5/2}(S^{1},G)$ and $h$ is a real vector field with $|h|_{3/2}<+\infty$,
then

$\pi(\gamma)^{*}\pi(X+ih)\pi(\gamma)=\pi(\text{\em
Ad}(\gamma^{-1})X)+iT(h)-\pi(h\gamma^{-1}\dot{\gamma})+ib(\gamma,X)+ib(\gamma,h)$
(19)

for some continuous real function $b(\gamma,h)$. In particular, by
$b(\gamma,Y)=c(\gamma^{-1},Y)$ we have

$b(\gamma,X)=-\ell\int_{0}^{2\pi}\braket{\dot{\gamma}\gamma^{-1},X}\frac{d\theta}{2\pi}\,,\quad
b(\gamma,h)=-\frac{\ell}{2}\int_{0}^{2\pi}h\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}\frac{d\theta}{2\pi}\,.$

##  3 Relative entropy and QNEC

Let $\mathcal{M}$ be a von Neumann algebra in standard form, and let $\varphi$
and $\psi$ be two faithful, normal and positive linear functionals on
$\mathcal{M}$ represented by vectors $\xi$ and $\eta$ in the natural cone. The
relative entropy is defined by [9]

$S(\varphi\|\psi)=-(\xi|\log\Delta_{\eta,\xi}\xi)\,,$

where the above scalar product has to be intended by applying the spectral
theorem to the relative modular operator $\Delta_{\eta,\xi}$. The relative
entropy is nonnegative, convex, lower semicontinuous in the
$\sigma(\mathcal{M}_{*},\mathcal{M})$-topology and monotone increasing with
respect to von Neumann algebras inclusions $\mathcal{N}\subseteq\mathcal{M}$
(Theorem 5.3. of [10]). If the relative entropy is finite, then

$S(\varphi\|\psi)=i\frac{d}{dt}\varphi((D\psi\colon
D\varphi)_{t})\bigg{|}_{t=0}=-i\frac{d}{dt}\varphi((D\varphi\colon
D\psi)_{t})\bigg{|}_{t=0}\,,$ (20)

where $(D\varphi\colon D\psi)_{t}=(D\psi\colon D\varphi)_{t}^{*}$ is the
Connes cocycle.

We now denote by
$\mathcal{A}_{\ell}=\\{\mathcal{A}_{\ell}(I)\\}_{I\in\mathcal{K}}$ the
conformal net associated to the level $\ell$ vacuum representation $\pi$ of
some loop group $LG$. We are interested in computing

$S(t)=S_{\mathcal{A}_{\ell}(t,+\infty)}(\omega_{\gamma}\|\omega)\,,$ (21)

where $\omega$ is the vacuum state represented by the vacuum vector $\Omega$
and $\omega_{\gamma}=\omega\cdot\text{Ad}\,\pi(\gamma)^{*}$ is represented by
$\pi(\gamma)\Omega$. To this purpose, we define the support of a loop $\gamma$
by

$\text{supp}\,\gamma=\overline{\\{\theta\in S^{1}\colon\gamma(\theta)\neq
e\\}}\,.$

We also introduce the groups of continuously differentiable and piecewise
smooth loops

$\begin{split}B(z_{1},\dots,z_{n})&=\big{\\{}\gamma\in
C^{1}_{ps}(S^{1},G)\colon\gamma(z_{i})=e\,,\dot{\gamma}(z_{i})=0\big{\\}}\,,\end{split}$
(22)

where by piecewise smooth we mean that right and left derivatives always exist
and that $\gamma$ is smooth except for a finite number of points. By standard
arguments such functions are in $H^{s}(S^{1},G)$ for $s<5/2$. Therefore, by
Theorem 9 we have that if $\gamma$ is a loop in $B(z,w)$ then in
$PU(\mathcal{H})$ we have

$\pi(\gamma)=\pi(\gamma_{(z,w)})\pi(\gamma_{(w,z)})\,,$ (23)

with $\text{supp}\,\gamma_{(z,w)}$ contained in the closure of $(z,w)$ and
similarly for $\gamma_{(w,z)}$. In particular $\pi(\gamma_{(z,w)})$ is in
$\mathcal{A}_{\ell}((z,w))$ and $\pi(\gamma_{(w,z)})$ is in
$\mathcal{A}_{\ell}((w,z))$, where $(z,w)$ denotes the interval of $S^{1}$
obtained by moving counterclockwise from $z$ to $w$. We also recall that by
Bisognano-Wichmann we have $\log\Delta=-2\pi D$ with
$D=-\frac{i}{2}(L_{1}-L_{-1})$, that is $\log\Delta=T(\delta)$ with $\delta$
the vector field generating $\delta(t).u=e^{-2\pi t}u$. We are therefore
interested in computing the vacuum expectation of

$\begin{split}\pi(\gamma)^{*}T(\delta)\pi(\gamma)&=T(\delta)+i\pi(\delta\gamma^{-1}\dot{\gamma})+b(\gamma,\delta)\,.\end{split}$
(24)

######  Lemma 19.

In every vacuum representation, for $X$ in $H^{3/2}(S^{1},\mathfrak{g})$ we
have $(\Omega|\pi(hX)\Omega)=0$ for every vector field $h$ in
$\mathfrak{sl}_{2}(\mathbb{C})$.

###### Proof.

We can suppose $X$ to be in $L\mathfrak{g}$. Notice that
$L\mathfrak{g}=\mathfrak{g}+\text{Ran}\,d$, since an element $X$ in
$L\mathfrak{g}$ with null average is the derivative of some element $Y$ in
$L\mathfrak{g}$. Therefore,

$\pi(hX)=\pi(h.Y)+\pi(ha)=[\pi(h),\pi(Y)]+\pi(ha)\,,\quad a\in\mathfrak{g}\,,$

and the thesis follows as the vacuum is $G$-invariant and Möb-invariant. ∎

######  Proposition 20.

Let $\gamma$ be a loop in $H^{3}(S^{1},G)$. Pick a not dense open interval $I$
of the circle and write $\gamma=\gamma_{I}\gamma_{I^{\prime}}$ as in (23). Set
$\Delta^{it}_{I}=e^{it\pi(\delta_{I})}$ and denote by $\delta_{I}$ the
generator of dilations of the interval $I$. If $\dot{\gamma}$ vanishes on the
boundary of $I$ then the Connes cocycle $(D\omega_{\gamma}\colon D\omega)_{t}$
of $\mathcal{A}_{\ell}(I)$ is given by

$(D\omega_{\gamma}\colon
D\omega)_{t}=e^{it(a+c(\gamma_{I},\delta_{I}))}e^{t(i\pi(\delta_{I})+\pi(\delta_{I}\dot{\gamma}_{I}\gamma_{I}^{-1}))}\Delta^{-it}_{I}\,,$
(25)

for some $a=a_{\gamma}$ in $\mathbb{R}$. In particular, $a$ depends only on
the values of $\gamma$ at the boundary of $I$ and $a_{\gamma}=0$ if
$\gamma(z)=e$ for $z$ in the boundary of $I$.

###### Proof.

First we check that $\delta_{I}\dot{\gamma}_{I}\gamma_{I}^{-1}$ is in
$H^{2}(S^{1},G)$ since it vanishes with its first derivative on the boundary
of $I$. Hence the right hand side of (25), which we denote by $u_{t}$, is a
well defined unitary operator which is in $\mathcal{A}_{\ell}(I)$ by the
Trotter product formula. To prove the existence of $a$ in $\mathbb{R}$ as in
the statement it suffices to check that $u_{t}$ verifies the relations

(i) $\sigma_{t}(x)=u_{t}\sigma_{t}(x)u_{t}^{*}\,,\quad
x\in\mathcal{A}_{\ell}(I)\,,$

(ii) $u_{t+s}=u_{t}\sigma_{t}(u_{s})\,.$

Here $\sigma_{t}$ and $\sigma^{\gamma}_{t}$ are the modular automorphisms
associated to the states $\omega$ and $\omega_{\gamma}$. The first relation
follows by noticing that

$\begin{split}\sigma^{\gamma}_{t}(x)&=\text{Ad}\,\Delta^{it}_{I,\gamma}(x)=\text{Ad}\,\pi(\gamma)\Delta_{I}^{it}\pi(\gamma)^{*}(x)\\\
&=\text{Ad}\,\pi(\gamma)\Delta_{I}^{it}\pi(\gamma)^{*}\Delta_{I}^{-it}\Delta_{I}^{it}(x)\\\
&=\text{Ad}\,u_{t}\cdot\sigma_{t}(x)\,,\end{split}$ (26)

where we used Lemma 3.(ii) of [11] and Theorem 17. The second relation can be
easily verified and thus $a$ does exist. Now we prove that $a=a_{\gamma}$
depends only on the values of $\gamma$ at the boundary of $I$. Consider $\eta$
in $H^{3}(S^{1},G)$ such that, $\eta(z)=e$ and $\dot{\eta}(z)=0$ for $z$ in
the boundary of $I$. Notice that $(D\omega_{\eta\gamma}\colon
D\omega)_{t}=\pi(\eta)(D\omega_{\gamma}\colon
D\omega)_{t}\sigma_{t}(\pi(\eta)^{*})$. Therefore, with the notation of
Corollary 18 we have

$\begin{split}a_{\eta\gamma}-b(\eta_{I}\gamma_{I},\delta_{I})&=-i\frac{d}{dt}\omega_{\eta\gamma}((D\omega_{\eta\gamma}\colon
D\omega)_{t})\bigg{|}_{t=0}=-i\frac{d}{dt}\omega_{\eta\gamma}(\pi(\eta)(D\omega_{\gamma}\colon
D\omega)_{t}\sigma_{t}(\pi(\eta)^{*}))\bigg{|}_{t=0}\\\
&=a_{\gamma}-b(\gamma_{I},\delta_{I})-b(\eta_{I}\gamma_{I},\delta_{I})+b(\gamma_{I},\delta_{I})\,,\end{split}$
(27)

and by the identity $a_{\eta\gamma}=a_{\gamma}$ the assertion is proved. If
$\gamma(z)=e$ for $z$ in the boundary of $I$ then $\pi(\gamma_{I})$ is in
$\mathcal{A}_{\ell}(I)$ and the last statement follows by Lemma 3.(iv) of
[11]. ∎

Now we arrive to the main theorem of this work, that is we will use the
previous results to prove the QNEC [4] on the loop groups models for the
states of the type $\omega_{\gamma}$. Namely, we want to prove that the
relative entropy (21) verifies

$S^{\prime\prime}(t)\geq 0\,.$ (28)

######  Theorem 21.

Let $\gamma$ be a loop in $H^{3}(S^{1},G)$. Then the relative entropy (21) is
finite and given by

$S(t)=-\frac{\ell}{2}\int_{t}^{\infty}(u-t)\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$
(29)

In particular, the QNEC (28) holds, with

$S^{\prime\prime}(t)=-\frac{\ell}{2}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}(t)\geq
0\,.$ (30)

###### Proof.

Since the vacuum is $G$-invariant then we can replace $\gamma$ with $\gamma g$
for any $g$ in $G$, thus we can suppose $\gamma(\infty)=e$. We first suppose
$\gamma$ to be in $B(\infty)$. We point out that if $\gamma(t)=e$ and
$\dot{\gamma}(t)=0$ then $S(t)$ is finite and given by (29) since we can use
equation (20) and Proposition 20. Now we prove that $S(t)$ is finite for any
$t$ real. Indeed, for any $t$ real we can pick a smooth loop $\eta$ with
$\text{supp}\,\eta\leq t$ and such that $\eta(t-k)=\gamma(t-k)^{-1}$ and
$\dot{(\eta\gamma)}(t-k)=0$ for some $k>0$. This implies that

$S_{\mathcal{A}_{\ell}(t,+\infty)}(\omega_{\gamma}\|\omega)=S_{\mathcal{A}_{\ell}(t,+\infty)}(\omega_{\eta\gamma}\|\omega)\leq
S_{\mathcal{A}_{\ell}(t-k,+\infty)}(\omega_{\eta\gamma}\|\omega)<+\infty\,,$

where the last relative entropy is finite by the argument used above. By
similar arguments we have that
$\bar{S}(t)=S_{\mathcal{A}_{\ell}(-\infty,t)}(\omega_{\gamma}\|\omega)$ is
finite for any $t$ real. Now we focus on the case $t=0$, since the general
case follows by covariance. We suppose $\dot{\gamma}(0)=0$ and we write
$\gamma=\gamma_{+}\gamma_{-}$, with $\gamma_{+}(u)=e$ for $u\leq 0$ and
$\gamma_{-}(u)=e$ for $u\geq 0$. By Proposition 20 we have

$S(0)=a_{\gamma}-\frac{\ell}{2}\int_{0}^{\infty}u\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$

Now we emulate some techniques used in [8] and we prove that $a_{\gamma}=0$.
Given $\lambda>0$ real, consider the function $f(u)=ue^{\lambda u}$. For $n>0$
integer, we consider a smooth diffeomorphism $\rho=\rho_{\lambda,n}$ of the
circle such that, in the real line picture, it verifies $\rho(u)=f(u)$ for
$0\leq u\leq n-\frac{1}{n}$ and $\rho(u)=f^{\prime}(n)u+(f(n)-nf^{\prime}(n))$
for $u\geq n$. We also suppose $\rho(u)/\rho^{\prime}(u)$ to be uniformly
bounded for $n-\frac{1}{n}\leq u\leq n$. Consider now the loop
$\gamma_{\lambda,n}(u)=\gamma(\rho^{-1}_{\lambda,n}(u))$. By the identity
$a_{\gamma}=a_{\gamma_{\lambda,n}}$ and by monotone convergence we have

$\begin{split}0&\leq\inf_{\lambda}S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{\lambda,n}}\|\omega)=a_{\gamma}-\frac{\ell}{2}\int_{n}^{\infty}(u-n)\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,,\end{split}$
(31)

and by monotone convergence we have $a_{\gamma}\geq 0$. Now we prove the other
inequality. Consider smooth paths $\zeta_{n}$ in $G$ with extremes
$\zeta_{n}(0)=e$ and $\zeta_{n}(1)=\gamma(0)$. We also suppose that
$\dot{\zeta}_{n}(0)=\dot{\zeta}_{n}(1)=0$, $\ddot{\zeta}_{n}(0)=0$ and
$\ddot{\zeta}_{n}(1)=\ddot{\gamma}(0)/n^{2}$. Lastly, we suppose that
$\zeta_{n}\to\zeta$ uniformly with its derivatives for some smooth path
$\zeta$ on $G$. We now define

$\gamma_{n}(u)=\begin{cases}\gamma(u)&u\geq 0\,,\\\ \zeta_{n}(nu+1)&-1/n\leq
u\leq 0\,,\\\ e&u\leq-1/n\,.\end{cases}$

By monotonicity
$S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma}\|\omega)=S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{n}}\|\omega)\leq
S_{\mathcal{A}_{\ell}(-1/n,+\infty)}(\omega_{\gamma_{n}}\|\omega)$, and after
a limit we have

$a_{\gamma}\leq-\frac{\ell}{2}\int_{0}^{1}u\braket{\dot{\zeta}\zeta^{-1},\dot{\zeta}\zeta^{-1}}du\,.$

However, if we now consider the function $g_{\lambda}(u)=ue^{\lambda(u-1)}$
and we define $\zeta_{\lambda}(u)=\zeta(g_{\lambda}^{-1}(u))$, then

$a_{\gamma}\leq-\frac{\ell}{2}\int_{0}^{1}u\braket{\dot{\zeta_{\lambda}}\zeta_{\lambda}^{-1},\dot{\zeta_{\lambda}}\zeta_{\lambda}^{-1}}du\leq-\frac{\ell}{2\lambda}\int_{0}^{1}u\braket{\dot{\zeta}\zeta^{-1},\dot{\zeta}\zeta^{-1}}du\to
0\,,\qquad\lambda\to+\infty\,.$

Finally, we have proved that $a_{\gamma}=0$ if $\dot{\gamma}(0)=0$. To remove
this condition, we notice that if $P$ is the generator of translations then
the average energy in the state $\omega_{\gamma}$ is finite and given by

$E_{\gamma}=(\pi(\gamma)\Omega|P\pi(\gamma)\Omega)=-\frac{\ell}{2}\int_{-\infty}^{+\infty}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}\frac{du}{2\pi}\,.$
(32)

Therefore we can apply Lemma 1. of [4], namely for every $t_{1}$ and $t_{2}$
in $\mathbb{R}$ we have

$(S(t_{1})-S(t_{2}))+(\bar{S}(t_{2})-\bar{S}(t_{1}))=(t_{2}-t_{1})2\pi
E_{\gamma}\,.$ (33)

This implies that $S(t)$ and $\bar{S}(t)$ are both Lipschitz functions.
Consider now a smooth real function $\rho(u)$ defined on $[0,1]$ and such that
$\rho(0)=0$ and $\rho(1)=1$. We also suppose
$\rho^{\prime}(0)=\rho^{\prime\prime}(0)=0$, $\rho^{\prime}(1)=1$ and
$\rho^{\prime\prime}(1)=0$. We define

$\gamma_{n}(u)=\begin{cases}\gamma(u)&u\geq 1/n\,,\\\ \gamma(\rho(nu)/n)&0\leq
u\leq 1/n\,,\\\ \eta(u)&u\leq 0\,,\end{cases}$

where $\eta$ is a smooth function such that $\gamma_{n}$ is in
$H^{3}(S^{1},G)$. Therefore, by (33) we have

$0\leq
S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{n}}\|\omega)-S_{\mathcal{A}_{\ell}(1/n,+\infty)}(\omega_{\gamma_{n}}\|\omega)\leq\frac{2\pi}{n}E_{\gamma_{n}}\to
0\,,$

and thus we have

$\begin{split}S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma}\|\omega)&=\lim
S_{\mathcal{A}_{\ell}(1/n,+\infty)}(\omega_{\gamma}\|\omega)=\lim
S_{\mathcal{A}_{\ell}(1/n,+\infty)}(\omega_{\gamma_{n}}\|\omega)\\\ &=\lim
S_{\mathcal{A}_{\ell}(1/n,+\infty)}(\omega_{\gamma_{n}}\|\omega)-S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{n}}\|\omega)+S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{n}}\|\omega)\\\
&=\lim S_{\mathcal{A}_{\ell}(0,+\infty)}(\omega_{\gamma_{n}}\|\omega)\\\
&=-\frac{\ell}{2}\int_{0}^{\infty}u\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.\end{split}$
(34)

The most of the work is done. Now we just have to remove the condition
$\dot{\gamma}(\infty)=0$. If we apply covariance to equation (34) then we have

$S_{\mathcal{A}_{\ell}(-\infty,0)}(\omega_{\gamma}\|\omega)=-\frac{\ell}{2}\int_{-\infty}^{0}u\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}$

for any $\gamma$ in $H^{3}(S^{1},G)$ such that $\dot{\gamma}(0)=0$. But this
condition can be removed as in (34), and by covariance we have that the above
expression of $S(0)$ holds for a generic loop $\gamma$ in $H^{3}(S^{1},G)$. ∎

As a remark, we add a physical characterization of the second derivative
$S^{\prime\prime}(t)$ which motivates this axiomatic definition of the QNEC.
Since the quantity $E=E_{\gamma}$ given by (32) is the average energy in the
state $\omega_{\gamma}$, then we can define the density of energy

$E(t)=-\frac{\ell}{4\pi}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}(t)\,.$

In particular, we have that the QNEC is satisfied with the equality

$E(t)=S^{\prime\prime}(t)/2\pi\geq 0\,.$

######  Corollary 22.

Let $\gamma$ be a loop in $H^{3}(S^{1},G)$. Then

$S_{\mathcal{A}_{\ell}(-r,r)}(\omega_{\gamma}\|\omega)=-\frac{\ell}{2}\int_{-r}^{r}\frac{1}{2r}(r-u)(r+u)\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$
(35)

In particular, if $E_{\gamma}$ is the energy (32) then we have the Bekenstein
Bound

$S_{\mathcal{A}_{\ell}(-r,r)}(\omega_{\gamma}\|\omega)\leq\pi rE_{\gamma}\,.$
(36)

###### Proof.

Since we proved Theorem 27 for a generic loop in $H^{3}(S^{1},G)$ then we can
just apply Möb-covariance to equation (38) to have the identity (35). See also
[9] for some similar results. ∎

###  3.1 QNEC on $LSU(n)$

In this section we focus on the case $G=SU(n)$ and we use a construction
illustrated in [15] to show that a positive energy representation of $LSU(n)$
can be extended to a positive energy representation of the Sobolev group
$H^{s}(S^{1},SU(n))$ for $s>1/2$. In particular, we will use this fact to
provide a simpler proof of the QNEC (28).

We begin by considering the natural action of $G=SU(n)$ on $V=\mathbb{C}^{n}$
and set $H=L^{2}(S^{1},V)$, or equivalently $H=L^{2}(S^{1})\otimes V$. We can
naturally define a continuous action $M$ of $LG$ on $H$ by
$M_{\gamma}f(\phi)=\gamma(\phi)f(\phi)$. We can also define an action of Rot
on $H$ by $R_{\theta}f(\phi)=f(\phi-\theta)$ with respect to the
representation of $LG$ is covariant, that is it satisfies
$R_{\theta}M_{\gamma}R_{\theta}^{-1}=M_{R_{\theta}\gamma}$. If $P$ is the
orthogonal projection onto the Hardy space $H_{+}$, namely

$H_{+}=\biggl{\\{}f\in L^{2}(S^{1},V)\colon f(\theta)=\sum_{k\geq
0}f_{k}e^{ik\theta}\text{ with }f_{k}\in V\biggr{\\}}\,,$

then we can define a new Hilbert space $H_{P}$ which is equivalent to $H$ as
real Hilbert space, but with complex structure given by $J=iP-i(1-P)$. The
Segal quantization criterion, which we now recall, allows us to define a
positive energy representation of $LG$ on the fermion Fock space
$\mathcal{F}_{P}=\Lambda H_{P}$ known as the fundamental representation of
$LSU(n)$. Since $\mathcal{F}_{P}(0)=\Lambda V$, the fundamental representation
of $LSU(n)$ is the direct sum of all the $n+1$ irreducible positive energy
representations of $LSU(n)$ of level $\ell=1$. The fundamental representation
contains the basic representation, that is the unique level one vacuum
representation.

######  Definition 23.

The _restricted unitary group_ is the topological group

$U_{P}(H)=\bigl{\\{}u\in U(H)\colon[u,P]\in L^{2}(H)\bigr{\\}}\,.$

The considered topology is the strong operator topology combined with the
metric given by $d(u,v)=\|[u-v,P]\|_{2}$.

Any $u\in U(H)$ gives rise to an automorphism of $CAR(H)$, called _Bogoliubov
automorphism_ , via $a(f)\mapsto a(uf)$. For every projection $P$ on $H$ there
is an irreducible representation of $CAR(H)$ on $\mathcal{F}_{P}$ which is
denoted by $\pi_{P}$. The Bogoliubov automorphism is said to be _implemented_
on $\mathcal{F}_{P}$ if $\pi_{P}(a(uf))=U\pi_{P}(a(f))U^{*}$ for some unitary
$U\in U(\mathcal{F}_{P})$.

######  Theorem 24.

Segal’s quantization criterion. [15] If $[u,P]$ is a Hilbert-Schmidt operator
then $u$ is implemented on $\mathcal{F}_{P}$ by some unitary operator $U_{P}$.
Moreover, $U_{P}$ is unique up to a phase and the constructed map $U_{P}(H)\to
PU(\mathcal{F}_{P})$ is continuous.

######  Proposition 25.

The fundamental representation of $LSU(n)$ can be extended to
$H^{s}(S^{1},SU(n))$ for any $s>1/2$. In particular, every positive energy
representation of $LSU(n)$ extends to a positive energy representation of
$H^{s}(S^{1},SU(n))$ for $s>1/2$.

###### Proof.

Notice that since a loop $\gamma$ in $LSU(n)$ is also a map from $S^{1}$ to
$M_{n}(\mathbb{C})$ then we can write $\gamma$ as a Fourier series
$\gamma(z)=\sum\widehat{\gamma}_{k}z^{k}$, where $\widehat{\gamma}_{k}\in
M_{n}(\mathbb{C})$. We consider on $H$ the basis $e_{j}^{k}(z)=z^{k}e_{j}$,
where $(e_{j})$ is the standard basis of $\mathbb{C}^{n}$. We define
$M_{pq}=\widehat{\gamma}_{p-q}$ and we note that
$M_{\gamma}e^{k}_{j}=\sum_{i}M_{ik}e^{i}_{j}$, so that
$(e^{p}_{i},M_{\gamma}e^{q}_{j})=(e_{i},M_{pq}e_{j})$. So $M_{\gamma}$ is
represented by a $\mathbb{Z}\times\mathbb{Z}$ matrix $(M_{pq})$ of
endomorphisms. We have

$\begin{split}\|[P,M_{\gamma}]\|_{2}^{2}&=\sum_{p\geq
0,q<0}\|M_{pq}\|_{2}^{2}+\sum_{p<0,q\geq 0}\|M_{pq}\|_{2}^{2}\\\
&=\sum_{k>0}k\|\widehat{\gamma}_{k}\|_{2}^{2}-\sum_{k<0}k\|\widehat{\gamma}_{k}\|_{2}^{2}\\\
&=\sum_{k\in\mathbb{Z}}|k|\|\widehat{\gamma}_{k}\|_{2}^{2}\leq\sum_{k\in\mathbb{Z}}(1+|k|)^{2s}\|\widehat{\gamma}_{k}\|_{2}^{2}\,,\\\
\end{split}$

for $s>1/2$. It is easy to verify that the map $\gamma\mapsto M_{\gamma}\in
U_{P}(H)$ is continuous. We also have that the rotation group acts on
$H^{s}(S^{1},G)$ by continuous operators (see Lemma A.3 of [2]). Therefore,
the thesis follows by the Segal quantization criterion, the complete
reducibility of positive energy representations (Thm. 9.3.1. of [12]),
Proposition 2.3.3. of [14] and remarks below. ∎

######  Proposition 26.

Let $\gamma$ be a loop in $H^{1}(S^{1},SU(n))$. Pick a not dense open interval
$I$ of the circle and write $\gamma=\gamma_{I}\gamma_{I^{\prime}}$ as in (23).
Then, in $PU(\mathcal{H})$ we have

$(D\omega_{\gamma}\colon
D\omega)_{t}=\pi(\gamma_{I}\delta_{I}(t).\gamma_{I}^{-1})\,,$ (37)

where $(D\omega_{\gamma}\colon D\omega)_{t}$ is the Connes cocycle of
$\mathcal{A}_{\ell}(I)$ and $\delta_{I}(t)$ denotes the dilation associated to
$I$.

###### Proof.

First we check that $\gamma_{I}\delta_{I}(t).\gamma_{I}^{-1}$ is in
$H^{1}(S^{1},SU(n))$ since it is continuous on the boundary of $I$, hence the
right hand side of (37) is well defined. With the same computations of
Proposition 20 we have that
$\sigma^{\gamma}_{t}(x)=\text{Ad}\,\pi(\gamma_{+}\delta_{I}(t).\gamma_{+}^{-1})\cdot\sigma_{t}(x)$
for $x$ in $\mathcal{A}_{\ell}(I)$. Therefore, we have that
$(D\omega_{\gamma}\colon D\omega)_{t}$ is equal to
$\pi(\gamma_{+}\delta_{I}(t).\gamma_{+}^{-1})$ up to a unitary $V$ in the
commutant of $\mathcal{A}_{\ell}(I)$, but $(D\omega_{\gamma}\colon
D\omega)_{t}$ and $\pi(\gamma_{+}\delta_{I}(t).\gamma_{+}^{-1})$ are both in
$\mathcal{A}_{\ell}(I)$ and thus $V$ is a scalar. ∎

######  Theorem 27.

Let $\gamma$ be a loop in $H^{1}(S^{1},SU(n))$. Suppose also that, in the real
line picture, the support of $\gamma$ is bounded from below. Then the relative
entropy (21) is finite and given by

$S(t)=-\frac{\ell}{2}\int_{t}^{\infty}(u-t)\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$
(38)

In particular, the QNEC (28) holds, with

$S^{\prime\prime}(t)=-\frac{\ell}{2}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}(t)\geq
0\,.$ (39)

###### Proof.

Since the vacuum is $SU(n)$-invariant then we can replace $\gamma$ with
$\gamma g$ for any $g$ in $SU(n)$, thus we can suppose $\gamma(\infty)=e$. As
above, if $\gamma(t)=e$ then $S(t)$ is finite and given by (38). We can prove
that $S(t)$ is finite for any $t$ real as in Theorem 21, and similarly we have
that $\bar{S}(t)=S_{\mathcal{A}_{\ell}(-\infty,t)}(\omega_{\gamma}\|\omega)$
is finite for any $t$ real. If $P$ is the generator of translations then the
average energy $E_{\gamma}$ in the state $\omega_{\gamma}$ is finite and given
by equation (32). Therefore we can apply Lemma 1. of [4], and equation (33)
holds. This implies that $S(t)$ and $\bar{S}(t)$ are both Lipschitz functions
and in particular they are absolutely continuous. The next step is an estimate
on $S^{\prime}(t)$. For simplicity we focus on the case $t=0$ and we write
$\gamma=\gamma_{+}\gamma_{-}$ with $\gamma_{+}(u)=e$ for $u\leq 0$ and
$\gamma_{-}(u)=e$ for $u\geq 0$. By Proposition 26 the Connes cocycle
$u^{\prime}_{s}=(D\omega\colon D\omega_{\gamma})_{s}$ on
$\mathcal{A}_{\ell}(-\infty,0)$ is equal in $PU(\mathcal{H})$ to
$\pi(\gamma_{-}\delta(-s).\gamma_{-}^{-1})^{*}$. But also the state
$\omega_{\gamma}\cdot\text{Ad}(u^{\prime}_{s})^{*}$ verifies the finiteness
conditions required to apply Lemma 1. and thus we have

$-S^{\prime}(0)\leq 2\pi E_{s}\,,\quad
E_{s}=(u^{\prime}_{s}\pi(\gamma)\Omega|Pu^{\prime}_{s}\pi(\gamma)\Omega)\,,\quad
s\in\mathbb{R}\,.$

However, one can simply prove that

$\inf_{s}2\pi
E_{s}=-\frac{\ell}{2}\int_{0}^{+\infty}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$

Therefore, by repeating the argument with any $t$ in $\mathbb{R}$ we have

$-S^{\prime}(t)\leq-\frac{\ell}{2}\int_{t}^{+\infty}\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,.$

Finally, if we define

$F(t)=-\frac{\ell}{2}\int_{t}^{\infty}(u-t)\braket{\dot{\gamma}\gamma^{-1},\dot{\gamma}\gamma^{-1}}{du}\,,$

then we can conclude that $S(t)=F(t)$ for any $t$ in $\mathbb{R}$. Indeed, if
the support of $\gamma$ is compact then $H(t)=S(t)-F(t)$ is an absolutely
continuous function with nonnegative derivative and going to $0$ as
$t\to+\infty$. If the support of $\gamma$ is contained in $(k,+\infty)$ then
by lower semicontinuity $S(t)\leq F(t)$ for every $t$ real, and we can
similarly deduce that $H(t)=0$ for every $t$ real. ∎

##  4 Solitonic representations from discontinuous loops

In this section we follow [6] and we construct proper solitonic
representations of the conformal net
$\mathcal{A}=\\{\mathcal{A}(I)\\}_{I\in\mathcal{K}}$ associated to some vacuum
positive energy representation $\pi$ of a loop group $LG$. We denote by $U$
the projective unitary continuous representation of $\textit{\em
Diff}_{+}(S^{1})$ and we always suppose $G$ to be simple, compact and simply
connected. We recall that $\mathcal{K}$ denotes the family of all the open,
nonempty and not dense intervals of the circle. Given $h$ in $G$, we define

$L_{h}G=\big{\\{}\zeta\in
C^{\infty}(\mathbb{R},G)\colon\zeta(x)^{-1}\zeta(x+2\pi)=h\big{\\}}\,.$

We denote by $\mathcal{I}_{\mathbb{R}}$ the set of all the open, nonempty and
not dense intervals of $S^{1}\setminus\\{-1\\}$. We then define
$\sigma_{\zeta}=\\{\sigma_{\zeta}^{I}\\}_{I\in\mathcal{I}_{\mathbb{R}}}$ as
the collection of maps given by

$\sigma_{\zeta}^{I}\colon\mathcal{A}(I)\to
B(\mathcal{H})\,,\quad\sigma^{I}_{\zeta}(x)=\text{Ad}\,\pi(\zeta_{I})(x)\,,$

where $\zeta_{I}$ is a loop in $LG$ such that
$\zeta_{I}(\theta)=\zeta(\theta)$ for $\theta$ in $I$ seen as a subinterval of
$(-\pi,\pi)$.

######  Proposition 28.

$\sigma_{\zeta}$ is an irreducible locally normal soliton with index $1$.

###### Proof.

Normality on each $\mathcal{A}(I)$ follows because on these local algebras
$\sigma_{\zeta}$ is given by the adjoint action by a unitary operator. The
compatibility property is clear, since if $I\subseteq J$ then
$\pi(\zeta_{I}\zeta_{J}^{-1})$ is in
$\mathcal{A}(I^{\prime})=\mathcal{A}(I)^{\prime}$. The index is $1$ since if
$I$ is in $\mathcal{I}_{\mathbb{R}}$ then
$\sigma_{\zeta}(\mathcal{A}(I))=\mathcal{A}(I)$, and for the same reason we
have that

$\bigvee_{I\in\mathcal{I}_{\mathbb{R}}}\sigma_{\zeta}(\mathcal{A}(I))=\bigvee_{I\in\mathcal{I}_{\mathbb{R}}}\mathcal{A}(I)=B(\mathcal{H})$

since the conformal net $\mathcal{A}$ is irreducible (see [6] for the
definitions). ∎

Since $\pi$ is irreducible if and only if it is irreducible as a projective
representation of $LG$, then $\sigma_{\zeta}$ is irreducible if $\pi$ is
irreducible (see also Corollary 1.3.3. of [14]). Notice that for $\zeta$ in
$L_{h}G$ we have that $\zeta_{t}(\phi)=\zeta(\phi)\zeta(\phi-t)^{-1}$ is in
$LG$ for any $t$ in $\mathbb{R}$. As before, we denote by
$\text{Rot}\cong\mathbb{T}$ the group of rotations of the circle. If $R_{t}$
is the unitary representation of Rot associated to $\pi$, then we can define
$R^{\zeta}_{t}=\pi(\zeta_{t})R_{t}$ in $PU(\mathcal{H})$ and notice that
$R^{\zeta}_{t}R^{\zeta}_{s}=R^{\zeta}_{t+s}$. However, in general
$R^{\zeta}_{2\pi}$ is not a scalar and therefore $\sigma_{\zeta}$ is not Rot-
covariant but only locally $\text{Rot}^{\sim}$-covariant. We recall the
definition.

######  Definition 29.

Let $\mathcal{G}$ be a topological group equipped with some homomorphism
$\iota\colon\mathcal{G}\to\textit{\em Diff}_{+}(S^{1})$. We say that a soliton
$\sigma=\\{\sigma_{I}\\}_{I\in\mathcal{I}_{\mathbb{R}}}$ is locally
$\mathcal{G}$-covariant if there is a unitary projective continuous
representation $U_{\sigma}$ of $\mathcal{G}$ which satisfies the following
property: if $I$ is in $\mathcal{I}_{\mathbb{R}}$ and $V$ is a connected
neighborhood of the identity in $\mathcal{G}$ such that $g.I$ is in
$\mathcal{I}_{\mathbb{R}}$ for every $g$ in $V$, then
$\text{Ad}\,U_{\sigma}(g)\sigma_{I}(x)=\sigma_{\iota(g).I}\text{Ad}\,U(\iota(g))(x)$
for every $x$ in $\mathcal{A}(I)$.

We can notice that if $\zeta$ is in $L_{g}G$ and $\eta$ is in $L_{h}G$ then
$\zeta\eta^{-1}$ is in $L_{h^{-1}g}G$ if $h^{-1}g$ is in $Z(G)$. In
particular, if $\zeta$ and $\eta$ are in both in $L_{h}G$ then
$\zeta\eta^{-1}$ is in $LG$ and $\sigma_{\zeta}$ is unitarily equivalent to
$\sigma_{\eta}$.

######  Theorem 30.

Let $\pi$ be a vacuum positive energy representation of $LG$ of level
$\ell\geq 1$. Given $\zeta$ in $L_{h}G$, the soliton $\sigma_{\zeta}$ extends
to a DHR representation if and only if $h$ is central.

###### Proof.

First we suppose $h$ to be in $Z(G)$. A quick computation shows that in this
case $R^{\zeta}_{2\pi}=\pi(h)$. By the identity
$\pi(h)e^{\pi(X)}\pi(h)^{*}=e^{\pi(X)}$ for any $X$ in $L\mathfrak{g}_{0}$ we
have that $R^{\zeta}_{2\pi}$ is a scalar since $\pi$ is irreducible. This
implies that $\sigma_{\zeta}$ is locally Rot-covariant and we have that
$\sigma_{\zeta}$ can be extended to a locally normal DHR representation by
using the arguments of Proposition 3.8. of [6]. Now we suppose $h$ to be not
central. By absurd, $\sigma_{\zeta}$ extends to a DHR representation and thus
it is Rot-covariant. Denote by $U^{\zeta}_{\theta}$ the corresponding
projective representation of the circle. Since $\sigma_{\zeta}$ is locally
normal, by using the additivity property one can show that

$\text{Ad}U_{\pi}^{\zeta}\cdot\sigma_{\zeta}(x)=\sigma_{\zeta}\cdot\text{Ad}R_{\pi}(x)=\sigma_{\zeta}\cdot\text{Ad}R_{-\pi}(x)=\text{Ad}R_{-\pi}^{\zeta}\cdot\sigma_{\zeta}(x)\,,\quad
x\in\mathcal{A}((0,\pi))\,.$

It follows that $U^{\zeta}_{\pi}R^{\zeta}_{\pi}$ is in
$\mathcal{A}((0,\pi))^{\prime}$, and by similarly proceeding on
$\mathcal{A}((-\pi,0))$ we have that $U^{\zeta}_{\pi}R^{\zeta}_{\pi}$ is in
$\mathcal{A}((-\pi,0))^{\prime}$. Therefore $R^{\zeta}_{\pi}=U^{\zeta}_{-\pi}$
up to a scalar and thus $R^{\zeta}_{2\pi}$ is a scalar. Now consider a maximal
torus $T\subset G$ containing $h$. Since $T$ is connected, we can suppose that
$\zeta(x)$ belongs to $T$ for any $x$ in $\mathbb{R}$, and by commutativity we
have that $R^{\zeta}_{2\pi}=\pi(h)$ in $PU(\mathcal{H})$. Therefore we have
that $h$ is a noncentral element acting on $\mathcal{H}$ as a scalar. If we
now consider the kernel

$N=\\{g\in G\colon\pi(g)\in\mathbb{T}\\}\,,$

then $N$ is a normal subgroup of $G$ which is not contained in the center. But
$G$ is simple and connected, hence we have that $N=G$, which is an absurd. ∎

We conclude this last section by studying the equivalence classes of the
solitons constructed above. The DHR representations $\sigma_{z}$ with $z$ in
$Z(G)$ correspond to inequivalent irreducible positive energy representations
$\pi_{z}$ of the same level than $\pi$ (see Remark 14 and Theorem 3.2.3. of
[14]). Now we pick a maximal torus $T$ in $G$. Consider $\zeta$ in $L_{s}G$
and $\eta$ in $L_{t}G$ for some $s$ and $t$ in $T$. We can suppose $\zeta$ and
$\eta$ to be both contained in $T$. It can be easily noticed that

$\sigma_{\zeta}\cdot\sigma_{\eta}=\sigma_{\zeta\eta}\,,\quad\zeta\eta\in
L_{st}G\,,\qquad\sigma_{\zeta}^{-1}=\sigma_{\zeta^{-1}}\,,\quad\zeta^{-1}\in
L_{s^{-1}}G\,.$

It follows that $\sigma_{\zeta}$ and $\sigma_{\eta}$ are unitarily equivalent
if and only if $s=t$, hence we have infinitely many inequivalent solitons. If
we consider two maximal tori $T$ and $T^{\prime}=gTg^{-1}$, then what we can
say is that we have the identity

$\sigma_{g\zeta
g^{-1}}=\text{Ad}\pi(g)\cdot\sigma_{\zeta}\cdot\text{Ad}\pi(g)^{*}\,.$

## Acknowledgements

I thank Simone Del Vecchio for suggesting me the problem, and Daniela Cadamuro
and Henning Bostelmann for the hospitality in the period spent in Leipzig.

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