Abstract
We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of \(H^{s}(S^1,G)\) for \(s>3/2\), where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product \(LG \rtimes R\), with R a one-parameter subgroup of \(\mathrm{Diff}_+(S^1)\), and we compute the adjoint action of \(H^{s+1}(S^1,G)\) on the stress energy tensor.
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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 [20, 21]. 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). In this work we prove the QNEC for some states of interest on loop group models. We refer to [26] for similar results on the Virasoro nets.
Classically, the Null Energy Condition (NEC) is a constraint on the stress energy tensor which states that \(T_{ab}k^a k^b \ge 0 \), where \(k^a\) is a null vector field. This constraint is motivated by the positivity of the energy and it is a necessary condition for the field \(k^a\) to have some physical meaning. However, quantum fields can violate all local energy conditions, including the NEC. At any point the energy density \(\langle {T_{kk}}\rangle \) can be made negative, with magnitude as large as we wish, by an appropriate choice of a quantum system [13]. In the study of relativistic QFT coupled to gravity, Bousso, Fisher, Leichenauer and Wall [3] establish a new and surprising link between quantum information and the stress energy tensor. In this work, a Quantum Null Energy Condition (QNEC) is defined as a null energy lower bound which is expected to be satisfied by most reasonable quantum fields. This lower bound is defined by using the relative entropy of Araki, an object typically used in quantum information theoretical contexts.
The first non commutative entropy notion, von Neumann’s quantum entropy, was originally designed as a Quantum Mechanics version of Shannon’s entropy: if a normal state \(\psi \) is given by a density matrix \(\rho _\psi \), then the von Neumann entropy is defined by
However, in Quantum Field Theory local von Neumann algebras are typically factors of type \(III_1\) [19], no trace or density matrix exists and the von Neumann entropy is undefined. Nontheless, the Tomita–Takesaki modular theory applies and one may consider the relative entropy of Araki [25]
between the normal states \(\varphi \) and \(\psi \). The relative entropy generalizes the classical Kullback-Leibler divergence and measures how \(\psi \) deviates from \(\varphi \). From an information theoretical viewpoint, \(S(\varphi \Vert \psi ) \) is the mean value in the state \(\varphi \) of the difference between the information carried by the state \(\psi \) and the state \(\varphi \).
Let now \(({\mathcal {A}} , U, \Omega ) \) be a local QFT on the Minkowski space \({\mathbb {R}}^{n+1}\) with vacuum state \(\omega \) and stress-energy tensor T. Denote by \(\mathfrak {A} = \overline{\bigcup _{\mathcal {O}} {\mathcal {A}} ({\mathcal {O}})}\) the \(C^*\)-algebra of quasi-local observables. Consider one of the two null directions u tangent to the standard Rindler wedge W. By Poincaré-covariance, we can associate a wedge \(W^u\) to any pair of a null direction u and a point x of the Minkowski space. Let \({\mathcal {M}}^u_t = {\mathcal {A}} (W^u_t)\) be the local algebra associated to \(W^u_t= W^u+ tu\). Given a locally normal state \(\psi \) of \(\mathfrak {A}\) represented on \( {\mathcal {A}} (W^u)\) by some vector \(\eta \), consider the relative entropy \( S(t) = S_{{\mathcal {M}}^u_t} (\psi \Vert \omega )\) and the averaged stress-energy density tensor \(\langle {T_{uu}(t)}\rangle = (\eta | {T_{uu}(x+tu)} \eta ) \). In natural units, we will say that the vector \(\eta \) satisfies the Quantum Null Energy Condition (QNEC) if for any u and x we have the null energy bound
for any \(t \ge 0\), u and x, assuming this inequality to have some distributional or classical meaning. The wedges \(W_t^u\) can be replaced with arbitrary regions deformed in the null direction u [22], but here we will not work in such a generality. In this context, the convexity of the relative entropy often appears, and for this reason in other works the QNEC is just stated as the convexity of the mentioned relative entropy [8]. This convexity property has been proved in a model independent setting for a very wide class of states.
Theorem 1
[7] Let \({\mathcal {N}} \subseteq {\mathcal {M}}\) be a ±hsm inclusion with standard vector \(\Omega \) giving the state \(\omega \). Denote by P the positive generator of translations and by \(({\mathcal {M}}_t)_{t \in {\mathbb {R}}}\) the associated flow of von Neumann algebras. If \(\psi (x) = (\eta | x \eta )\) is a vector state with finite null energy, namely such that
then the relative entropy \( S(t) = S_{{\mathcal {M}}_t}(\psi \Vert \omega ) \) is convex. Furthermore, if \(S(t_0)\) is finite then we have
where \(C'_t\) is the set of all the isometries \(w'\) in \({\mathcal {M}}_t'\) such that the complement relative entropy \( \bar{S}_{w'}(t) = S_{{\mathcal {M}}'_t}(\psi _{w'} \Vert \omega ) \) is finite, with \(\psi _{w'}(x) = (w' \eta | x w' \eta )\). The identity (3) is satisfied at each point such that \(S'(t)\) exists and on such points it can be computed by
In the above notation, we have
and \(\widehat{P}(t) = P_{\widehat{\eta }_t } \), where \(\widehat{\eta }_t \) is the unique vector in the natural cone of \({\mathcal {M}}_t\) representing the state \(\psi \).
Actually, the provided proof refers to -hsm inclusions, but the +hsm case can be similarly proved. It is also shown that the null energies (5) are finite and that the infimum (4) is obtained as \(s \rightarrow \pm \infty \) if the inclusion is \(\mp \)hsm. Of course, in principle such a characterization can be used for computational purposes. However, determining the Connes cocycle (5) for each deformation parameter t is a not trivial issue. For this reason, the proof of the QNEC (1) on loop group models led us to the study of Sobolev extensions of Positive Energy Representations of loop groups. We describe our work a bit more in detail.
The first result is the construction of solitonic representations \(\sigma _{\gamma }\) of the conformal net associated to a vacuum PER \(\pi \) of a loop group LG. In general, these solitons are induced by a path \(\gamma \) in \(C^\infty ([- \pi , \pi ], G)\). If the path \(\gamma \) satisfies some periodicity conditions on its derivatives, then it can be extended to what we here call a discontinuous loop. Discontinuous loops are defined as elements of
where h is a generic element of G. If the discontinuity h of \(\zeta \) is in Z(G), then it is already known that the obtained soliton \(\sigma _\zeta \) extends to a DHR representation which corresponds to a PER \(\zeta _* \pi \) of same level as \(\pi \) [29]. What we show here is that this condition is also necessary: the soliton \(\sigma _\zeta \) extends to a DHR representation if and only if the discontinuity h is central. The proof follows by a contradiction argument, since a locally normal DHR representation is automatically \(\text {Rot}\)-covariant [10].
Consider now the conformal net \(({\mathcal {A}}, U, \Omega )\) associated to some vacuum PER \(\pi \) of a loop group LG. In the real line picture, the von Neumann algebras associated to half-lines define a hsm inclusion. Since on these local algebras the solitonic states \(\omega _\gamma \) of above have a representing vector satisfying (2), the convexity of the relative entropy is satisfied by Theorem 17. We then use some intermediate results used in the proof of Theorem 1 to explicitly compute the relative entropy and we show the QNEC (1) to be satisfied with an equality. Furthermore, the obtained formula allow us to prove the Bekenstein Bound in a very simple way. The purpose of doing the mentioned proof led us to the second main result of this work, that is the extension of a PER of a loop group LG to a PER of the Sobolev loop group \(H^{s}(S^1, G)\) for \(s>3/2\). Such a Sobolev extension allows us to compute the adjoint action of \(H^{s+1}(S^1,G)\) on the stress energy tensor. Similar issues involving the smearing with non-smooth functions in conformal nets contexts have been treated in [5, 23] and [31].
Unexpectedly, it turns out that our extension result follows by a more general theorem proved in [24], which implies that a Positive Energy Representation of a loop group LG can be extended to a PER of \(H^{1}(S^1, G)\) by using holomorphic induction methods. However, even if not completely general, the proof presented here is mathematically simpler and original. Without knowing the results of [24] we were not able to prove the QNEC (1) by directly using the expression (5), and for this reason we used an approach more similar to that ones of [17] and [26]. Nevertheless, by using explicit constructions of [30], in the case \(G=SU(n)\) we are able to show that the Sobolev extension holds up to \(H^s(S^1, SU(n))\) for \(s>1/2\). This technical improvement allows us to show that the identity (4) is indeed verified. The exhibited proof is not just an application of Theorem 1 or of [24] and it is thought in such a way to make this work as self contained as possible.
2 Mathematical Background
We mainly follow [29]. 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
where the torus \({\mathbb {T}}\cong \text {Rot} \) acts on LG by rotations \(R_\theta . \gamma (\phi ) = \gamma (\phi - \theta )\) and U is a strongly continuous unitary representation inducing an isotypical decomposition \({\mathcal {H}}= \bigoplus _{n \ge n_0} {\mathcal {H}}(n)\) for some integer \(n_0\). Without loss of generality, we can suppose that \(n_0=0\) and that \({\mathcal {H}}(0)\) is not zero-dimensional. A PER is said to be of finite type if \( \dim {\mathcal {H}}(n) < + \infty \) for every n. 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
Let X(n) be the map \(\theta \mapsto X e^{i n \theta } \) for X in \(\mathfrak {g}\) and n integer. Then \([X(n),Y(m)]=[X,Y](n+m)\), showing that the space spanned by these elements, which we will denote by \(L^\text {pol}\mathfrak {g}\), forms a Lie algebra. On \(L^\text {pol}\mathfrak {g}\) we can define an involution by \(X(n)^*=X^*(-n)\). Moreover, if \({\mathcal {H}}^{\text {fin}}\) is the subspace of finite energy vectors, namely the algebraic sum of the subspaces \({\mathcal {H}}(n)\), then we can define a projective representation \(\pi \) of \(L^\text {pol}\mathfrak {g}\) on \({\mathcal {H}}^{\text {fin}}\) in such a way to verify the commutation relations ([29], Theorem 1.2.1.)
We point out that the existence of such a representation of \(L^\text {pol}\mathfrak {g}\) is not a trivial issue, since these commutation relations do not uniquely determine the projective representation of \(L^\text {pol}\mathfrak {g}\), and also the representation of LG cannot be differentiated in a straightforward way as in finite dimensional cases. If d is the generator of rotations, namely \(U(R_\theta ) =e^{i \theta d}\), then we have that \( [d, \pi (X)] = i \pi (\dot{X}) \) where \(\dot{X}(\theta ) = \frac{d}{d \theta }X(\theta )\). The above operators are all closable and we also have the formal adjunction property \(\pi (X)^* = \pi (X^*) \) on \({\mathcal {H}}^\text {fin}\). Furthermore, the projective representation \(\pi \) of \(L^\text {pol}\mathfrak {g}\) on \({\mathcal {H}}^\text {fin}\) can be lifted to a projective representation \(\pi \) of \(L\mathfrak {g}= C^{\infty }(S^1, \mathfrak {g})\) on \({\mathcal {H}}^\infty \) in such a way to verify all the previous relations, where \({\mathcal {H}}^\infty \) is the Fréchet space of smooth vectors for Rot. We recall that by definition \({\mathcal {H}}^\infty = \bigcap _{s} {\mathcal {H}}^s\), where \(s \in {\mathbb {R}}\) and \({\mathcal {H}}^s\) is the scale space, that is the completion of \({\mathcal {H}}^\text {fin}\) with respect to the Sobolev norm \(\Vert \xi \Vert _s = \Vert (1+d)^s \xi \Vert \). Notice that the projective representation \(\pi \) of \(L\mathfrak {g}\) is actually a representation if restricted on \(\mathfrak {g}\), since the projective representation of G lifts to a unitary representation. Also, the subspaces \({\mathcal {H}}(n)\) are G-invariant. The adjoint action of LG on the mentioned operators is given by [29]
where the real constants \(c(\gamma , X)\) and \(c(\gamma , d)\) are explicitly given by
Here \(\langle {\cdot , \cdot }\rangle \) denotes the basic inner product, namely the Killing form normalized on the highest root \(\theta \) in such a way that \(\langle {\theta , \theta }\rangle =2\). The elements \(\gamma ^{-1} \dot{\gamma } \) and \(\dot{\gamma } \gamma ^{-1}\) of \(L\mathfrak {g}\) are the left logarithmic derivative and the right logarithmic derivative of \(\gamma \), respectively defined by
We will use the following notation:
We can define a representation of the Virasoro algebra \(\text {Vir}\)
by Sugawara construction, that is such a representation is given by defining on \({\mathcal {H}}^\text {fin}\) the operator
where we used the Einstein convention on summations and the normal ordering notation, namely the symbol \(:x(n) y(m) :\) stands for x(n)y(m) if \(n \le m\) and for y(m)x(n) if \(n > m \). The elements \(\{x_i \}\) and \(\{x^i \}\) appearing in (8) can be arbitrary dual basis with respect to the basic inner product, namely \(\langle {x^i, x_j}\rangle = \delta _{ij}\), and g is the dual Coxeter number, that is
where \(\alpha ^\vee _i\) are the simple coroots and \({a}^\vee _i \) are strictly positive. By the assumption \(\langle {\theta , \theta }\rangle =2\) it can be shown that the dual Coxeter number is half the Casimir of the adjoint representation, namely we have \([X_i, [X^i,Y]] = 2g Y \) for Y in \(\mathfrak {g} \). Notice that if \(X_i\) are in \(\mathfrak {g}_0\) and such that \(- \langle {X_i, X_j}\rangle = \delta _{ij}\) then \(X^i = - X_i\). The constant c uniquely determined by (7) is called the central charge of the representation. If the PER is irreducible then we have \(L_0 = d + h\) for some rational number h, where h is therefore the lowest eigenvalue of \(L_0\) and it is called the trace anomaly. In any irreducible PER, the central charge and the trace anomaly are given by [15]
where \(C_\lambda \) is the Casimir associated to the basic inner product \(\langle {\cdot , \cdot }\rangle \) and to the null energy space \({\mathcal {H}}(0)= {\mathcal {H}}_\lambda \), which is the irreducible highest weight representation of \(\mathfrak {g}\) associated to some dominant integral weight \(\lambda \) satisfying
The set of dominant integral weights \(\lambda \) satisfying condition (10) is called the level \(\ell \) alcove. We will say that \(\pi \) is a vacuum positive energy representation, or simply a vacuum representation, if \({\mathcal {H}}(0) \) is one-dimensional. If \({\mathcal {H}}(0) = {\mathbb {C}}\Omega \) with \((\Omega | \Omega )=1\), then the state \(\omega \) associated to \(\Omega \) is called the vacuum state. Notice that \(\pi \) is a vacuum representation if and only if irreducible and with \(h=0\). More in general, if \({\mathcal {H}}(0)= {\mathcal {H}}_\lambda \) then the trace anomaly can be computed by taking in account that
where \(\rho \) is the Weyl vector, that is the sum of all the fundamental weights. Equivalently, the Weyl vector can be defined as half the sum of all the positive roots.
Consider now \(\mathrm{Diff}_+(S^1)\), the Fréchet Lie group of the orientation preserving diffeomorphisms of the circle. The natural action of \(\mathrm{Diff}_+(S^1)\) on LG is smooth. Furthermore, every PER \(\pi \) of LG is \(\mathrm{Diff}_+(S^1)^\sim \)-covariant, namely there is a projective unitary representation U of the universal covering \(\mathrm{Diff}_+(S^1)^\sim \) such that \(U(\tilde{\rho })\pi (\gamma )U(\tilde{\rho })^* = \pi (\rho . \gamma )\) [14, 15]. Consider now \({\mathcal {H}}^{0,\text {fin}}\), the algebraic direct sum of the eigenspaces of \(L_0\). On the infinitesimal level, in general the space \({\mathcal {H}}^{0, \text {fin}}\) is a direct sum of infinitely many unitary irreducible representations \(V(c,h_i) \) of the Virasoro algebra. Such a representation integrates to a unitary projective representation U of \(\mathrm{Diff}_+(S^1)^\sim \), and if the appearing highest weights \(h_i\) differ only by integers then U reduces to a unitary projective representation of \(\mathrm{Diff}_+(S^1)\) [11, 14]. Now we briefly study the irreducible unitary representations V(c, h) of the Virasoro algebra appearing from an irreducible PER of level \(\ell \) of LG. If \(\ell =0\) then \(\lambda =0\), and by \(c=h=0\) we have the trivial representation of \(\text {Vir}\). If \(\ell \ge 1\), then V(c, h) belongs to the continuous series, namely we have \(h \ge 0\) and \(c \ge 1\). The estimate on the central charge follows by the inequality \(g+1 \le \dim \mathfrak {g}\), which can be noticed by studying the following table [18]:
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 |
Lemma 2
[18] \([L_n, x(k)]=-kx(n+k)\) on \({\mathcal {H}}^{0, \text {fin}}\).
As a corollary of Lemma 2, 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 \({\mathcal {H}}^{0, \text {fin}}\) for every X in \(L^{\text {pol}}\mathfrak {g}\), where \(h.X(\theta ) = h(\theta ) \frac{d}{d \theta }X(\theta )\).
The Lie algebra \(L^{\text {pol}}\mathfrak {g}\) can be completed to a Banach Lie algebra \(L\mathfrak {g}_t \) for any \(t \ge 0\). Indeed, given \(X = \sum _k X_k(k)\) in \(L^{\text {pol}}\mathfrak {g}\), we can define \(L\mathfrak {g}_t \) as the completion of \(L^{\text {pol}}\mathfrak {g}\) with respect to the norm
We have norm continuous embeddings with dense range \( C^{\lceil t \rceil +1} (S^1, \mathfrak {g})\hookrightarrow L\mathfrak {g}_t \hookrightarrow C^{\lfloor t\rfloor } (S^1, \mathfrak {g}) \), and for any \(t \ge n\) we have \( \Vert X^{(n)}\Vert _\infty \le |X|_t \). Notice that, in general, we can similarly define the Banach Lie algebra \(L\mathfrak {g}_{s,p} \) as the completion of \(L^{\text {pol}}\mathfrak {g}\) with respect to the norm
We now set \({\mathcal {S}}_t = L{\mathbb {C}}_t\), namely the space of continuous complex functions h on \(S^1\) satisfying
Notice that we can naturally identify \({\mathcal {S}}_t \) with a space of Sobolev vector fields on the circle. We can define two continuous 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 \( (1+|n+k|)^t \le (1+|n|)^t(1+|k|)^t \) we have \(|h.X|_{s,p} \le |h|_s |X|_{s+1,p} \) and \(|hX|_{s,p} \le |h|_s |X|_{s,p} \).
We conclude this preliminary section with some standard notions about the stress energy tensor. Let \(h(\theta ) \frac{d}{d \theta }\) be a smooth vector field on the circle and define \({\mathcal {V}} = \bigcap _{k \ge 0} {\mathcal {D}} (L_0^k)\). The stress energy tensor
is a closable operator with \({\mathcal {H}}^{0,\text {fin}}\) as a dense core [5, 11, 17]. Moreover, by the energy bounds [16]
with \(k \ge 0 \) any natural number and \(\xi \) in \({\mathcal {H}}^{0,\text {fin}}\), we have that T(h) is well defined and closable for every h in \({\mathcal {S}}_{3/2}\), since
for every \(k \ge 0\) natural and \(\xi \) in \({\mathcal {D}} (L_0^{k+1})\). It follows that \({\mathcal {V}} \) is a dense invariant domain for T(h) if h is smooth. Furthermore, we have that \(T(\bar{h})^* = \overline{T(h)}\) for every h in \({\mathcal {S}}_{3/2}\) and therefore T(h) is essentially selfadjoint if h is real valued. For further properties of the stress-energy tensor, see [14].
3 Solitonic Representations from Discontinuous Loops
In this section we follow [11] and we construct proper solitonic representations of the conformal net associated to some vacuum positive energy representation of a loop group. We begin by briefly recalling some basic definitions about conformal nets. We refer to [11, 15, 29] for further treatments of the topic.
Let \({\mathcal {K}}\) be the set of open, nonempty and non dense intervals of the circle. For I in \({\mathcal {K}}\), \(I'\) denotes the interior of the complement. The Möbius group Möb acts on the circle by linear fractional transformations. A Möbius covariant net \(({\mathcal {A}}, U, \Omega )\) consists of a family \(\{{\mathcal {A}}(I) \}_{I \in {\mathcal {K}}}\) of von Neumann algebras acting on a separable Hilbert space \({\mathcal {H}}\), a strongly continuous unitary representation U of Möb and a vector \(\Omega \) in \({\mathcal {H}}\), called the vacuum vector, satisfying the following properties:
-
(i)
\({\mathcal {A}}(I_1) \subseteq {\mathcal {A}}(I_2)\) if \(I_1 \subseteq I_2\) (isotony),
-
(ii)
\({\mathcal {A}}(I_1) \subseteq {\mathcal {A}}(I_2)'\) if \(I_1 \subseteq I_2'\) (locality),
-
(iii)
\(U(g) {\mathcal {A}}(I) U(g)^* = {\mathcal {A}}(g.I)\) for every g in Möb and I in \({\mathcal {K}}\) (Möbius covariance),
-
(iv)
The representation U has positive energy, namely the generator of rotations has non-negative spectrum (positivity of the energy),
-
(v)
\(\Omega \) is cyclic for the von Neumann algebra \(\bigvee _{I \in {\mathcal {K}}} {\mathcal {A}}(I)\), and up to a scalar \(\Omega \) is the unique Möb-invariant vector of \({\mathcal {H}}\) (vacuum).
By the Howe-Moore vanishing theorem, it follows by the axioms (iv) and (v) that the vacuum vector \(\Omega \) is, up to a phase, the only vector fixed by the subgroups of rotations, translations and dilations defined below (12). With these assumptions, the following properties automatically hold [11, 15]:
-
(vi)
\({\mathcal {A}}(I') = {\mathcal {A}}(I)'\) for every I in \({\mathcal {K}}\) (Haag duality),
-
(vii)
\({\mathcal {A}}(I) \subseteq \bigvee _{\alpha } {\mathcal {A}}(I_\alpha )\) if \(I \subseteq \bigcup _\alpha I_\alpha \) (additivity),
-
(viii)
If \(I_+\) is the upper half of the circle and \(\Delta \) is the modular operator associated to \({\mathcal {A}}(I_+)\) and \(\Omega \), then for every t in \({\mathbb {R}}\) we have
$$\begin{aligned} \Delta ^{it} = U (\delta _{-2 \pi t}) \,, \end{aligned}$$(11)where \(\delta _t.u=e^t u\) is the one parameter group of dilations (Bisognano–Wichmann),
-
(ix)
Each local algebra \({\mathcal {A}}(I)\) is a type III factor and \(\bigvee _{I \in {\mathcal {I}}_{\mathbb {R}}} {\mathcal {A}}(I) = B({\mathcal {H}})\), with \({\mathcal {I}}_{\mathbb {R}}\) the set of all the open, nonempty and non dense intervals of \( S^1 \setminus \{-1 \}\) (irreducibility).
Definition 3
By a conformal net, or also a \(\mathrm{Diff}_+(S^1)\)-covariant net, we shall mean a Möb-covariant net \(({\mathcal {A}}, U, \Omega )\) which satisfies the following condition:
(x) U extends to a projective unitary representation of \(\mathrm{Diff}_+(S^1)\) on \({\mathcal {H}}\) such that
for every I in \({\mathcal {K}}\). Furthermore, if \(\text {supp}\, \rho \subset I'\), with \(\text {supp}\, \rho \) the closure of the complement of the set of the points z such that \(\rho (z)=z\), then we have
A Möbius covariant net is either also conformal or not, but if it is, the extension of the of the representation to \(\mathrm{Diff}_+(S^1)\) is unique [5, 6]. In a conformal net, the following is automatic [23]:
(xi) if \(\bar{I} \subset J\) then there is a type I factor \({\mathcal {R}}\) such that \({\mathcal {A}}(I) \subset {\mathcal {R}} \subset {\mathcal {A}}(J)\) (split property).
If the interval I does not contain \(-1\), one can pass from the circle picture to the real line picture [11, 15]. Explicitly, by the changes of variables \(z=C(u)\) with \(C(u) = (1+iu)/(1-iu)\) the Cayley transform, we can identify I with an interval of the real line. In the real line picture the point \(-1\) of \(S^1\) corresponds to \(\infty \), with \({\mathbb {R}}\cup \{ \infty \}\) the Alexandroff compactification of \({\mathbb {R}}\). With this identification, we can define the one parameters groups of rotations, dilations and translations mentioned above:
If \(\alpha \) is a Möbius transformation mapping the upper half of the circle \(I_+\) onto I, then by conjugating (12) by \(\alpha \) we can define the groups \(R_I\), \(\delta _I\) and \(\tau _I\) of rotations, dilations and translations of the interval I.
Now we give some definitions about the representation theory of conformal nets. A (locally normal) DHR (Doplicher-Haag-Roberts) representation of a conformal net \(({\mathcal {A}}, U, \Omega )\) is a family \(\rho = \{ \rho _I \}_{I \in {\mathcal {K}}}\) of normal representations \(\rho _I\) of the von Neumann algebras \({\mathcal {A}}(I)\) on some Hilbert space \({\mathcal {H}}_\rho \) such that \(\rho _I =\rho _J \vert _{{\mathcal {A}}(I)}\) if \(I \subseteq J\). We say that two DHR representations \(\rho _1\) and \(\rho _2\) are equivalent if there is some unitary operator U from \({\mathcal {H}}_{\rho _1}\) to \({\mathcal {H}}_{\rho _2}\) such that \(U \rho _{1, I}(x) = \rho _{2, I}(x)U\) for every x in \({\mathcal {A}}(I)\) and I in \({\mathcal {K}}\). The DHR representation induced by the identity is called the vacuum representation. A DHR representation \(\rho \) is said to be irreducible if \(\bigvee _{I \in {\mathcal {K}}} \rho _I({\mathcal {A}}(I)) = B({\mathcal {H}}_\rho )\). If a topological group \(\mathcal {G}\) acts continuously on \(S^1\) by elements of \(\mathrm{Diff}_+(S^1)\), a DHR representation \(\rho \) is said to be \(\mathcal {G}\)-covariant if there exists a strongly continuous unitary projective representation \(U_\rho \) of \(\mathcal {G}\) such that
for all g in \(\mathcal {G}\) and I in \({\mathcal {K}}\), where \(\iota :\mathcal {G} \rightarrow \mathrm{Diff}_+(S^1)\) is the induced homomorphism. A locally normal DHR representation is automatically Möb-covariant [10].
A similar class of representations of a conformal net is given by the so-called solitonic representations. In the following, we will denote by \({\mathcal {I}}_{\mathbb {R}}\) the set of all the open, nonempty and non dense intervals of \( S^1 \setminus \{-1 \}\). A (locally normal) soliton \(\sigma \) of a conformal net \(({\mathcal {A}}, U, \Omega )\) is a family of maps \(\sigma = \{ \sigma _I \}_{I \in {\mathcal {I}}_{\mathbb {R}}}\) where \(\sigma _I\) is a normal representation of the von Neumann algebra \({\mathcal {A}}(I)\) on a fixed Hilbert space \({\mathcal {H}}_\sigma \) such that \(\sigma _I = \sigma _J \vert _{{\mathcal {A}}(I)} \) if \(I \subseteq J\). Every DHR representation gives rise to a solitonic representation by restriction, while the converse is in general not true. If \({\mathcal {G}}\) is a topological group equipped with some homomorphism \( \iota :{\mathcal {G}} \rightarrow \mathrm{Diff}_+(S^1)\), then we will say that a soliton \(\sigma \) 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 \( \text {Ad} \, U_\sigma (g) \sigma _I(x) =\sigma _{\iota (g).I}(\text {Ad} \, U (\iota (g) )\cdot x) \) for every x in \({\mathcal {A}}(I) \). With \({\mathbb {R}}_\pm \) considered as elements of \({\mathcal {I}}_{\mathbb {R}}\), the index of a soliton \(\sigma \) is the Jones index of the inclusion \(\sigma ({\mathcal {A}}({\mathbb {R}}_+)) \subseteq \sigma ({\mathcal {A}}({\mathbb {R}}_-))'\).
Finally, we have concluded our brief overview on conformal nets. We now consider a vacuum positive energy representation \(\pi \) of level \(\ell \) of some loop group LG. We always suppose G to be simple, compact and simply connected. It can be shown that the family of von Neumann algebras
is a conformal net [15, 29], where \(\tilde{\pi }\) is the lift of \(\pi \) described below in Remark 16. We will denote by U the projective unitary continuous representation of \(\mathrm{Diff}_+(S^1)\) verifying the covariance property. Consider now a smooth path \(\gamma :[- \pi , \pi ] \rightarrow G\). We suppose \(\gamma \) to admit, at all orders, finite right derivatives in \(- \pi \) and finite left derivatives in \(\pi \). We then define \(\sigma _\gamma =\{\sigma _\gamma ^I \}_{I \in {\mathcal {I}}_{\mathbb {R}}}\) as the collection of maps given by
where \(\gamma _I\) is a loop in LG such that \(\gamma _I (\theta ) =\gamma (\theta )\) for \(\theta \) in I seen as a subinterval of \((-\pi , \pi )\).
Proposition 4
\(\sigma _\gamma \) is an irreducible locally normal soliton with index 1.
Proof
Normality on each \({\mathcal {A}}_\ell (I)\) follows because on these local algebras \(\sigma _\gamma \) is given by the adjoint action by a unitary operator. The compatibility property is clear, since if \(I \subseteq J\) then \(\pi (\gamma _I \gamma _J^{-1})\) is in \({\mathcal {A}}_\ell (I') = {\mathcal {A}}_\ell (I)'\). The index is 1 since if I is in \({\mathcal {I}}_{\mathbb {R}}\) then \(\sigma _\gamma ({\mathcal {A}}_\ell (I)) = {\mathcal {A}}_\ell (I)\), and for the same reason we have that
since the conformal net \({\mathcal {A}}_\ell \) is irreducible. \(\quad \square \)
We will now focus on a smaller class of solitons. Given h in G, we define a discontinuous loop as an element of the group
The restriction of a discontinuous loop on \([- \pi , \pi ]\) clearly induces a soliton \(\sigma _\zeta \). In the following, we will study the equivalence classes of such solitonic representations.
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 [29]). 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 \( {\mathbb {R}}\). We now denote by \(\text {Rot}^\sim \) the universal covering of \(\text {Rot} \cong {\mathbb {T}}\), the group of rotations of the circle. If \(U_t\) is the unitary representation of \(\text {Rot}\) associated to \(\pi \), then we can define \(V^\zeta _t = \pi (\zeta _t) U_t\) in \(PU({\mathcal {H}})\) for t in \({\mathbb {R}}\) and notice that \(V^\zeta _t V^\zeta _s = V^\zeta _{t+s}\). However, in general \(V^\zeta _{2 \pi }\) is not a scalar and therefore \(\sigma _\zeta \) is not \(\text {Rot}\)-covariant but only locally \(\text {Rot}^\sim \)-covariant. We can notice that if \(\zeta \) is in \(L_gG \) and \(\eta \) is in \(L_hG \) 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 both in \(L_hG\) then \(\zeta \eta ^{-1}\) is in LG and \(\sigma _\zeta \) is unitarily equivalent to \(\sigma _\eta \).
Theorem 5
Let \(\pi \) be a vacuum positive energy representation of LG of level \(\ell \ge 1\). Given \(\zeta \) in \(L_hG\), 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 \(V^\zeta _{2 \pi } = \pi (h)\). By the identity \(\pi (h) e^{\pi (X)} \pi (h)^* = e^{\pi (X)}\) for any X in \(L \mathfrak {g}_0\) we have that \(V^\zeta _{2 \pi } \) is a scalar since \(\pi \) is irreducible. This implies that \(\sigma _\zeta \) is locally \(\text {Rot}\)-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 [11]. Now we suppose h to be not central. By absurd, \(\sigma _\zeta \) extends to a DHR representation and thus it is \(\text {Rot}\)-covariant [10]. Denote by \(U^\zeta _{\theta }\) the corresponding intertwining projective representation of the circle. If we define the DHR representation
then by construction \(\rho _\zeta \) is implemented by the unitary \(U_\pi U^\zeta _{- \pi } V^{\zeta }_\pi U_{- \pi }\). Since \(\sigma _\zeta \) is a locally normal DHR representation, by using the additivity property one can show that \(\rho _\zeta (x) =x \) for x in \({\mathcal {A}}((0, \pi )) \) and for x in \({\mathcal {A}}((0, \pi ))' \). It follows that \(U^\zeta _{- \pi } V^\zeta _\pi \) is a scalar and thus \(V^\zeta _{2 \pi }\) 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 \(V^\zeta _{2 \pi } = \pi (h)\) 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
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. \(\quad \square \)
We conclude this section by studying the equivalence classes of the solitons constructed above. If z is in Z(G), then the DHR representations \(\sigma _\zeta \) with \(\zeta \) in \(L_zG\) correspond to inequivalent irreducible positive energy representations \(\zeta _* \pi \) of the same level as \(\pi \) (see Remark 16 and Theorem 3.2.3. of [29]). 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
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 \(T'=gTg^{-1}\), then what we can say is that we have the identity
that is the solitons \(\sigma _{g \zeta g^{-1}}\) and \(\sigma _\zeta \) are equivalent up to some inner automorphism.
4 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 \( L^kG = C^{k}(S^1, G) \). The exponential map \(\exp _{LG} :L\mathfrak {g}_0 \rightarrow LG \) is naturally defined by \( \exp _{LG}(X) = \exp _G \cdot X \) and is a local homeomorphism near the identity [27]. Here we define and describe some properties of Sobolev loop groups.
Let M be a Riemannian manifold. Suppose M to be isometrically embedded in \({\mathbb {R}}^\nu \) for some \(\nu >0\). Define, for \(1 \le p < \infty \) and \(0 \le s < \infty \), the fractional Sobolev space [2, 12]
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 \( \Vert f \Vert _{s,p} = \Vert \Delta ^{s/2} f \Vert _p + \Vert f \Vert _p\), where \(\Delta \ge 0\) is the closure on \(L^p(S^1, {\mathbb {R}}^\nu )\) of the laplacian seen as an operator on \(C^\infty (S^1, {\mathbb {R}}^\nu )\) [9]. We recall that the closure of an operator between linear subspaces of Banach spaces (and not only Hilbert spaces) is its smallest closed extension, and that the fractional Laplacian \(\Delta ^\alpha \) for \(0< \alpha < 1\) can be defined by the Fourier transform [12].
In the following, every compact Lie group G will be considered as a Riemannian Lie group with respect to the unique Riemannian structure extending \(-\langle {\cdot , \cdot }\rangle \), namely 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 \(\rho :G \rightarrow U(n)\) induces an isometric 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 \( \lambda \text {tr}(\rho (x)^*\rho (y)) = - \langle {x , y}\rangle \) 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 6
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 p and sp in \((1, \infty )\). Its Banach Lie algebra is \(W^{s,p}(S^1, \mathfrak {g}_0)\), the exponential map exists and it is a local homeomorphism. 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 [9]
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
This map is well defined since \(\exp _G \cdot X = e^X\) is an absolutely convergent series and it is also a local homeomorphism. 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 [2]), 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 [29], 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, where the continuity of the appearing homogeneous polynomials is guaranteed by equation (15). The theorem is proved. \(\quad \square \)
Corollary 7
Every loop \(\gamma \) in \(W^{s,p}(S^1, G)\) is a finite product of exponentials, since the exponential map is a local homeomorphism and \(W^{s,p}(S^1, G)\) is connected.
Remark 8
Theorem 6 still holds if the circle \(S^1\) is replaced with a torus \({\mathbb {T}}^m\). This follows from the fact that the mentioned density theorem [29] is verified for a generic cube \(Q^m\), that \({\mathbb {T}}^m\) can be defined as a quotient of \(Q^m\) and that the convolution with a smooth function preserves the periodicity.
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 PER of LG. The definition of Positive Energy Representation of a Sobolev loop group can be given just by replacing LG with \(W^{s,p}(S^1, G)\) in the definition given above.
Proposition 9
Let \(\iota :G \rightarrow H\) and \(\pi :G \rightarrow U\) be two homomorphisms of topological groups. We suppose H to be connected and \(\iota (G)\) to be dense in H. Suppose the existence of a neighborhood V of the identity in H and of a continuous function \(p_0 :V \rightarrow U\) such that \(\pi (g_\alpha ) \rightarrow p_0(v)\) whenever \(\iota (g_\alpha ) \rightarrow v\), with \((g_\alpha ) _{\alpha \in A}\) a net in G and v in V. Then, \(p_0\) extends to a continuous homomorphism \(p :H \rightarrow U\) such that \(\pi = p \cdot \iota \).
Proof
By the connectedness of H we have that \(H = \cup _n V^n\). We show by induction that p can be well defined on \(V^n\) for every n. We set \(p=p_0\) on V. Suppose the thesis true for \(V^n\), and consider elements w in \(V^n\) and v in V. Pick a net \((h_\beta )_{\beta \in B}\) such that \( \iota (h_\beta ) \rightarrow v\). By inductive hypothesis the limit
is well defined and does not depend on the net \((g_\alpha )_{\alpha \in A}\) such that \(\iota (g_\alpha ) \rightarrow wv\). Hence p is a well defined group homomorphism. The continuity of p follows by induction as well, and the identity \(\pi = p \cdot \iota \) is satisfied by construction. \(\quad \square \)
Proposition 10
Let \(\pi \) be a PER of a loop group LG. If X is in \(W^{s,p}(S^1, \mathfrak {g}_0)\) for \(1 \le p \le 2\) and \(s>3/2 + 1/p\), then \(\pi (X)\) is a closable operator which is essentially skew-adjoint on any core of \(L_0\).
Proof
We first notice that by the Sugawara formula we have \(L_0 \ge 0\), since we have \(L_0 = d + h_i\) for some \(h_i \ge 0\) on each irreducible summand \(\pi _i\) of \(\pi \). If \({\mathcal {H}}^{0,\text {fin}}\) is the algebraic direct sum of the eigenspaces of \(L_0\), then we will denote by \({\mathcal {H}}^{0,s}\) the completion of \({\mathcal {H}}^{0,\text {fin}}\) with respect to the Sobolev norm \(\Vert \xi \Vert _{0,s} = \Vert (1+L_0)^s \xi \Vert \). By Lemma 2 and Proposition 1.2.1. of [29], for \(\xi \) in \({\mathcal {H}}^{0,\text {fin}}\) and X in \(L^\text {pol}\mathfrak {g}\) we have
for any s in \({\mathbb {R}}\). By density one extends \(\pi \) to \((L\mathfrak {g})_{|s|+3/2}\) in such a way to still verify the same estimates for \(\xi \) in \({\mathcal {H}}^{0,s+ 1/2}\). It follows that if X is in \(L\mathfrak {g}_{3/2}\) then both \(\pi (X)\) and \( [1+L_0,\pi (X)] \) are bounded operators from \({\mathcal {H}}^{0,1/2}\) to \({\mathcal {H}}\). By the Nelson commutator theorem [28, Thm. X.36] we have that if X is in \((L\mathfrak {g}_0)_{3/2}\) then the restriction of \(\pi (X)\) on
is a closable operator on \({\mathcal {H}}\) which is essentially skew-adjoint on any core of \(L_0\) such as \({\mathcal {H}}^{0,\text {fin}}\). 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 \(X (\theta )= \sum _k X_k e^{i k \theta }\) then by the Hölder inequality
where \(A_{s,p} \) and \(B_{s,p} \) exist and are finite by construction and by Riesz-Thorin respectively. 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 skew-symmetric operator on \({\mathcal {H}}^{0,\textit{fin}}\) which is essentially skew-adjoint on any core of \(L_0\).
Propositions 9 and 10 can be used to extend a strongly continuous projective representation of LG to a strongly continuous projective representation of \(W^{s,p}(S^1, G)\). However, for convenience in the following we will focus on \(H^s(S^1, G)=W^{s,2}(S^1, G)\). We show how a different approach can improve the results of Proposition 10.
Proposition 11
If \(\pi \) is a PER of a loop group LG, the induced projective representation \(\pi \) of \(L\mathfrak {g}\) can be extended to \(H^s(S^1, \mathfrak {g})\) for \(s>3/2\), with \(\pi (X)\) closable and such that
for some \(C_s>0\). Moreover, \( \pi (X)^* = \overline{\pi (X^*)} \), and in particular \( \pi (X) \) is essentially skew-adjoint if X is skew-adjoint.
Proof
We use some techniques shown in [5]. Given \(X= \sum _n X_n(n)\) in \(L \mathfrak {g}_{1,1}\), the operator \(\pi (X)\) is well defined on \({\mathcal {H}}^{0,1/2} \) and (16) follows by the previous estimates since for \(t>1/2\) and \(s=1+t\) we have
It is also closable since \(\pi (X^*) \subseteq \pi (X)^*\). Notice also that since \({\mathcal {H}}^{0,\text {fin}}\) is a core for \((1+L_0)^{1/2}\) then \(\pi (X^*)\) is the formal adjoint of \(\pi (X)\) on the associated scale space \({\mathcal {H}}^{0,1/2}\) for any X in \(H^{3/2}(S^1, \mathfrak {g})\). Now we define on \({\mathcal {H}}^{0,1/2}\) the operator
which is well defined since \(e^{- \epsilon L_0} :{\mathcal {H}}\rightarrow {\mathcal {H}}^{0,\infty } \subseteq {\mathcal {H}}^{0,1/2}\). By \( - R_{X^*, \epsilon } \subseteq R^*_{X, \epsilon }\) we have that \(R_{X, \epsilon }\) is closable. Notice that if \(L_0 v_k = k v_k\) then
We will now show that \(\Vert R_{x(n), \epsilon } \Vert ^2 \le 2(\ell + g) | x(n) |_{1,1} \). The case \(n=0\) is trivial and we can suppose \(n<0\) as \( - R_{X^*, \epsilon } \subseteq R^*_{X, \epsilon }\). By simple analysis techniques one can prove that
for any \( \epsilon \ge 0 \) and \(k \ge 0\). Therefore if \(v=\sum _{k \ge 0} v_k\) is in \({\mathcal {H}}^{0,\text {fin}}\) then we have
It follows that \(\Vert R_{X, \epsilon } \Vert ^2 \le 2(\ell + g) | X|_{1,1} \) for every X in \(L \mathfrak {g}_{1,1}\) and that \(R_{X,\epsilon } \rightarrow 0\) strongly as \(\epsilon \rightarrow 0\). Moreover, by the identity \(R^*_{X,\epsilon } = - \overline{R_{X^*, \epsilon }} \) we have that \(R^*_{X,\epsilon } \rightarrow 0\) strongly as well. Now we arrive to the crucial point: if v is in \({\mathcal {D}}(\pi (X)^*)\) then
and this concludes the proof since \(e^{- \epsilon L_0}v \rightarrow v\) as \(\epsilon \rightarrow 0\). \(\quad \square \)
Theorem 12
Let \(\pi :LG \rightarrow PU({\mathcal {H}})\) be a positive energy representation of LG. Then \(\pi \) can be extended to a positive energy representation of \(H^{s}(S^1,G)\) for \(s>3/2\).
Proof
We consider an open neighborhood U in \(H^s(S^1, \mathfrak {g}_0)\) on which the exponential map of \(H^s(S^1, G)\) is a homeomorphism and set \(V=\exp _{H^s}(U)\). For \(\gamma = \exp _{H^s}(X)\) in V we define in \(PU({\mathcal {H}})\)
The neighborhood V verifies Proposition 9, since if \(\gamma _\alpha = \exp (X_\alpha )\) converges to \(\gamma = \exp (X)\) in V then the estimate (16) implies that \(\pi (X_\alpha )\xi \) is a Cauchy net for every \(\xi \) in \({\mathcal {H}}^{0,1/2}\). 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 [28]), thus \(\pi \) can be continuously extended. Finally, since the rotation group acts on \(H^s(S^1, G)\) by continuous operators (see Lemma A.3 of [4]) and since LG is dense in \(H^s(S^1, G)\), we have that \(\pi \) is actually a Positive Energy Representation since it is \(\text {Rot}\)-covariant. \(\quad \square \)
Proposition 13
[29] Let \(\rho _s = \exp _{\mathrm{Diff}_+(S^1)}(sh)\) be a smooth diffeomorphism of \(S^1\), with h a smooth real vector field of the circle. Set \(R_h=\{\rho _s\}_{s \in {\mathbb {R}}}\). Then the exponential map \(L \mathfrak {g}_0 \rtimes {\mathbb {R}}h \rightarrow LG \rtimes R_h\) is well defined and continuous. Moreover, if \(X_\alpha = \rho _\alpha . X = X \cdot \rho _\alpha ^{-1}\) then
Proof
To compute the exponential map, we fix \(X+ \alpha h\) in \(L\mathfrak {g}_0 \rtimes {\mathbb {R}}h\) and look for \(f :{\mathbb {R}}\rightarrow LG \rtimes R_h \) which satisfies \((X + \alpha h)f = \dot{f}\) and \(f(0)=1\). We suppose f to be of the form \(f_t = \gamma ^t \rho _{\phi (t)}\) with \(\gamma \) in LG. As a manifold, \(LG \rtimes R_h\) is the product of LG and \(R_h\), thus \(s \mapsto \exp _{LG}(sX)\rho _{s\alpha }\) is the integral curve for \(X + \alpha h\) at the identity. Therefore, with the notation \( \gamma _s (\theta ) = \gamma (\rho _s^{-1}(\theta ) ) \) we have
whence \(\phi (t) = \alpha t\), and we must solve
Now we notice that if \(\gamma _0^t\) is a solution of the equation \(\frac{d}{dt} \gamma ^t_0 = X_{- \alpha t} \gamma _0^t\) with initial condition \(\gamma _0^0=1\), then \(\gamma ^t=(\gamma _0^t)_{\alpha t}\) is the solution of (18) we were looking for. Therefore, if we embed G in a space of matrices \(M_m({\mathbb {C}})\) and we consider LG as a closed subspace of \(C^\infty (S^1, M_m({\mathbb {C}}))\), then by Theorem 1.4.1. of [29] we have
where the right side of (19) converges in each \(C^k(S^1, M_m({\mathbb {C}}))\) and hence in LG. Finally, equation (17) follows from \(\gamma ^1 = (\gamma _0^1)_\alpha \), and the continuity of \(\exp _{LG \rtimes R_h}\) follows from Theorem 1.4.1. of [29]. \(\quad \square \)
Corollary 14
The following holds in \(PU({\mathcal {H}})\):
Proof
By the Trotter product formula and Proposition 13 we have the following identities in \(PU({\mathcal {H}})\):
where we used the identities \(e^{iT(h)} = \pi ( \exp _{R_h}(h))\) and \(e^{\pi (X)} = \pi ( \exp _{LG}(X))\) which hold in \(PU({\mathcal {H}})\). \(\quad \square \)
Lemma 15
Let \(\pi :G \rightarrow PU({\mathcal {H}})\) be a strongly continuous projective representation of a topological group G. Then the map
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 \(U({\mathcal {H}})\), since the short exact sequence given by \(U({\mathcal {H}}) \rightarrow PU({\mathcal {H}})\) admits local continuous sections [1]. 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. \(\quad \square \)
Remark 16
A continuous projective representation \(\pi :G \rightarrow PU({\mathcal {H}})\) can be naturally lifted to a continuous unitary representation \(\tilde{\pi }\) of \( \widetilde{G} = \big \{(g,u) \in G \times U({\mathcal {H}}) :\pi (g) = [u] \big \} \) given by \(\tilde{\pi }(g,u)= u\).
Theorem 17
If \(\gamma \) is in \(H^{s}(S^1, G)\) and X is in \(H^{s}(S^1, \mathfrak {g}_0)\) for some \(s>3/2\), then
for some continuous real function \(c(\gamma , X)\). Moreover, if \(\gamma \) is in \(H^{1+s}(S^1, G)\) and h is a real vector field \({\mathcal {S}}_{s} \), then
for some continuous real function \(c(\gamma ,h)\).
Proof
We first prove (21) in the smooth case. We will identify \({\mathbb {R}}h\) with \(i {\mathbb {R}}h\) for formal convenience. 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 \(PU({\mathcal {H}})\):
and consequently \(\pi (\gamma ) e^{t \pi (Y)}\pi (\gamma )^* = \lambda (t) e^{ t \pi (\text {Ad}(\gamma )Y)} \) for some function \(\lambda :{\mathbb {R}}\rightarrow {\mathbb {T}}\). But \(\lambda :{\mathbb {R}}\rightarrow {\mathbb {T}}\) is a continuous homomorphism and therefore \(\lambda (t) = e^{i a t }\) 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 \(c(\gamma , ih) = c(\gamma , h) \) for simplicity. Therefore, the claimed expression follows by the Stone’s theorem and by using the product rule for the derivative on the identity \(1 = \gamma _t \cdot \gamma _t^{-1}\).
Now we prove (20) in the Sobolev case. Consider \((\gamma _\alpha , X_\alpha )\) in \(LG \times L\mathfrak {g}_0\) converging to \((\gamma , X)\) in \(H^{s}(S^1, G) \times H^{s}(S^1, \mathfrak {g}_0)\). 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 \(e^{i c(\gamma _\alpha , X_\alpha )}\) converges to \(e^{i c(\gamma , X )}\), that is \(e^{i c(\gamma , X )}\) 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 (21) in the Sobolev case. Consider \(\gamma \) in \(H^{1+s}(S^1, G)\) and h real in \({\mathcal {S}}_{s} \). Notice that \(i\pi (h)\) and \(\pi (h \dot{\gamma }\gamma ^{-1})\) are both essentially skew-adjoint. Consider now smooth approximating nets \(\gamma _\alpha \rightarrow \gamma \), \(X_\alpha \rightarrow X\) and \(h_\alpha \rightarrow h\) as before. By the previous propositions, the approximating right hand side of (21) 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
strongly for every t 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. \(\quad \square \)
Corollary 18
The scale space \({\mathcal {H}}^\alpha \subseteq {\mathcal {H}}\) is \(H^{s}(S^1, G)\)-invariant for \(\alpha \ge 0\) and \(s>5/2\). Moreover, for any integer n such that \(n \le \lfloor s-1\rfloor \), the corresponding map \(H^{s}(S^1, G) \times {\mathcal {H}}^n \rightarrow {\mathcal {H}}^n/{\mathbb {T}}\) is continuous.
Proof
Since \({\mathcal {D}}(u^*Au) = u^*{\mathcal {D}}(A)\) for every unitary u and every self-adjoint operator A, then
Since \({\mathcal {D}}((1+d)^\alpha ) = {\mathcal {H}}^\alpha \) for \(\alpha \ge 0\), the \({\mathcal {H}}^s\)-invariance follows. Now we prove the second statement, where we can suppose \(n \ge 1\). By Proposition 1.5.3. of [29] we have \(\Vert \pi (\gamma ) \xi \Vert _n \le (1+M_{n-1})^n \Vert \xi \Vert _n\), 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 [29]. \(\quad \square \)
Theorem 19
With the hypotheses of Theorem 17, we have
Proof
We follow Theorem 1.6.3. of [29], 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
If \(\gamma ^t = \exp _{LG}(tX)\), then the map \(t \mapsto c(\gamma ^t, Y) \) is differentiable at \(t=0\) since \( LG \times R_h \rightarrow LG \) is smooth. In particular, we have that
and so
By using (24) we have that \(c(\gamma ^t,h)\) is differentiable everywhere, with
or more compactly
We naturally expect the solution of the ODE to be given by the Duhamel formula
Using \(\frac{d}{dt} \text {Ad}(\gamma _t)Y = [X, \text {Ad}(\gamma _t)Y]\), it is easy to verify that (26) defines a \(C^1({\mathbb {R}}, (L\mathfrak {g}\rtimes i {\mathbb {R}}h)^*)\) solution of (25) with initial condition \(c_0=0\). The solution is unique. Finally, one can use (26) and Corollary 1.6.2. of [29] to obtain the claimed expressions in the smooth case. By the continuity of \(c(\gamma , Y)\) shown in Theorem 17 the thesis is proved.
\(\square \)
Corollary 20
By repeating the proof of Theorem 17, one can show that if \(\gamma \) is in \(H^{s}(S^1, G)\) and X is in \(H^{s}(S^1, \mathfrak {g}_0)\) for some \(s>3/2\), then
for some continuous real function \(b(\gamma , X)\). Similarly, if \(\gamma \) is in \(H^{s+1}(S^1, G)\) and h is a real vector field in \({\mathcal {S}}_{s} \), then
for some continuous real function \(b(\gamma ,h)\). In particular, by \(b(\gamma , Y ) = c(\gamma ^{-1}, Y)\) we have
5 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 [22]
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 (Theorem 5.3. of [25]). If the relative entropy is finite, then
where \((D \varphi :D \psi )_t = (D \psi :D \varphi )_t^*\) is the Connes cocycle.
We now denote by \({\mathcal {A}}_\ell = \{ {\mathcal {A}}_\ell (I) \}_{I \in {\mathcal {K}}}\) the conformal net associated to a level \(\ell \) vacuum representation \(\pi \) of some loop group LG. As before, we will denote by \({\mathcal {K}}\) the set of all the open, non empty and non dense intervals of the circle. To each interval I in \({\mathcal {K}}\) we associate the von Neumann algebra
where \(\tilde{\pi }\) is the lift of \(\pi \) described in Remark 16 and the support of a loop \(\gamma \) is defined by
If the interval I does not contain \(-1\), then it can be identified with an open interval of the real line through the Cayley transform. On the real line picture the stress energy tensor is given by \(\Theta (f) = T(C_* f)\), with \(C_*f\) the push-forward of the vector field \(f(u) \frac{d}{du}\) by the Cayley transform \(C(u) = (1+iu)/(1-iu)\) [17]. Notice that even if we can identify the function f with the vector field \(f(u) \frac{d}{du}\), the push-forward \(C_*f\) has to be intended as a pushforward of vector fields and not just a composition of functions.
We now come back to our work. We are interested in computing
where \(\omega \) is the vacuum state represented by the vacuum vector \(\Omega \) and \(\omega _\gamma = \omega \cdot \text {Ad}\, \pi (\gamma )^*\) is represented by \(\pi (\gamma )\Omega \) for some loop \(\gamma \) in LG. More in general, the same result will apply to the solitonic states given by the solitons (13) of above. We introduce the groups of Sobolev loops
By standard arguments, continuously differentiable and piecewise smooth loops are in \(H^s(S^1, G)\) for \(s < 5/2\) [11, 17], where we say that \(\gamma \) is piecewise smooth if right and left derivatives always exist and if \(\gamma \) is smooth except on a finite number of points. If there is no ambiguity, we will use a similar notation to denote the groups (32) in the real line picture. Consider now the interval \(I = (z, w)\) of \(S^1\) obtained by moving counterclockwise from z to w. We will denote by \(\gamma _I\) the map such that \(\gamma _I = \gamma \) on [z, w) and \(\gamma _I = e\) on [w, z), so that we have the identity
By Theorem 12 we have that if \(\gamma \) is a loop in B(z, w) then in \(PU({\mathcal {H}})\) we have \(\pi (\gamma ) = \pi (\gamma _{(z, w)} ) \pi (\gamma _{(w,z)}) \). In particular, in this case \(\pi (\gamma _{(z, w)}) \) is in \({\mathcal {A}}_\ell ((z,w))\) and \(\pi (\gamma _{(w,z)}) \) is in \({\mathcal {A}}_\ell ((w,z))\). We also recall that by the Bisognano-Wichmann theorem (11) we have the identity \(\log \Delta = - 2 \pi D\) with \(D=-\frac{i}{2}(L_1 - L_{-1})\), that is \(\log \Delta =- 2 \pi T(\delta )\) with \(\delta \) the vector field generating \(\delta (s).u = e^{ s}u\). Notice also that the vacuum expectation of
is given by the real constant \(b(\gamma ,\delta )\) described in Corollary 20.
Proposition 21
Let \(\gamma \) be a loop in \(H^3(S^1, G)\). Pick a non dense open interval \(I=(z,w)\) of the circle and write \(\gamma = \gamma _I \gamma _{I'} \) as in (33). Denote by \(\delta _I\) the generator of dilations of the interval I and set \(\Delta ^{it}_I = e^{-2 \pi it T (\delta _I)}\). If \(\dot{\gamma }\) vanishes on the boundary of I then the Connes cocycle \((D \omega _\gamma :D \omega )_{t}\) of \({\mathcal {A}}_\ell (I)\) is given by
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 \(H^2(S^1, \mathfrak {g}_0)\) since it vanishes with its first derivative on the boundary of I. Hence the right hand side of (35), 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)
\(\sigma ^\gamma _t(x)= u_t \sigma _t (x) u_t^*\,, \quad x \in {\mathcal {A}}_\ell (I) \,,\)
-
(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
where we used Lemma 3.(ii) of [26] and Theorem 19. 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 \( (D \omega _{\eta \gamma } :D \omega )_{t} = \pi (\eta ) (D \omega _{\gamma } :D \omega )_{t} \sigma _t(\pi (\eta )^*) \). Therefore, with the notation of Corollary 20 we have
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 [26]. \(\quad \square \)
Remark 22
If \(\gamma \) is an element of \(H^2([- \pi , \pi ], G)\), then as in the smooth case we can consider the soliton \(\sigma _\gamma \) given above by (13). In particular, Proposition 21 still holds for the solitonic states \(\omega _\gamma = \omega \cdot \sigma _\gamma ^{-1}\) with \(\gamma \) in \(H^3([-\pi , \pi ], G)\). This follows from the fact that if \(\eta \) is a loop in \(H^3(S^1, G)\) such that \(\gamma = \eta \) on I, then \(\omega _{\eta } =\omega _{\gamma }\) on \({\mathcal {A}}(I)\).
Now we arrive to the main part of this work, that is we will use the previous results to prove the QNEC (1) on loop groups models for the solitonic states \(\omega _\gamma = \omega \cdot \sigma _\gamma ^{-1}\) given by (13). In the real line picture, the path \(\gamma \) corresponds to an element of \(H^2({\mathbb {R}}, G)\).
Theorem 23
Let \(\omega _\gamma = \omega \cdot \sigma _\gamma ^{-1}\) be a solitonic state corresponding, in the real line picture, to some element \(\gamma \) of \(H^2({\mathbb {R}}, G)\). Then the relative entropy (31) is finite for every t in \({\mathbb {R}}\) and explicitly given by
Proof
As discussed in Remark 22, we can suppose \(\gamma \) to be the real line parametrization of some element of \(H^2(S^1, G)\). Since the vacuum is G-invariant, we can replace \(\gamma \) with \(\gamma g\) for any g in G, thus we can suppose \(\gamma (\infty ) =e\). By using the real line picture notation for the groups (32), 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 (37) since we can use equation (29), Proposition 21 and the continuity of \(\omega _{\gamma }((D \omega _{\gamma } :D\omega )_t)\) with respect to \(\gamma \) in \(H^2(S^1, G)\). 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 \( \text {supp} \, \eta \le t\) and such that \( \eta (t-k) = \gamma (t-k)^{-1} \) and \(\dot{(\eta \gamma )}(t-k) =0\) for some \(k>0\). This implies that
where the last relative entropy is finite by the argument used above. By similar arguments we have that
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 \(u \le 0\) and \(\gamma _-(u)=e\) for \(u \ge 0\). By Proposition 21 we have
Now we emulate some techniques used in [17] 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 \(0 \le u \le n - \frac{1}{n}\) and \(\rho (u) =f'(n)u+(f(n)-nf'(n))\) for \(u \ge n\). We also suppose \(\rho (u)/\rho '(u)\) to be uniformly bounded for \(n - \frac{1}{n} \le u \le n\). 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 once more we have
and by monotone convergence we have \(a_\gamma \ge 0\). Now we prove the other inequality. Consider a smooth path \( \zeta _n\) in G with extremes \(\zeta (0) = e\) and \(\zeta (1) = \gamma (0)\). We also suppose that \(\dot{\zeta }(0) = \dot{\zeta }(1) = 0\). We now define
By monotonicity \( S_{{\mathcal {A}}_\ell (0, + \infty )}(\omega _{\gamma } \Vert \omega ) = S_{{\mathcal {A}}_\ell (0, + \infty )}(\omega _{\gamma _n} \Vert \omega ) \le S_{{\mathcal {A}}_\ell (-1/n, + \infty )}(\omega _{\gamma _n} \Vert \omega ) \), so that after a limit we have the inequality
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
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
Therefore we can apply Lemma 1. of [7], namely for every \(t_1\) and \(t_2\) in \({\mathbb {R}}\) we have
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 \(\rho '(0) = \rho ''(0)=0\), \(\rho '(1)=1\) and \(\rho ''(1)=0\). We define
where \(\eta \) is a smooth function such that \(\gamma _n\) is in \(H^2(S^1, G)\). Therefore, by (41) we have
and thus we have
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 (42) then we have
for any \(\gamma \) in \(H^2(S^1, G)\) such that \(\dot{\gamma }(0)=0\). But this condition can be removed as in (42), and by covariance we have that the above expression of S(0) holds for a generic loop \(\gamma \) in \(H^2(S^1, G)\). \(\quad \square \)
Corollary 24
If \(E_\gamma \) is the null energy (40), then we have the Bekenstein Bound
where
Furthermore, the complement relative entropy (38) is given by
Proof
As in the previous theorem, it is not restrictive to suppose \(\gamma \) to be in \(H^2(S^1, G)\). The statement then follows by Möb-covariance, since in general we have
for every interval (a, b) of the real line, with \(D_{(a,b)}(u)\) the density of the dilation operator of (a, b). \(\quad \square \)
Now we use the previous theorem to discuss the QNEC (1). In this case we have that the second derivative \(S''(t)\) exists everywhere in the classical sense with \(S''(t) \ge 0\). If P is the generator of translations, then by the Sugawara formula we have \(P = \Theta (\frac{d}{du})\), hence the quantity \(E=E_\gamma \) given by (40) is an averaged stress energy tensor in the null direction u in the state \(\omega _\gamma \).
Theorem 25
Let \(\gamma \) be an element of \(H^2({\mathbb {R}}, G)\) as in Theorem 23. If we consider the null energy density
then the states \(\omega _\gamma \) verify the QNEC (1), with
Similarly, \( E_\gamma (t) = \bar{S}''(t)/2 \pi \ge 0 \), where \(\bar{S}(t) \) is the complement relative entropy (38) given by (45).
This theorem is the main result of this work. However, definition (47) may seem not rigorous to the reader, since we can arbitrarily add a function with null average to the integral (40). For this reason, we will now motivate our definition of stress energy tensor density. Given h in \({\mathcal {S}}_{3/2}\), the notation
is widely used. Therefore, in general we can consider two vectors \(\xi \) and \(\eta \) in \({\mathcal {V}} = \bigcap _{k \ge 0} {\mathcal {D}} (L_0^k)\), recall that
and define \((\eta | T(z)\xi )\) as the kernel of some tempered distribution. This definition is consistent by defining \((\eta | T(z) \xi )\) by using (49) if \(\xi \) and \(\eta \) are in \({\mathcal {H}}^{0, \text {fin}}\). By Corollary 20, in the real line picture in our case we have
for every h, and this motivates (47) since in general we expect the identity
to hold. In addition, we can also show a different way to recover (47) by using some results of [7].
Let \({\mathcal {N}} \subseteq {\mathcal {M}}\) be a -hsm inclusion with corresponding family of von Neumann algebras \(({\mathcal {M}}_t)_{t \in {\mathbb {R}}}\). We denote by \(P \ge 0\) the generator of translations and by \(\omega \) the faithful normal state given by the common standard vector \(\Omega \). Given two real parameters \(t < t'\), consider a normal state \(\psi \) of \({\mathcal {M}}_t\) with representing vector \(\eta \). If u is some isometry, then we will denote by \(\psi _u \) the vector state represented by \(u \eta \). We will use the notation \(P_\eta = (\eta | P \eta )\). We define
where \(C'_t\) is the family of all the isometries \(w'\) in \({\mathcal {M}}'_t\) such that \(P_{w' \eta }\) and \(S_{{\mathcal {M}}'_t}(\psi _{w'} \Vert \omega ) \) are both finite, and similarly \(C_{t'}\) is the family of all the isometries w in \({\mathcal {M}}_{t'}\) such that \(P_{w \eta }\) and \(S_{{\mathcal {M}}_{t'}}(\psi _{w} \Vert \omega ) \) are finite. Notice that \(E_\psi (t,t')\) is well defined as a state-dependent quantity, since any two vectors which represent \(\psi \) on \({\mathcal {M}}_t\) differ by an isometry of \({\mathcal {M}}'_t\).
Proposition 26
Let \({\mathcal {A}}\) be a von Neumann algebra on \({\mathcal {H}}\) and let \(U_s=e^{-isP}\) be a one parameter strongly continuous unitary group such that \(U_{-s} {\mathcal {A}} U_s \subseteq {\mathcal {A}}\) for \(s \ge 0\). If u and \(u'\) are isometries in \({\mathcal {A}}\) and \({\mathcal {A}}'\), then for every vector \(\xi \) in \({\mathcal {H}}\) we have
under the assumption that the quantities \(|P|_{uu' \xi }\), \(|P|_{u \xi }\), \(|P|_{u' \xi }\), and \(|P|_{\xi }\) are all finite.
Proof
Take \(s>0\) and consider
Note that \(({uu' \xi |U_{-s}uu' \xi }) =( {\xi |(u^*U_{-s}uU_{-s})(u')^*U_{-s}u' \xi } )\), where we used the fact that \(u^*U_{-s}uU_{s}\) belongs to \({\mathcal {A}}\) for \(s>0\). Thanks to this remark, we can write \(D= D_1 + D_2 + D_3 + D_4\), where
and so we have the estimate \(|D| \le |D_1| + |D_2| + |D_3| + |D_4|\). We can bound all of these terms as \(|D_i| \le \Vert (U_s - 1) \eta _1 \Vert \, \Vert (U_s - 1) \eta _2 \Vert \), where \(\eta _1 , \eta _2 \in \{ \xi , u \xi , u'\xi \}\). For \(\zeta \) in \(\{ \xi , u \xi , u'\xi \}\) we can use the spectral representation of P to write, for \(s>0\), the identity
By \(|1-e^{-is \lambda }| /s \le |\lambda |\) and by the finiteness of \(|P|_\zeta = (\zeta | |P| \zeta )\) we can use the dominated convergence theorem, and so we have
It follows that
and so by the estimates above we finally obtain that \(D/s \rightarrow 0\) for \(s \rightarrow 0^+\). Therefore
and the thesis follows. \(\quad \square \)
The previous proposition is an intermediate result used in the proof of Theorem 1. Finally, by using the proof of Theorem 1 and Proposition 26 we have the following fact.
Proposition 27
Given two real parameters \(t < t'\), consider a normal state \(\psi \) of \({\mathcal {M}}_t\) with representing vector \(\eta \) such that \(P_\eta < + \infty \). Consider the Connes cocycles
If the relative entropies \(S(t)=S_{{\mathcal {M}}_t} (\psi \Vert \omega )\) and \(\bar{S}(t')=S_{{\mathcal {M}}'_{t'}} (\psi \Vert \omega )\) are finite, then
In other words, what we did was just to notice by the proof of Theorem 1 that, under some finiteness assumptions, the null energies of all the representing vectors for a normal state are minimized by the Connes cocycles. Notice also that by Theorem 1 and (41) we have
Finally, we can define
After this premise, we can show that the density (47) is actually given by the limit (53). In this step we will use the results of [24], which ensures us that a PER of a loop group LG can be extended to a PER of \(H^1(S^1, G)\). In particular, this implies that Theorem 23 and Theorem 25 are still true in this generality. The same argument applies to Proposition 31 below as well. Furthermore, as shown later in Proposition 31, we can compute (50) by using (52). By doing so we have
This tells us that the null energy density (47) can be recovered by using (53).
5.1 QNEC on LSU(n)
In this section we focus on the case \(G=SU(n)\) and we use a construction illustrated in [30] 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 (48).
We begin by considering the natural action of \(G=SU(n) \) on \(V={\mathbb {C}}^n\) and we 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 \(\text {Rot} \) 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
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) [27, 30]. Notice that \({\mathcal {F}}_P(0) = \Lambda V\) is the fundamental representation of SU(n). 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 28
The restricted unitary group is the topological group
where the considered topology is the strong operator topology combined with the metric given by the distance \(d(u,v)=\Vert [u-v,P]\Vert _2\).
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 \(U \in U({\mathcal {F}}_P)\) [30].
Theorem 29
Segal’s quantization criterion. [30] 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 \(U_P(H) \rightarrow PU({\mathcal {F}}_P)\) is continuous.
Proposition 30
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 \(\gamma (z)=\sum \widehat{\gamma }_kz^k\), where \(\widehat{\gamma }_k \in M_n({\mathbb {C}})\). We consider on H the basis \(e_j^k(z)=z^ke_j\), 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
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 [4]), and by \([R_\theta ,P]=0\) we have that the projective representation of \(\text {Rot} \) is actually a strongly continuous unitary representation. Therefore, the thesis follows by the Segal quantization criterion, the complete reducibility of positive energy representations (Thm. 9.3.1. of [27]), Proposition 2.3.3. of [29] and remarks below. \(\quad \square \)
Proposition 31
Let \(\gamma \) be a loop in \(H^1(S^1, SU(n))\). Pick a non dense open interval \(I=(z,w)\) of the circle and write \(\gamma = \gamma _I \gamma _{I'} \) as in (33). Then, in \(PU({\mathcal {H}})\) we have
where \((D \omega _\gamma :D \omega )_t\) 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 (54) is well defined. With the same computations of Proposition 21 we have that \(\sigma ^\gamma _t(x) = \text {Ad}\, \pi (\gamma _I \delta _I(- 2 \pi t). \gamma _I^{-1}) \cdot \sigma _t (x) \) for x in \({\mathcal {A}}_\ell (I)\). Therefore, we have that \((D \omega _\gamma :D \omega )_{t} \) is equal to \(\pi (\gamma _I \delta _I(- 2 \pi t). \gamma _I^{-1})\) up to a unitary V in the commutant of \({\mathcal {A}}_\ell (I)\), but \((D \omega _\gamma :D \omega )_t \) and \(\pi (\gamma _I \delta _I(- 2 \pi t). \gamma _I^{-1})\) are both in \({\mathcal {A}}_\ell (I)\) and thus V is a scalar. \(\quad \square \)
Theorem 32
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 (31) is finite and given by
In particular, the QNEC (1) is satisfied as shown above in Theorem 25.
Proof
Since the vacuum is SU(n)-invariant, 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 (55). We can prove that S(t) is finite for any t real as in Theorem 23, and similarly we have that \(\bar{S}(t) = S_{{\mathcal {A}}_\ell (- \infty , t)}(\omega _\gamma \Vert \omega )\) 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 (40). Therefore we can apply Lemma 1. of [7], and equation (41) 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 of \(S'(t)\). For simplicity we focus on the case \(t=0\) and we write \(\gamma = \gamma _+ \gamma _- \) with \(\gamma _+(u)=e\) for \(u \le 0\) and \(\gamma _-(u)=e\) for \(u \ge 0\). By Proposition 31 the Connes cocycle \(u'_s = (D \omega :D \omega _\gamma )_s\) on \({\mathcal {A}}_\ell (- \infty ,0)\) is equal in \(PU({\mathcal {H}})\) to \(\pi (\gamma _- \delta (2 \pi s).\gamma _-^{-1})^*\). But also the state \(\omega _\gamma \cdot \text {Ad}(u'_s)^*\) verifies the finiteness conditions required to apply Lemma 1. and thus we have \(- S'(0) \le 2 \pi E_s \), where \(E_s =( u'_s \pi (\gamma )\Omega | P u'_s \pi (\gamma )\Omega )\) for s real. However, one can simply prove that
Therefore, by repeating the argument with any t in \({\mathbb {R}}\) we have
Finally, if we define
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 \(|t| \rightarrow + \infty \). If the support of \(\gamma \) is contained in \((k, +\infty )\) then by lower semicontinuity \(S(t) \le F(t)\) for every t real, and we can similarly deduce that \(H(t)=0\) for every t real. The QNEC inequality (48) can be proved as above, with the only exception that in this case we do not have to use [24] to compute (47). \(\quad \square \)
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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. I also thank the reviewers for the many useful corrections.
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Panebianco, L. Loop Groups and QNEC. Commun. Math. Phys. 387, 397–426 (2021). https://doi.org/10.1007/s00220-021-04170-3
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DOI: https://doi.org/10.1007/s00220-021-04170-3