aa r X i v : . [ m a t h - ph ] O c t Rotational KMS states and type I conformal nets
Roberto Longo ∗ and Yoh Tanimoto † Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”Via della Ricerca Scientifica 1, 00133 Rome, ItalyEmail: [email protected] , [email protected] Abstract
We consider KMS states on a local conformal net on S with respect to rotations.We prove that, if the conformal net is of type I, namely if it admits only type I DHRrepresentations, then the extremal KMS states are the Gibbs states in an irreduciblerepresentation. Completely rational nets, the U(1)-current net, the Virasoro nets andtheir finite tensor products are shown to be of type I. In the completely rational case,we also give a direct proof that all factorial KMS states are Gibbs states. QFT, Quantum Field Theory, was originally designed to describe finitely many quantum,relativistic particles, with particle creation/annihilation due to the interaction. In this view,statistical mechanics aspects due to an infinitely many particle distribution are absent. Thereare however extreme situations where QFT shows a thermodynamical behaviour, a mostimportant one being the black hole background Hawking radiation, that lead to considerthermal states in QFT.As is known, thermal equilibrium states at infinite volume in quantum statistical mechan-ics are characterized by the KMS condition for the dynamical flow, a one-parameter auto-morphism group α t of the observable C ∗ -algebra A . A state ϕ , i.e. a positive linear functionalon A normalized with ϕ (1) = 1, satisfies the KMS condition w.r.t. τ at inverse temperature β > x, y ∈ A , there is a function F xy analytic in the strip S β = { < Im z < β } ,bounded and continuous on the closure S β , such that F xy ( t ) = ϕ (cid:0) xα t ( y ) (cid:1) ,F xy ( t + iβ ) = ϕ (cid:0) α t ( y ) x (cid:1) , At finite volume, where the degrees of freedom are finite, KMS states are Gibbs states: ϕ ( x ) = Tr( e − βH x ) / Tr( e − βH ) with H the Hamiltonian; at infinite volume, Gibbs states might ∗ Supported in part by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures andModels”, PRIN-MIUR, GNAMPA-INdAM and Alexander von Humboldt Foundation. † Supported by the JSPS fellowship for research abroad. e − βH is not necessarily trace class, yet the KMS condition is preserved under theinfinite volume limit.From the mathematical viewpoint, KMS states are of most importance, being related tothe Tomita-Takesaki modular theory of von Neumann algebras. The KMS condition mea-sures, in a sense, the deviation of the state ϕ from the tracial property ϕ ( xy ) = ϕ ( yx ). Inview of an infinite-dimensional quantum index theorem, one expects QFT to be the under-lying framework and the role of the (super)-trace to be played by (super)-KMS states. Thedescription of the KMS states then turns out a natural problem with different motivations.This paper concerns KMS states in low dimensional CFT, Conformal Quantum FieldTheory. On one hand the mathematical structure of CFT is much better understood, withvery interesting connections with other mathematical subjects, and in particular the OperatorAlgebraic approach is powerful and deep. On the other hand CFT is of much interest inPhysics in various situations, e.g. Critical Phenomena or AdS/CFT correspondence.CFT in (1 + 1)-dimensions is an extension of the tensor product of two one-dimensional(one could say (cid:0) + (cid:1) -dimensional) CFT, so initially one has to study CFT on the real lineor on its compactification S . The real line and the circle pictures are equivalent, however,the physical Hamiltonian as QFT is the one associated to the translation flow in the real linepicture. The conformal Hamiltonian is the one associated with the rotation flow in the circlepicture and one can usually extract more easily information from the conformal Hamiltoniansince its spectrum is discrete.An analysis of the KMS states w.r.t. the translation flow has been given in [CLTW12a,CLTW12b]. The main result in [CLTW12a] is that, in the completely rational case, for everyfixed inverse temperature β >
0, there exists a unique KMS state w.r.t. translations, thegeometric KMS state. In the non-rational case, however, there might be uncountably manyKMS states. They are all described for the U (1)-current net and possibly all for the Virasoronets [CLTW12b].The purpose of this paper is to investigate the KMS states with respect to the rotationflow. In the rotational case, the first point to clarify is the choice of the C ∗ -algebra thatsupports the rotational flow and on which the KMS state is to be defined. Such a choice isnatural and well known in the translation case: the C ∗ -algebra generated by the local vonNeumann algebras associated to bounded intervals of the real line. On the other hand, theintervals of the circle do not form an inductive family and a more thoughtful construction isnecessary. A universal C ∗ -algebras was defined by Fredenhagen, and a different constructionis in [Fre90, GL92]. We shall explain in detail the construction as we need it.We shall first give a general, complete description in the completely rational case: everyextremal KMS state is a Gibbs state in some irreducible representation. We shall make useof the structure of the universal C ∗ -algebra in this case [CCHW13]; a similar descriptionfor super-KMS states in this case is due to Hillier [Hil15]. Rotational KMS states in thecompletely rational case were also studied in [Iov15].Our results are not restricted to the rational case. Indeed, we shall prove that any extremalrotational KMS state on a large class of non-rational conformal nets is a Gibbs state insome irreducible representation. The point is that, in general, the GNS representation withrespect to a KMS state might be of type II or III and could not be decomposed uniquelyinto irreducible (type I) representations. We exclude this possibility for many importantconformal nets. 2ctually, we prove that some conformal nets are of type I, namely they do not have typeII or III representations at all. Moreover, at the moment, no example of conformal net notof type I is known. It is possible that diffeomorphism covariance implies the type I property.One can understand how general the type I property is by the following. Suppose A is aconformal net such that, for any given λ >
0, there exists at most countably many irreduciblerepresentations ρ of A such that λ if the lowest eigenvalue of the conformal Hamiltonian L ρ of ρ . Then A is of type I. Many conformal nets are then immediately shown to be of type Iby this criterion. Among them are the Virasoro nets and the U (1)-current net. Their finitetensor products can be shown to be of type I by a separate argument.This paper is organized as follows. In Section 2, we recall our operator-algebraic setting forconformal field theory and introduce our main dynamical system, the universal C ∗ -algebra.The fundamental examples of KMS state, the Gibbs states, are also introduced. In Section 3,we present our classification result of KMS states. First we are concerned with the completelyrational case where the structure of the universal C ∗ -algebra is completely understood, thenwe pass to the general case. We determine that an extremal KMS state on a type I netis a Gibbs state, and prove that some well-known nets are of type I. The problem of thepossible occurrence of type II and III representations naturally arises here and we make someobservations. In Section 4, we discuss possible applications of our results. Let us recall our mathematical framework for conformal field theory on the compactifiedone-dimensional spacetime S . See also [CLTW12a].Let I be the set of open, connected, non-empty and non-dense subsets (intervals) of thecircle S . A (local) M¨obius covariant net is a triple ( A , U, Ω) where A is a map thatassigns to each I ∈ I a von Neumann algebra A ( I ) on a common Hilbert space H andsatisfies the following requirements:1. (Isotony) If I ⊂ I , then A ( I ) ⊂ A ( I ).2. (Locality) If I ∩ I = ∅ , then A ( I ) and A ( I ) commute.3. (M¨obius covariance) U is a strongly continuous unitary representation of the M¨obiusgroup M¨ob = PSL(2 , R ) on H and for any g ∈ M¨ob and any interval I ∈ I we haveAd U ( g )( A ( I )) = A ( gI ) . (Positivity of energy) The generator L of the rotation one-parameter subgroup ispositive ( U ( R t ) = e itL with R t the rotation by t ).5. (Vacuum vector) Ω is a unit vector of H , which is the unique (up to a scalar) U -invariant vector; Ω cyclic for S I ∈I A ( I ).From these assumptions, the following automatically follow, see [FJ96, Section 3]3. (Additivity) If I ⊂ S κ I κ , then A ( I ) ⊂ W κ A ( I κ ), where W κ M κ denotes the vonNeumann algebra generated by {M κ } .7. (Reeh-Schlieder property) Ω is cyclic for each local algebra A ( I ).A representation of a M¨obius covariant net A is a family ρ = { ρ I } I ∈I , where ρ I is aunital ∗ -representation of A ( I ), on a common Hilbert space H ρ such that ρ I | A ( I ) = ρ I for I ⊂ I . We say that ρ is locally normal if each ρ I is normal. We say ρ is factorial if W I ∈I ρ I ( I ) is a factor.A M¨obius covariant net ( A , U, Ω) is called a conformal net if the representation U of theM¨obius group extends to a strongly continuous projective representation of the group Diff( S )of orientation-preserving diffeomorphisms of S , that is covariant, namely Ad U ( g )( A ( I )) = A ( gI ), and Ad U ( g ) acts trivially on A ( I ) if g is acts identically on I .We say that the net A has the split property if for each pair I , I of intervals such that I ⊂ I , there is a type I factor N ( I , I ) such that A ( I ) ⊂ N ( I , I ) ⊂ A ( I ). The splitproperty follows from the conformal covariance [MTW16]. C ∗ -algebra For a given M¨obius covariant net A , Fredenhagen [Fre90] proposed to consider a C ∗ -algebrawhich is universal in the sense that any representation of the net A can be regarded as arepresentation of this algebra. This notion has been used widely in the study of superselectionsectors in conformal field theories, and we will take it as the algebra of our physical system.Yet, there seems to be a confusion in the literature about the construction. The firstpaper which introduced the universal C ∗ -algebra was [Fre90, Section 2]. We take a slightvariation of it: one considers the free ∗ -algebra A generated by {A ( I ) } , modulo the relationsdue to the inclusions A ( I ) ⊂ A ( I ), for I , I ∈ I , I ⊂ I . Clearly a representation ρ of A defines a representation of A , still denoted by ρ . For a given x ∈ A , one defines theseminorm by sup ρ ∈ Γ k ρ ( x ) k , where Γ is the class of all representations. In the Zermelo-Fraenkel set theory with the axiomof Choice (ZFC), Γ is not a set (an intuitive explanation would be the following: on eachset with cardinality larger or equal to the cardinality of the continuum, one can define astructure as a Hilbert space. Hence the class of all Hilbert spaces is “as large as” the classof all sets (with cardinality larger or equal to the cardinality of the continuum), and wouldcause Russell’s paradox. A precise reason is that the sets in ZFC are only those which areconstructed by axiom schemas). However, the above supremum can be justified in ZFC asfollows : For a given x ∈ A , we consider the following: { s ∈ R : there is a representation ρ of A such that s = k ρ ( x ) k} , which is a subset of R in ZFC by the axiom schema of separation (see standard textbooks We owe this observation to Sebastiano Carpi. The axiom schema of separation reads, for a given predicate F ( x ) with a variable x as follows:( ∃ B )( ∀ x )( x ∈ B ↔ x ∈ A & F ( x )). In words, it states that for a set A there is a subset B which con-sists of all elements of A which satisfy F .
4n axiomatic set theory, e.g. [Sup60, Jec78]). Hence one can take the supremum and the restfollows.Another commonly cited paper [GL92, Section 8] has a problem, because one has to takethe direct sum parametrized by “all the representations”, which is definitely not a set.Let us also provide a construction of the universal algebra which is closer to that of[GL92]. We consider as before the free ∗ -algebra A generated by {A ( I ) } modulo the inclusionrelations as above. We denote by ι I the embedding of A ( I ) into A . Let S be the set ofstates (positive, unital linear functionals in the sense ϕ ( x ∗ x ) ≥ x ∈ A ) ϕ on A . Bydefinition of A , the GNS representation ρ ϕ of A with respect to ϕ satisfies for I ⊂ Jρ ϕ ◦ ι J | A ( I ) = ρ ϕ ◦ ι I . Note that, for any ϕ , an element x ∈ A ( I ) is represented by a bounded operator ρ ϕ ( x ).Indeed, x ∗ x ≤ k x k in A ( I ), hence ρ ϕ ( x ∗ x ) ≤ k x k because ρ ϕ | A ( I ) is a representation ofa von Neumann algebra and a representation of a C ∗ -algebra is order preserving.Now let x be an element of A ; then x is a finite sum P k Q l x k,l of finite products ofelements of A ( I k,l ), I k,l ∈ I .Let ρ be a representation of the net A . Then ρ gives rise to a representation of A . With x = P k Q l x k,l as above, we have k ρ ( x ) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ X k Y l x k,l !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k Y l ρ ( x k,l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Y l ρ ( x k,l ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X k Y l k ρ ( x k,l ) k ≤ X k Y l k x k,l k , where k ρ ( x k,l ) k is the norm of r ( x k,l ) in A ( I k,l ). Thus k ρ ( x ) k ≤ C x < ∞ , where the constant C x does not depend on ρ .We define a seminorm on A by k x k = sup ϕ ∈S k ρ ϕ ( x ) k , which is finite since k x k ≤ C x , and we take the C ∗ -completion (modulo null elements), thatwe denote by C ∗ ( A ).Let us remark that this construction avoids the set-theoretical problem: while “the classof all representations” is too large to be a set, one can consider the set of all states, becauseit is a subset of all maps from A into C with linearity, positivity and unitarity, which canbe formulated again by the axiom schema of separation.Now, as C ∗ ( A ) is not defined through the supremum over all representations, we have tocheck the universal property. Proposition 2.1.
For each representation { ρ I } of the net A , there is a representation ρ ofthe algebra C ∗ ( A ) constructed above such that ρ I = ρ ◦ ι I .Proof. { ρ I } gives rise to a representation ρ of A . In order to prove that ρ extends to C ∗ ( A )we have to show that ρ is continuous w.r.t. the norm of C ∗ ( A ), namely k ρ ( x ) k ≤ k x k , x ∈ A . This follows because every representation of a C ∗ -algebra is direct sum of cyclicrepresentations, thus k ρ ( x ) k is the supremum of ρ ϕ ( x ) with ϕ running in a family of states.5ow we may properly call C ∗ ( A ) the universal C ∗ -algebra of the net A . By the veryuniversal property, it is unique up to an isomorphism.Actually, we are mostly interested in locally normal representations, hence it is naturalto take account of locally normal representations only. This has been done by [CCHW13]:we take our C ∗ ( A ) and consider the locally normal universal representation ρ ln , which is thedirect sum of all GNS representations over all states ϕ on C ∗ ( A ) such that ρ ϕ is locally normal.The universal property can be again proven by decomposing an arbitrary representation intocyclic representations. We take the quotient C ∗ ln ( A ) := C ∗ ( A ) / ker ρ ln and call it the locallynormal universal C ∗ -algebra of the net A . The properties of C ∗ ln ( A ) claimed in [CCHW13]can be restored without any modification, since the actual construction is not needed in theproofs but only the universality is used.If the net A is conformal, any locally normal representation ρ is covariant with respectto the universal cover g M¨ob of the M¨obius group and one can take the unique implementingoperators from ρ ( C ∗ ln ( A )), and indeed they are finite products of local elements [DFK04,Theorem 6]. From this it follows that the action of g M¨ob on C ∗ ln ( A ) is inner. Proposition 2.2.
Let A be a M¨obius covariant net with the split property and ρ be a locallynormal representation of C ∗ ln ( A ) with a cyclic vector Φ . Then the representation space H ρ isseparable.Proof. As in [KLM01, Appendix C], we consider the set I Q of intervals with rational endpoints, an intermediate type I factor N ( I , I ) between A ( I ) ⊂ A ( I ) , I , I ∈ I Q , I ⊂ I (we just choose one N ( I , I ) for each pair I ⊂ I , not necessarily the canonical choice of[DL84]), let K ( I , I ) be the algebra of compact operators in N ( I , I ) (under the identifica-tion N ( I , I ) ∼ = B ( H )) and denote by A the C ∗ -algebra generated by {K ( I , I ) } . Note that A is a separable C ∗ -algebra.As ρ is locally normal, for each I we have ρ ( A ( I )) ⊂ ρ ( A ) ′′ . Indeed, if I ⊂ I ⊂ I , I , I ∈I Q , then K ( I , I ) ⊂ A ( I ) and as I tends to I , any element in A ( I ) can be approximatedfrom A in the σ -weak topology. Then the claim follows from the local normality of ρ , and italso follows that ρ ( C ∗ ln ( A )) ⊂ ρ ( A ) ′′ .Now, by assumption there is a cyclic vector Φ for ρ ( C ∗ ln ( A )), hence it is also cyclic for ρ ( A ) ′′ . As ρ ( A ) is a C ∗ -algebra, ρ ( A ) ′′ is the closure of ρ ( A ) in the strong operator topologyand we have ρ ( A )Φ = ρ ( A ) ′′ Φ = H ρ . As ρ ( A ) is separable, H ρ is also separable. Remark . The converse of Prop. 2.2 is also true. If A is a M¨obius covariant net and ρ arepresentation of C ∗ ln ( A ) with separable H ρ , then ρ is locally normal. Indeed the A ( I )’s aretype III factors, and every representation of a σ -finite type III factor on a separable Hilbertspace is normal [Tak02, Theorem V.5.1], while the local algebras A ( I ) are automatically σ -finite by the Reeh-Schlieder property: the vacuum state is faithful [Tak02, PropositionII.3.9]. Let A be a C ∗ -algebra and α a one-parameter automorphism group of A (not necessarilypointwise norm-continuous). 6 KMS state of A w.r.t. α at inverse temperature β ∈ R + is a state ϕ on A such thatfor any pair of elements x, y ∈ A there is a bounded analytic function F xy on R + i (0 , β ),which is continuous on R + i [0 , β ], such that F xy ( t ) = ϕ ( α t ( x ) y ) , F xy ( t + iβ ) = ϕ ( yα t ( x )) . Given a conformal net A , we are interested in states on the universal C ∗ -algebra C ∗ ( A )w.r.t. the rotation one-parameter automorphism group. Any state ψ on C ∗ ( A ) gives riseto a GNS representation ρ ψ of C ∗ ( A ), whose restriction to {A ( I ) } (i.e. { ρ ψ ι I } I ∈I ) is arepresentation of the net. We say that ψ is locally normal if its restriction to each localalgebra A ( I ) is normal. We do not know whether this implies in general that the GNSrepresentation ρ ψ is locally normal. Yet, for KMS states, we have the following Lemmas.The proof of the first one is essentially the same as that of [TW73, Theorem 1], one shouldonly note that the funnel structure is not necessary. Lemma 2.4.
Let A be a C ∗ -algebra which contains a σ -finite properly infinite von Neumannalgebra M , and ϕ a state on A such that the GNS vector Φ (for the GNS representation ρ ϕ with respect to ϕ ) is separating for ρ ϕ ( A ) ′′ . Then ϕ | M is normal and ρ ϕ | M is normal.Proof. As Φ is separating for ρ ϕ ( A ) ′′ , ρ ϕ ( A ) ′′ is σ -finite, hence ρ ϕ ( M ) ′′ is σ -finite as well. Thenthe restriction ρ ϕ to a properly infinite algebra M is normal [Tak02, Theorem V.5.1]. Lemma 2.5.
Let ϕ be a KMS state on C ∗ ( A ) with respect to the rotation flow α . Then ϕ islocally normal and its GNS representation ρ ϕ is locally normal.Proof. The local algebras in the vacuum representation have a separating vector Ω, hencethey are σ -finite, and are known to be of type III [GL96, Proposition 1.2]. Now the claimfollows from Lemma 2.4 and the fact that the GNS vector Φ is separating for ρ ϕ ( A ) ′′ for anyKMS state ϕ (see [BR97, Lemma 5.3.8 and Corollary 5.3.9]. The pointwise norm-continuityassumption of α is not necessary for this result).Thanks to these Lemmas, we do not have to distinguish C ∗ ( A ) and C ∗ ln ( A ) as long as weare interested in KMS states. Remark . In the real line case, the GNS representation of every locally normal state (i.e.normal on each local algebra) is locally normal. To see this, let ϕ be a locally normal stateof the quasi-local C ∗ -algebra A ≡ S I ⋐ R A ( I ) k·k with GNS triple ( H , ρ, Φ) and fix an interval I ∈ I . The restriction of ρ I to H I ≡ ρ ( A ( I ))Φ is normal as it is the GNS representation of anormal state. For any larger interval ˜ I ⊃ I we have ρ I = ρ ˜ I (cid:12)(cid:12) A ( I ) , so ρ I is normal on H ˜ I too.Since the H ˜ I ’s form an inductive family whose union is dense in H by the cyclicity of Φ, itfollows that ρ I is normal on H .Let ρ be a locally normal, rotation-covariant, irreducible representation of A in which e − βL ρ is trace class for β >
0, where L ρ is the generator of the one-parameter unitary groupof rotations. This is a typical situation that holds true in most important cases. Then onecan define the following Gibbs state on C ∗ ( A ): ϕ ρ,β ( x ) = Tr (cid:0) e − βL ρ ρ ( x ) (cid:1) Tr( e − βL ρ ) . (1)7 ρ,β is a (locally normal) rotational β -KMS state of C ∗ ( A ). Thus we have a natural classof KMS states. The relation between the KMS condition and the Gibbs states at the giventemperature can be found in [Haa96]. An early consideration of rotational Gibbs states canbe found in [Sch94].For some important class of nets, the structure of the irreducible representations is wellunderstood. This is the case, in particular, for the class of completely rational nets, whichwe will consider in Section 3.1. In such cases, we shall see that all extremal KMS states areGibbs states as in (1). Our main question is whether this is always true. We show this to betrue under a mild condition in Section 3.2.3, but the question remains open in general. The stress energy density in a Gibbs state can be computed through the character formula.For a test function f with support in an interval I ∈ I , the stress energy tensor T in thevacuum representation, smoothed with f , is an unbounded operator T ( f ) affiliated to A ( I ).If ρ is an irreducible representation of A , we may define T ρ ( f ) = ρ ( T ( f )), making use thatbounded functions, e.g. the resolvent, of T ( f ) belong to A ( I ). The expectation value of thestress-energy tensor in the Gibbs state is then ϕ ρ,β ( T ( f )) = Tr( e − βL ρ T ρ ( f ))Tr( e − βL ρ ) . This is indeed finite if, for example, there is ǫ > e − ( β − ǫ ) L ρ ) is finite because thepolynomial energy bound holds for the Virasoro algebra [CW05, Lemma 4.1] (which impliesthat e − ǫL ρ T ( f ) is bounded). This condition is quite generic.Furthermore, in such a case, one can compute this value by expanding T ρ ( f ) = P f n L ρn ,where the f n = π R π − π f ( e it ) e − int dt are the Fourier modes of f . As L ρn , n = 0 changes theenergy eigenvalues while Tr can be computed by expanding along a basis of L ρ eigenvectors,all the contributions from L ρn , n = 0 drop out and we have ϕ ρ,β ( T ( f )) = f Tr( e − βL ρ L ρ )Tr( e − βL ρ ) = f − dχ ρ ( e − s ) /ds | s = β χ ρ ( e − β ) = − π Z π − π f ( e it ) dt · dχ ρ ( e − s ) /dsχ ρ ( e − s ) (cid:12)(cid:12)(cid:12) s = β , where χ ρ ( q ) = Tr q L ρ is known as the character of the representation ρ . Thus ϕ ρ,β ( T (1)) = − π dχ ρ ( e − s ) /dsχ ρ ( e − s ) (cid:12)(cid:12)(cid:12) s = β = q π dχ ρ ( q ) /dqχ ρ ( q ) (cid:12)(cid:12)(cid:12) q = e − β = q π d log( χ ρ ( q )) dq (cid:12)(cid:12)(cid:12) q = e − β . The characters for some specific examples can be found in the literature, e.g. [KR87].
In this section we determine all KMS states in the completely rational case.8et A be a M¨obius covariant net on S . Following [KLM01], one defines the µ -index µ A of A as the Jones index of the 4-interval inclusion: µ A ≡ (cid:2) ( A ( I ) ∨ A ( I )) ′ : A ( I ) ∨ A ( I ) (cid:3) , where I k ∈ I , k = 1 ...
4, are disjoint intervals in S whose union is dense in S and I k , I k +2 , k = 1 ,
2, have no common boundary point. A is said to be completely rational if µ A < ∞ and A satisfies the split property and thestrong additivity property [KLM01]. The split property follows from the trace class propertyof e − βL , for all β > µ -index [LX04]. Thus, for a local conformal net A , theonly condition for A to be completely rational is µ A < ∞ .If A is completely rational, then µ A = X k d ( ρ k ) where { ρ k } is a complete family of irreducible inequivalent representations of A and d ( ρ k ) isthe dimension of ρ k . It follows that A has only finitely many irreducible representations, allof them have finite index and every representation is a direct sum of irreducible finite indexrepresentations [KLM01].As shown in [CCHW13], the locally normal universal C ∗ -algebra C ∗ ln ( A ) takes a particu-larly simple form in the completely rational case. Theorem 3.1. [CCHW13] If A is a completely rational net, then C ∗ ln ( A ) is isomorphic to afinite direct sum of type I factors: C ∗ ln ( A ) = F ⊕ F ⊕ · · · ⊕ F n , with F k = B ( H k ) , where H k , k = 0 , , . . . n corresponds to inequivalent irreducible represen-tations of the net A . In particular, C ∗ ln ( A ) is a von Neumann algebra and its center is finitedimensional. The minimal central projections e k of C ∗ ln ( A ) are thus in one-to-one correspondence withthe irreducible representations ρ k of C ∗ ln ( A ): ρ k ( x ) = xe k , x ∈ C ∗ ln ( A ) , (2)say with ρ the vacuum representation.Recall that, as a completely rational net, it admits only finitely many irreducible represen-tations (up to equivalence), so any representation is M¨obius covariant (see [GL92, Corollary7.2], and the modification to the circle is straightforward). With U the unitary representa-tion of M¨ob associated with the net A , the adjoint action of U on the net A gives, by theuniversal property of C ∗ ln ( A ), an automorphism group of C ∗ ln ( A ) that acts trivially on thecenter. In view of Theorem 3.1, U = U ⊕ · · · ⊕ U n , U k the covariance unitary representation of g M¨ob in the representation ρ k .Let now ϕ be an extremal β -KMS state of C ∗ ln ( A ) w.r.t. the rotation one-parameter group α t . As the GNS representation of ϕ acts on a separable Hilbert space by Proposition 2.2and Lemma 2.5, it must either be faithful on or annihilate the components B ( H k ). Beingextremal, the support of ϕ is e k for some k , namely ϕ ( e j ) = δ jk . Thus ϕ can be viewed as anormal state on B ( H k ) and we have: Theorem 3.2.
Let A be a completely rational net as above, and ϕ an extremal, rotational β -KMS state. Then there exists an irreducible representation ρ of A such that ϕ ( x ) = Tr (cid:0) e − βL ρ ρ ( x ) (cid:1) Tr( e − βL ρ ) , x ∈ C ∗ ln ( A ) , In particular e − βL ρ is trace class.Proof. By the above discussion, ρ is equal to a ρ k given in (2), so the proof follows by thefollowing lemma, which is essentially known. Lemma 3.3.
Let R = B ( H ) be a type I factor, ϑ a one-parameter automorphism group, and ϕ a normal β -KMS state of R w.r.t. ϑ . Then there exists a positive, non-singular, selfadjointoperator H on H (thus affiliated to R ) such that ϕ ( x ) = Tr (cid:0) e − βH x (cid:1) Tr( e − βH ) , x ∈ R . We have
Tr( e − βH ) < ∞ and ϑ t ( x ) = Ad e itH ( x ) , x ∈ R , t ∈ R .Proof. Since R is a factor, ϕ is faithful due to the KMS property: this follows from thefaithfulness of the GNS representation and the fact that the GNS vector is separating for aKMS state [BR97, Lemma 5.3.8 and Corollary 5.3.9]. As R is a type I factor, there existsa positive, non-singular trace class operator T with trace one [BR97, Proposition 2.4.3] suchthat ϕ ( x ) = Tr( T x ). We may write T = e − βH with a self-adjoint operator H and, as T isbounded, the spectrum of H is bounded below. By adding a scalar, we may assume that H ispositive, but then the trace Tr( e − βH ) is no longer 1, so we have the formula ϕ ( x ) = Tr( e − βH x )Tr( e − βH ) .Then t Ad e − iβtH is the modular group of ϕ [BR97, Example 2.5.16]. Therefore, wehave Ad e itH = ϑ t as there is a unique one-parameter automorphism group which satisfiesthe KMS condition with respect to the state ϕ [Tak03a, Theorem VIII.1.2]. Let A be a M¨obius covariant net. We say that a (locally normal) representation ρ of A is of type I if ρ ( C ∗ ( A )) ′′ is a type I von Neumann algebra. Let A be a M¨obius covariant net with the split property and ϕ a β -KMSstate on C ∗ ( A ) with respect to the rotation flow α . Then ϕ can be decomposed a.e. uniquelyas follows: ϕ = Z ⊕ X dµ ( λ ) ϕ λ , here the GNS representation ρ ϕ λ with respect to ϕ λ is factorial. If ρ ϕ λ is type I, then ϕ λ ( x ) = Tr (cid:0) e − βL λ ρ ϕ λ ( x ) (cid:1) Tr( e − βL λ ) , where L λ is the conformal Hamiltonian in the representation ρ ϕ λ .Proof. By Lemma 2.5, the GNS representation ρ ϕ is locally normal, and by Proposition 2.2 ρ ϕ acts on a separable Hilbert space. By considering the central disintegration of ρ ϕ ( C ∗ ( A )) ′′ ,we also obtain the disintegration of the representation of ρ | A , with A any separable, suitablychosen C ∗ -subalgebra of C ∗ ( A ), by a similar argument as [KLM01, Proposition 56] (see also[Dix77, Theorem 8.4.2], [Tak02, Theorem IV.8.21 and Section V.1]): ρ ϕ | A = Z X dµ ( λ ) ρ ϕ λ | A and ρ ϕ λ are locally normal for almost all λ . According to this disintegration, the GNS vectorΦ ϕ disintegrates Φ ϕ = Z X dµ ( λ ) Φ ϕ λ and the state h Φ , · Φ i on ρ ϕ ( C ∗ ( A )) ′′ gets the disintegration [Tak02, Proposition IV.8.34]: ϕ ( x ) = h Φ ϕ , ρ ϕ ( x )Φ ϕ i = Z ⊕ X dµ ( λ ) h Φ ϕ λ , ρ ϕ ( x ) λ Φ ϕ λ i . Hence we can define ϕ λ ( x ) = h Φ ϕ λ , ρ ϕ ( x ) λ Φ ϕ λ i first for x ∈ A and then extend it to C ∗ ( A )by local normality, which is the first statement. ϕ λ are again KMS states with respect torotations for almost all λ , by considering the disintegration of the modular operator.If ρ ϕ λ is of type I, then it follows that the state ϕ λ is given by the Gibbs state by Lemma3.3. If we assume conformal covariance, type III representations do not occur since the rotationsare inner. Furthermore, for type II states on a conformal net, a Gibbs-like formula is validby replacing Tr by the unique tracial weight τ , c.f. Lemma 3.3. Lemma 3.5. If A is conformal, then for any KMS states ϕ , ρ ϕ ( C ∗ ( A )) ′′ contains no typeIII component.Proof. As we saw in Section 2.2, g M¨ob, especially the rotations, is inner. Thus, the modularautomorphisms of ρ ϕ ( C ∗ ( A )) ′′ with respect to ϕ are inner, hence the ρ ϕ ( C ∗ ( A )) ′′ cannothave a type III component (see [Tak02, Theorem IV.8.21, Section V.1], [Tak03a, TheoremVIII.3.14]).Let A be a conformal net and ρ a representation of A on H ρ . As we recalled in Section 2.2,by conformal covariance, there is a canonical inner implementation U ρ on H ρ with U ρ ( g ) ∈ ρ ( C ∗ ( A )) ′′ of g M¨ob. The generator of the associated unitary rotation one-parameter subgroup11f U ρ is positive [Wei06], which we denote by L ρ and we call it the conformal Hamiltonianof ρ . Of course, in case ρ is irreducible, this gives the usual definition of the conformalHamiltonian. Lemma 3.6.
Let A be a conformal net and ϕ an extremal, rotational β -KMS state. Supposethe GNS representation ρ ϕ of ϕ is of type II, namely ρ ϕ ( C ∗ ( A )) ′′ is a type II factor . Let τ denote the semi-finite trace of ρ ϕ ( C ∗ ( A )) ′′ . Then there is a positive self-adjoint operator L ρ affiliated to ρ ϕ ( C ∗ ( A )) ′′ as above and we obtain ϕ ( x ) = τ (cid:0) e − βL ρ ρ ( x ) (cid:1) τ ( e − βL ρ ) , x ∈ C ∗ ( A ) , In particular τ ( e − βL ρλ ) < ∞ .Proof. Set
M ≡ ρ ϕ ( C ∗ ( A )) ′′ . By the KMS property, the GNS vector ξ ϕ is cyclic and sepa-rating for M and Ad e − iβtL ρλ is the modular group of M w.r.t. to the state ¯ ϕ ≡ h ξ ϕ , · ξ ϕ i on M . By the Radon-Nikodym theorem, ¯ ϕ = τ ( h · ) with h a positive operator on H ρ affiliatedto M , and τ ( h ) = 1. The modular group of ¯ ϕ is then equal to Ad h it . Then h is proportionalto e − βL ρλ , thus h = e − βL ρλ /τ ( e − βL ρλ ) and the Lemma follows. We say that a M¨obius covariant net is of type I if it admits only locally normal representa-tions ρ such that ρ ( C ∗ ( A )) ′′ is of type I.Some important conformal nets turn out to be type I, therefore, any extremal KMS stateis the Gibbs state in one of the irreducible representations. Theorem 3.7.
If a conformal net A is of type I, then any rotational β -KMS state ϕ is aconvex combination (integration) of the Gibbs states in irreducible representations.Proof. Immediate from Lemma 3.3 and Proposition 3.4 (note that the split property followsfrom conformal covariance [MTW16]).As we recalled in Section 2.2, for a conformal net the representatives of g M¨ob are innerand unique, hence any locally normal representation ρ of the net (or the universal algebra C ∗ ( A )) is g M¨ob-covariant. The implementation is unique if we assume that the representativesbelong to ρ ( C ∗ ( A )) ′′ . With this unique inner implementation, the lowest eigenvalue l of thegenerator L ρ of rotations is non-negative [Wei06, Theorem 3.8]. Proposition 3.8.
Let A be a conformal net and assume that there are only countably manyequivalence classes of locally normal irreducible representations with a specified lowest eigen-value of the generator of rotations. Then A is of type I.Proof. By local normality and its disintegration restricted to A as in Proposition 3.4, it isenough to treat factorial representations. Let us consider a locally normal factorial repre-sentation ρ of A . The implementation of the 2 π -rotation commutes with any local element, In this case, it would be necessarily type II ∞ as the local algebras are of type III. ρ ( C ∗ ( A )) ′′ , on the other hand, it belongs to ρ ( C ∗ ( A )) ′′ by our choice that it isinner. When ρ ( C ∗ ( A )) ′′ is a factor, the implementation is then a scalar. This applies to anyinteger-multiple of 2 π , hence the spectrum of L ρ must be included in N + l , where l ≥ N is the set of non-negative integers.Now, we consider any disintegration of ρ | A into irreducible representations where A isthe separable C ∗ -subalgebra of C ∗ ( A ) as in Proposition 3.4 (this is possible, by choosinga maximally abelian algebra in ρ ( C ∗ ( A )) ′ because we have the split property: see [KLM01,Proposition 56] for disintegration and [MTW16] for the implication of the split property fromconformal covariance): ρ A = Z ⊕ X ρ λ dµ ( λ ) , where X is a certain index set. Let us assume, by contradiction, that ρ ( C ∗ ( A )) ′′ is a factorof not type I. Then by [KLM01, Proposition 57, Corollary 58], for a fixed λ , ρ λ is locallynormal, hence extends to C ∗ ( A ) and must be inequivalent to ρ λ ′ for almost all λ ′ , and thereare uncountably many such λ ′ ’s. But on the other hand, the inner implementation of g M¨obalso disintegrates and the lowest eigenvalue of L remains in N + l for each λ . By assumption,there are only countably many such inequivalent representations, which contradicts the aboveuncountable family of representations. This concludes the proof that ρ is type I.We have two basic examples with this property. Example . The U(1)-current net A U(1) : In two-dimensional spacetime, the naively definedmassless free field is plagued by the infrared problem. Yet it is possible to consider itsderivative. Its chiral components are called the U(1)-current. See [BMT88, Lon08] for itsoperator-algebraic formulation.The algebra is generated by the Fourier modes { J n } of the current which satisfy thefollowing relations [ J m , J n ] = mδ m + n, . This algebra has a distinguished representation withthe vacuum vector Ω such that J m Ω = 0 for m ≥ J ∗ m = J − m . For a smooth function f on S , one defines the Weyl operator W ( f ) by W ( f ) = exp (cid:16) i P m ˆ f m J m (cid:17) , where ˆ f m are theFourier components of f ( z ) = P m ˆ f m e imz .One defines the net by A U(1) ( I ) = { W ( f ) : supp f ⊂ I } ′′ . It turns out that this net isconformally covariant. The generator of rotations is given by the Sugawara formula L = 12 J + X m> J − m J m . For each q ≥
0, there are irreducible representations of the net A U(1) given by the state Ω q such that J m Ω = 0 for m > J Ω q = q Ω [BMT88].It can be proved that they are indeed all irreducible locally normal representations[CW16]. By their local energy bounds, { J m } can be also defined in any locally normalrepresentation. In each ρ of these representations, L ρ is again given by the above Sugawaraformula and the lowest eigenvalue is q . Namely, only two values q and − q share the samelowest energy. By Proposition 3.8 and Theorem 3.7, all KMS states with respect to rotationsare a direct integral of Gibbs states.We also note that the regular KMS states (namely, those in whose GNS representationthe generators { J m } can be defined) have been classified by [BMT88].13 xample . Virasoro nets Vir c : the net generated by the conformal covariance itself iscalled the Virasoro net. More precisely, one considers the group Diff( S ) and its projectiveunitary representations. There is a natural action of rotations, and if this action is alsoimplemented by unitary operators and the generator is positive, then we call such a projectiverepresentation of Diff( S ) a positive-energy representation. Such positive-energy irreduciblerepresentations have been classified by the so-called central charge c > h ≥ c and h are: c = 1 − m ( m +1) and h = (( m +1) r − ms ) − m ( m +1) , where m = 2 , , , · · · and r = 1 , , , · · · , m − s = 1 , , , · · · , r , or c ≥ h ≥ π c with h = 0, one can construct thecorresponding Virasoro net by Vir c ( I ) = { π c ( g ) : supp g ⊂ I } ′′ and it constitutes a conformalnet (see [Car04]). If c <
1, Vir c is completely rational [KL04].Let us consider c ≥
1. To any irreducible (hence type I) locally normal irreduciblerepresentation of Vir c there corresponds a positive-energy representation of Diff( S ) with c ≥ h ≥ c ≥ , h ≥ π ch of Vir c [BSM90][Car04,Section 2.4][Wei16].Therefore, our Proposition 3.8 applies to any value of c ≥ c whose GNS representation is factorial is the Gibbs state corresponding to thevalue h , and all such h ≥ A , U, Ω), one can naturally consider the tensor product(
A ⊗ A , U ⊗ U, Ω ⊗ Ω). Any finite tensor product of these nets has again the same property.Indeed we have the following.
Proposition 3.11.
A M¨obius covariant net with the split property A is of type I if and onlyif any factorial locally normal representation of A ⊗ A is of the form ρ ⊗ ρ .Proof. Suppose that A has only locally normal type I representations. Take a locally normalfactorial representation ˜ ρ of A ⊗ A . We show that the center Z (cid:0)W I ∈I ˜ ρ ( A ( I ) ⊗ C ) (cid:1) istrivial. Indeed, on one hand we have W I ∈I ˜ ρ ( A ( I ) ⊗ C ) ⊂ ˜ ρ ( C ∗ ( A ⊗ A )) ′′ . On the otherhand, let us take p ∈ Z _ I ∈I ˜ ρ ( A ( I ) ⊗ C ) ! = _ I ∈I ˜ ρ ( A ( I ) ⊗ C ) ! ∩ _ I ∈I ˜ ρ ( A ( I ) ⊗ C ) ! ′ . By additivity of the net and local normality of ˜ ρ , we have p ∈ W I ∈I , | I | < π ˜ ρ ( A ( I ) ⊗ C ). Anyelement in the latter algebra commutes with ˜ ρ ( C ⊗ A ( I κ )), where | I κ | < π , because for anypair of two intervals I , I shorter than π , one can find an interval which contains both, andit follows that the images ˜ ρ ( A ( I ) ⊗ C ) and ˜ ρ ( C ⊗ A ( I )) commute. Again by additivity, p commutes with ˜ ρ ( C ⊗ A ( I )) for any I and, therefore, p ∈ ˜ ρ ( C ∗ ( A ⊗ A )) ′ . Namely, p ∈ Z (cid:0)W I ∈I ˜ ρ ( A ( I ) ⊗ C ) (cid:1) ⊂ Z ( ˜ ρ ( C ∗ ( A ⊗ A )) ′′ ) = C as ˜ ρ is factorial. This implies thatthe restriction of ˜ ρ to A ⊗ C is already factorial, and by assumption, it is of type I, namely Actually, the split property is not necessary for the “only if” part. B ( H ) ⊗ C , where H ˜ ρ = H ⊗ H . As the image W I ∈I ˜ ρ ( C ⊗ A ( I ))commutes with W I ∈I ˜ ρ ( A ( I ) ⊗ C ) = B ( H ) ⊗ C by the same argument as above, we have W I ∈I ˜ ρ ( C ⊗ A ( I )) ⊂ C ⊗ B ( H ). In other words, ˜ ρ is a product representation of the form ρ ⊗ ρ .To show the converse under the split property, we take a non-type I factorial representation ρ of A and construct ¯ ρ ( x ) = J ρ ρ ( J xJ ) J ρ , where J ρ is the modular conjugation of ρ ( A ) ′′ withrespect to a certain faithful normal weight and J is an antilinear conjugation which mapslocal algebras to local algebras, for example, the modular conjugation of an interval withrespect to the vacuum state. We define ˜ ρ ( x ⊗ y ) = ρ ( x ) ¯ ρ ( y ). We first show that this isa locally normal representation. By the split property, A ( I ) is included in a type I factor N ( I , I ), where I ⊂ I . As ρ is locally normal, the image ˜ ρ ( N ( I , I ) ⊗ C ) = ρ ( N ( I , I ))is again a type I factor. The image ˜ ρ ( C ⊗ A ( I )) commutes with this, therefore, ˜ ρ is locally atensor product representation, and therefore, locally normal. The consistency condition for˜ ρ is obvious, hence it is a locally normal representation of A ⊗ A . Yet its image is B ( H ρ ),and its restriction is not of type I, thus ˜ ρ cannot be a product representation. As we saw, important classes of conformal nets are of type I. It is an open problem whetherthere exists a M¨obius covariant net not of type I. The situation is quite different from thecase of nets on the real line, where any translation KMS state on the quasilocal algebra isof type III [CLTW12a] or of nets on the Minkowski space where one can have any type ofrepresentation [DS82, DS83, BD84, DFG84].One concrete open case is the cyclic orbifold [LX04]. Take a conformal net A , make thetensor product A ⊗ A and consider the fixed point net (
A ⊗ A ) flip with respect to the flipbetween two components. If A is completely rational, then ( A ⊗ A ) flip is again completelyrational and all the sectors can be explicitly written in terms of sectors of A and twistedsectors. On the other hand, if A is not completely rational, then we do not have a completeclassification of sectors of ( A ⊗ A ) flip . In particular, we are not able to exclude the possibilityof non-type I representations, although all known sectors are of type I.Another candidate for a net with non-type I representations would be an infinite tensorproduct. Recall that (see e.g. [CW05, Section 6]) for a given countable family of M¨obiuscovariant nets { ( A k , U k , Ω k ) } , one can define a M¨obius covariant net by A ( I ) := O A k ( I ) , U ( g ) := O U k ( g ) , with respect to the reference vector Ω = N Ω k (see e.g. [Tak03b, Section XIV.1]). Let usassume that each A k admits a representation ρ k which is converging to the vacuum represen-tation in some sense. Then one may hope that the infinite tensor product of representations N ρ k could make sense. Even if each ρ k is of type I, the resulting product could be of nontype I. However, this discussion depends on the nature of the sequence ρ k and a detailedanalysis is needed.We note that the type I property for rotational β -KMS states can be characterized by acompactness criterion similar to the Haag-Swieca compactness condition (and the Buchholz-Wichmann nuclearity condition, see [Haa96]).15 roposition 3.12. Let A be a local conformal net and ϕ a rotational, factorial β -KMS stateof C ∗ ( A ) . Then ρ ≡ ρ ϕ is of type I if and only if the closure of e − β L ρ ρ ϕ ( C ∗ ( A ) )Ψ is compactin the norm topology of H for some, hence for every, non-zero vector Ψ of the GNS Hilbertspace H of ϕ . Here the suffix 1 denotes the unit ball.In this case e − sL ρ ρ ϕ ( C ∗ ( A ) )Ψ is compact for every s > , Ψ ∈ H .Proof. Let M be the weak closure of ρ ( C ∗ ( A )). By assumption M is a factor. Moreover, M Ψ = C ∗ ( A ) Ψ by Kaplansky density theorem. Note that e − sL ρ ∈ M for s >
0. Let T ( s )Ψ : M → H be the map x e − sL ρ x Ψ. Clearly T ( s )Ψ is compact if and only if e − sL ρ M Ψis compact. Now if e − sL ρ M Ψ compact, then e − sL ρ M Ψ ′ is compact for any other vector Ψ ′ in the linear span of { xx ′ Ψ : x ∈ M , x ′ ∈ M ′ } , which is a dense subspace of H as M is afactor. Since k T Ψ − T Ψ k = k T Ψ − Ψ k ≤ k e − sL ρ k k Ψ − Ψ k ≤ k Ψ − Ψ k ( k e − sL ρ k ≤ L ρ is positive), T ( s )Ψ ′ is then compact for all Ψ ′ ∈ H .Assume first that M is of type I. As M is in the standard form, we may identify M ≃B ( K ) ⊗ C where H = K ⊗ K , and further H with the Hilbert space HS( K ) of the Hilbert-Schmidt operators, so every vector Ψ ∈ H with a Hilbert-Schmidt operator S . In thisidentification, an element x ∈ M acts by left multiplication on HS( K ). Thus e − sL ρ M Ψ isidentified with e − sL ρ B ( K ) S , whose closure is compact because e − βL ρ is of trace class (Lemma3.3), hence e − sL ρ is compact for any s > s = β , thus x ∈ B ( K ) e − β L ρ xS iscompact (c.f. [BDL90]).On the other hand, assume now that T ( β )Ψ is compact for some non-zero Ψ, then by theabove argument T ( β )Φ ′ is compact for any vector Φ ′ . As the rotation one-parameter groupis inner, the modular operator ∆ of M w.r.t. Ψ is given by ∆ = e − βL ρ J e βL ρ J with J themodular conjugation of ( M , Φ). Thus the map x ∈ M 7→ e − β L ρ J e β L ρ J x
Φ = T ( β )Φ ′ ( x ),with Φ ′ = J e β L ρ J Φ is compact (Φ belongs to the domain of
J e sL ρ J if s < β ). As this isthe modular nuclearity map x ∈ M 7→ ∆ x Φ ∈ H , M is of type I by [BDL90, Corollary2.9]. Although the conformal Hamiltonian is not the physical Hamiltonian, namely it does notimplement the time translation QFT flow, there is some physical interest in consideringrotational KMS states in CFT.One example comes from the three-dimensional quantum gravity. If the cosmologicalconstant is assumed to be negative, one should then look at the solutions of the Einsteinequation which are asymptotically close to the AdS spacetime. Different solutions havedifferent boundary data and such solutions (with certain fall-off conditions) have been clas-sified in [GL14]. Two copies of the Virasoro group make the transformations between thesesolutions. Such an action of the Virasoro group is called a coadjoint action [Wit88]. Maloneyand Witten [MW10] tried to compute the partition function of the AdS gravity, but theyarrived at an expression which cannot be interpreted as a trace over a Hilbert space of theexponential of a self-adjoint operator. It has been proposed to study each orbit of the Vi-rasoro group first, e.g. [GL14]. In particular, one can consider the so-called BTZ black hole16olutions [BnTZ92]. In a hypothetical quantum theory, the Virasoro group should appear asa symmetry of the theory, while the black hole should be in a thermal state. Furthermore,the energy, hence the mass, of the black hole corresponds to the conformal Hamiltonian (see[GL14, eq. (50)]). In this way, KMS states on the Virasoro nets with respect to rotationsshould appear naturally. From our results, one can conclude that all such KMS states canbe represented on the direct sum or integral of the Verma module.Besides, it is an interesting question to make sense of quantum entropy of such black holestates from the operator-algebraic point of view. Acknowledgement
We thank Sebastiano Carpi for his remark on the axiomatic set theory, Alan Garbarz andMauricio Leston for inspiring discussions on three-dimensional quantum gravity and Mih´alyWeiner for informing us of the classification of irreducible sectors of the U(1)-current net.R.L. thanks the organisers of the “von Neumann algebras” program at the HausdorffInstitute in Bonn for the hospitality extended to him during May and July 2016, when partof this paper has been written.
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