Thermal States in Conformal QFT. II
aa r X i v : . [ m a t h - ph ] A ug Thermal States in Conformal QFT. II
Paolo Camassa, Roberto Longo, Yoh Tanimoto, Mih´aly Weiner ∗ Universit`a di Roma “Tor Vergata”, Dipartimento di MatematicaVia della Ricerca Scientifica, 1 - 00133 Roma, Italy
Abstract
We continue the analysis of the set of locally normal KMS states w.r.t. thetranslation group for a local conformal net A of von Neumann algebras on R . In thefirst part we have proved the uniqueness of KMS state on every completely rationalnet. In this second part, we exhibit several (non-rational) conformal nets whichadmit continuously many primary KMS states. We give a complete classificationof the KMS states on the U (1)-current net and on the Virasoro net Vir with thecentral charge c = 1, whilst for the Virasoro net Vir c with c > A ⊂ B and A is the fixedpoint of B w.r.t. a compact gauge group, then any locally normal, primary KMSstate on A extends to a locally normal, primary state on B , KMS w.r.t. a perturbedtranslation. Concerning the non-local case, we show that the free Fermi model admitsa unique KMS state. Dedicated to Rudolf Haag on the occasion of his 90th birthday
Research supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Con-formal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry”MRTN-CT-2006-0031962.Email: [email protected], [email protected], [email protected], [email protected] ∗ Permanent address: Budapest University of Technology and Economics Department of Analyses, Pf.91, 1521 Budapest, Hungary Introduction
We continue here our study of the thermal state structure in Conformal Quantum FieldTheory, namely we study the set of locally normal KMS states on a local conformal netof von Neumann algebras on the real line with respect to the translation automorphismgroup.As is known, local conformal nets may be divided in two classes [27] that reflect thesector (equivalence class of representations on the circle) structure. For a local conformalnet A , to be completely rational (this condition is characterized intrinsically by the finite-ness of the global index [21]) is equivalent to the requirement that A has only finitely manyinequivalent irreducible sectors and all of them have finite index. If A is not completelyrational then either A has uncountably many inequivalent irreducible sectors or A has atleast one irreducible sector with infinite index.Faithful KMS states of A w.r.t. translations are locally normal on the real line (assum-ing the general split property) and are associated with locally normal GNS representationsof the restriction of the net A to the real line. One may wonder whether the structure ofthese representations, i.e. of the KMS states, also strikingly depends on the rational/non-rational alternative.In the first part of our work [7] we have indeed shown the general result that, if A isa completely rational local conformal net, then there exists only one locally normal KMSstate with respect to translations on A at any fixed inverse temperature β >
0. Thisstate is the geometric KMS state ϕ geo which is canonically constructed for any (rational ornon-rational) local conformal (diffeomorphism covariant) net.In this paper we examine the situation when A is not completely rational. In contrast tothe completely rational case, we shall see that there are non-rational nets with continuouslymany KMS states.We shall focus our attention on two important models. The first one is the free field,i.e. the net generated by the U (1)-current. In this model we manage to classify all KMSstates. We shall show that the primary (locally normal) KMS states of the U (1)-currentnet are in one-to-one correspondence with real numbers q ∈ R ; as we shall see, each state ϕ q is uniquely and explicitly determined by its value on the current. The geometric KMSstate is ϕ geo = ϕ and any other primary KMS state is obtained by composition of thegeometric one with the automorphisms γ q of the net (see Section 4.2): ϕ q = ϕ geo ◦ γ q . The second model we study is the Virasoro net Vir c , the net generated by the stress-energy tensor with a given central charge c . This net is fundamental and is containedin any local conformal net [20]. If c is in the discrete series, thus c <
1, the net Vir c iscompletely rational, so there exists a unique KMS state by the first part of our work [7].In the case c = 1 we are able to classify all the KMS states. The primary (locallynormal) KMS states of the Vir net w.r.t. translations are in one-to-one correspondencewith positive real numbers | q | ∈ R + ; each state ϕ | q | is uniquely determined by its value onthe stress-energy tensor T : ϕ | q | ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx. q = 0, because it is the restriction of the geometricKMS state on the U (1)-current net, and the corresponding value of the ‘energy density’ π β + q is the lowest in the set of the KMS states. We construct these KMS states bycomposing the geometric state with automorphisms on the larger U (1)-current net.We mention that, as a tool here, we adapt the Araki-Haag-Kastler-Takesaki theoremto locally normal systems with the help of split property. We show that, if we have aninclusion of split nets with a conditional expectation, then any extremal invariant stateon the smaller net extends to the larger net. The original theorem will be discussed indetail, since we need an extension of a KMS state on the fixed point subnet to the wholenet. Furthermore, we warn the reader that the original proof of the theorem appears tobe incomplete (see Appendix A), yet we are able to give a complete proof for the case ofsplit nets (Corollary 3.11), which suffices for our purpose.Then we consider the case c >
1. In this case we produce a continuous family which isprobably exhaustive. While we leave open the problem of the completeness of this family,we mention that the formulae on polynomials of fields should be useful. There is a setof primary (locally normal) KMS states of the Vir c net with c > | q | ∈ R + ; each state ϕ | q | can beevaluated on the stress-energy tensor ϕ | q | ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx and the geometric KMS state corresponds to q = β q π ( c − and energy density πc β . It iseven possible to evaluate ϕ | q | on polynomials of the stress-energy tensor and these valuesare already determined by the value above on T ( f ), hence by the number | q | . This shouldgive an important information for the complete classification.We shall also consider a non-local rational model. We will see that there is only oneKMS state at each temperature in the free Fermi model. This model contains the Virasoronet Vir c with c = , which is completely rational [20]. Then by a direct application of theresults in Part I, we obtain the existence and the uniqueness of KMS state in this case.We end this introduction by pointing out that our results are relevant for the construc-tion of Boundary Quantum Field Theory nets on the interior of the Lorentz hyperboloid.As shown in particular in [26], one gets such a net from any translation KMS state on aconformal net on the real line, so our results directly apply. Here we collect basic notions and technical devices regarding nets of observables and ther-mal states. Although our main result in this paper is the classification of KMS states oncertain conformal nets on S , we need to adapt standard results on C ∗ -dynamical systemsto our locally normal systems. Since these materials can be stated for more general netsof von Neumann algebras, we first formulate the problems without referring to the circle.3 .1 Net of von Neumann algebras on a directed set Let I be a directed set. We always assume that there is a countable subset { I i } i ∈ N ⊂ I with I i ≺ I i +1 of indices such that for any index I there is some i such that I ≺ I i . A net (of von Neumann algebras) A on I assigns a von Neumann algebra A ( I ) to eachelement I of I and satisfies the following conditions: • (Isotony) If I ≺ J then A ( I ) ⊂ A ( J ). • (Covariance) There is a strongly-continuous unitary representation U of R and anorder-preserving action of R on I such that U ( t ) A ( I ) U ( t ) ∗ = A ( t · I ) , and for any index I and for any compact set C ⋐ R , there is another index I C suchthat t · I ≺ I C for t ∈ C .Since the net A is directed, it is natural to consider the norm-closed union of { A ( I ) } I ∈ I .We simply denote A = [ I ∈ I A ( I ) k·k and call it the quasilocal algebra . Each algebra A ( I ) is referred to as a local algebra . Ifeach local algebra is a factor, then we call A a net of factors . The adjoint action Ad U ( t )naturally extends to an automorphism of the quasilocal algebra A . We denote by τ t thisaction of R and call it translation (note that in this article τ t is a one-parameter familyof automorphisms, although in Part I [7, Section 2.3], where we assumed diffeomorphismcovariance, we denoted it by Ad U ( τ t ) to unify the notation).An automorphism of the net A (not just of A ) is a family { γ I } of automorphisms oflocal algebras { A ( I ) } such that if I ≺ J then γ J | A ( I ) = γ I . Such an automorphism extendsby norm continuity to an automorphism of the quasilocal algebra A which preserves all thelocal algebras. Conversely, any automorphism of A which preserves each local algebra canbe described as an automorphism of the net A .A net A is said to be asymptotically γ -abelian if there is an automorphism γ of thequasilocal C ∗ -algebra A implemented by a unitary operator U ( γ ) such that • γ is normal on each local algebra A ( I ) and maps it into another local algebra A ( γ · I ),where we consider that the automorphism acts also on the set I of indices by a littleabuse of notation. • for any pair of indices I, J there is a sufficiently large n such that A ( I ) and A ( γ n · J ) = U ( γ ) n A ( J )( U ( γ ) ∗ ) n commute, • γ and τ t commute. 4t is also possible (and in many cases more natural) to consider a one-parameter group { γ s } of automorphisms for the notion of asymptotic γ -abelianness (and weakly γ -clustering,see below). In that case, we assume that { γ s } is implemented by a strongly-continuousfamily { U ( γ s ) } and the corresponding conditions above can be naturally translated.We say that a net A is split if, for the countable set { I i } in the definition of the net,there are type I factors { F i } such that A ( I i ) ⊂ F i ⊂ A ( I i +1 ). Note that in this case theargument in the appendix of [21] applies. The definition of nets looks quite general, but we have principally two types of examplesin mind.The first comes from the nets on the circle S which we have studied in Part I. For thereaders’ convenience, we recall the axioms. A conformal net A on S is a map from thefamily of intervals I of S to the family of von Neumann algebras on H such that:(1) Isotony. If I ⊂ I , then A ( I ) ⊂ A ( I ).(2) Locality. If I ∩ I = ∅ , then [ A ( I ) , A ( I )] = 0.(3) M¨obius covariance.
There exists a strongly continuous unitary representation U of the M¨obius group PSL(2 , R ) such that for any interval I it holds that U ( g ) A ( I ) U ( g ) ∗ = A ( gI ) , for g ∈ PSL(2 , R ) . (4) Positivity of energy.
The generator of the one-parameter subgroup of rotations inthe representation U is positive.(5) Existence of vacuum.
There is a unique (up to a phase) unit vector Ω in H whichis invariant under the action of U , and cyclic for W I ∈ I A ( I ).(6) Conformal covariance.
The representation U extends to a projective unitary rep-resentation of Diff( S ) such that for any interval I and x ∈ A ( I ) it holds that U ( g ) A ( I ) U ( g ) ∗ = A ( gI ) , for g ∈ Diff( S ) ,U ( g ) xU ( g ) ∗ = x, if supp( g ) ⊂ I ′ . Strictly speaking, a net is a pair ( A , U ) of a family of von Neumann algebras A and agroup representation U , yet for simplicity we denote it simply by A .We identify S and the one-point compactification R ∪ {∞} by the Cayley transform.If A is a conformal net on S , we consider the restriction A | R with the family of all finiteintervals in R as the index set. The translations in the present setting are the ordinarytranslations. If we take a finite translation as γ , this system is asymptotically γ -abelian.To consider split property, we can take the sequence of intervals I n = ( − n, n ). It is known516] that each local algebra of a conformal net is a (type III ) factor . This property isexploited when we extend a KMS state on a smaller net to a larger net.The second type is a net of observables on Minkowski space R d (see [17] for a generalaccount). In this case the index set is the family of bounded open sets in R d . The group oftranslations in some fixed timelike direction plays the role of ”translations”, while a fixedspacelike translation plays the role of γ . The net satisfies asymptotic γ -abelianness.In both cases, it is natural to consider the continuous group γ s of (space-)translationsfor the notion of γ -abelianness. For a C ∗ -algebra A and a one-parameter automorphism group { τ t } , it is possible to considerKMS states on A with respect to τ . Since our local algebras are von Neumann algebras, itis natural to consider locally normal objects. Let ϕ be a state on the quasilocal algebra A .It is said to be locally normal if each restriction of ϕ to a local algebra A ( I ) is normal.A β -KMS state ϕ on A with respect to τ is a state with the following properties: for any x, y ∈ A , there is an analytic function f in the interior of D β := { ≤ ℑ z ≤ β } where ℑ means the imaginary part, continuous on D β , such that f ( t ) = ϕ ( xτ t ( y )) , f ( t + iβ ) = ϕ ( τ t ( y ) x ) . (1)The parameter β is called the temperature. In Part I we considered only the case β = 1since our main subject were the conformal nets, in which case the phase structure isuniform with respect to β . Furthermore, we studied completely rational models and provedthat they admit only one KMS state at each temperature. Also in this Part II the mainexamples are conformal, but these models admit continuously many different KMS statesand it should be useful to give concrete formulae which involve also the temperature.A KMS state ϕ is said to be primary if the GNS representation of A with respect to ϕ is factorial, i.e., π ϕ ( A ) ′′ is a factor. Any KMS states can be decomposed into primarystates [38, Theorem 4.5] in many practical situation, for example if the net is split or ifeach local algebra is a factor. Hence, to classify KMS states of a given system, it is enoughto consider the primary ones.If the net A comes from a conformal net on S , namely if we assume the diffeomorphismcovariance, we saw in Part I that there is at least one KMS state, the geometric state ϕ geo [7, Section 2.8]. It is easy to obtain a formula for ϕ geo with general temperature β .We exhibit it for later use: let ω := h Ω , · Ω i be the vacuum state, then ϕ geo := ω ◦ Exp β ,where, for any I ⋐ R , Exp β | A ( I ) = Ad U ( g β,I ) | A ( I ) and g β,I is a diffeomorphism of R withcompact support such that for t ∈ I it holds that g β,I ( t ) = e πtβ .If ϕ is γ -invariant (invariant under an automorphism γ or a one-parameter group { γ s } )and cannot be written as a linear combination of different locally normal γ -invariant states,then it is said to be extremal γ -invariant .We denote the GNS representation of A with respect to ϕ by π ϕ , the Hilbert spaceby H ϕ and the vector which implements the state ϕ by Ω ϕ . If ϕ is invariant under the6ction of an automorphism τ t (respectively γ , γ s ), we denote by U ϕ ( t ) (resp. U ϕ ( γ ), U ϕ ( γ s ))the canonical unitary operator which implements τ t (resp. γ , γ s ) and leaves Ω ϕ invariant.If ϕ is locally normal, the GNS representation π ϕ is locally normal as well, namely therestriction of π ϕ to each A ( I ) is normal. Indeed, let us denote the restriction ϕ i := ϕ | A ( I i ) .The representation π ϕ i is normal on A ( I i ). The Hilbert space is the increasing union of H ϕ i and the restriction of π ϕ to A ( I i ) on H ϕ j ( i ≤ j ) is π ϕ j , hence is normal. Then π ϕ | A ( I i ) is normal.Furthermore, the map t U ϕ ( t ) is weakly (and hence strongly) continuous, since theone-parameter automorphism τ t is weakly (or even *-strongly) continuous and U ϕ ( t ) isdefined as the closure of the map π ϕ ( x )Ω ϕ π ϕ ( τ t ( x ))Ω ϕ . Thus the weak continuity of t U ϕ ( t ) follows from the local normality of π ϕ and bound-edness of U ϕ ( t ), which follows from the invariance of ϕ . By the same reasoning, if there isa one-parameter family γ s , the GNS implementation U ϕ ( γ s ) is weakly continuous.If for any locally normal γ -invariant state ϕ the algebra E π ϕ ( A ) E is abelian, where E is the projection onto the space of U ϕ ( γ )-invariant (resp. { U ϕ ( γ s ) } ) vectors, then thenet A is said to be γ -abelian .A locally normal state ϕ on A is said to be weakly γ -clustering if it is γ -invariantand lim N →∞ N N X n =1 ϕ ( γ n ( x ) y ) = ϕ ( x ) ϕ ( y ) . for any pair of x, y ∈ A . For a one parameter group { γ s } , we define γ -clustering bylim N →∞ N Z N ϕ ( γ s ( x ) y ) ds = ϕ ( x ) ϕ ( y ) . At the end of this subsection, we remark that, in our principal examples coming fromconformal nets on S , KMS states are automatically locally normal by the following generalresult [38, Theorem 1]. Theorem 2.1 (Takesaki-Winnink) . Let A be a net such that A ( I i ) are σ -finite properlyinfinite von Neumann algebras. Then any KMS-state on A is locally normal. If A is a conformal net on S defined on a separable Hilbert space, then each localalgebra A ( I ) is a type III factor, in particular it is properly infinite, and obviously σ -finite,hence Theorem 2.1 applies. Let A and B be two nets with the same index set I acting on the same Hilbert space. Iffor each index I it holds A ( I ) ⊂ B ( I ), then we say that A is a subnet of B and writesimply A ⊂ B . We always assume that each inclusion of algebras has a normal conditionalexpectation E I : B ( I ) → A ( I ) such that 7 (Compatibility) For I ≺ J it holds that E J | B ( I ) = E I . • (Covariance) τ t ◦ E I = E t · I ◦ τ t , andSee [25] for a general theory on nets with a conditional expectation.Principal examples come again from nets of observables on S . As remarked in PartI [7, Section 2.2], if we have an inclusion of nets on S there is always a compatible andcovariant family of expectations.Another case has a direct relation with one of our main results. Let A be a net on I and assume that there is a family of *-strongly continuous actions α I,g of a compact Liegroup G on A ( I ) such that if I ⊂ J then α J,g | A ( I ) = α I,g and τ t ◦ α I,g = α t · I,g ◦ τ t . By thefirst condition (compatibility of α ) we can extend α to an automorphism of the quasilocal C ∗ -algebra A , and by the second condition (covariance of α ) α and τ commute. Then foreach index I we can consider the fixed point subalgebra A ( I ) G =: A G ( I ). Then A G isagain a net on I . Furthermore, since the group is compact, there is a unique normalizedinvariant mean dg on G . Then it is easy to see that the map E ( x ) := R G α g ( x ) dg is alocally normal conditional expectation A → A G . The group G is referred to as the gaugegroup of the inclusion A G ⊂ A .The *-strong continuity of the group action is valid, for example, when the groupaction is implemented by weakly (hence strongly) continuous unitary representation of G .In fact, if g n → g , then U g n → U g strongly, hence α g n ( x ) = Ad U g n ( x ) → Ad U g ( x ) and α g n ( x ∗ ) = Ad U g n ( x ∗ ) → Ad U g ( x ∗ ) strongly since { U g n } is bounded. This is the case, asare our principal examples, when the net is defined in the vacuum representation (see [7,Section 2.1]) and the vacuum state is invariant under the action of G .If the net A is asymptotically γ -abelian, then we always assume that γ commutes with α g . C ∗ -dynamical systems A pair of a C ∗ -algebra A and a pointwise norm-continuous one-parameter automorphismgroup α t is called a C ∗ -dynamical system . The requirement of pointwise norm-continuityis strong enough to allow extensive general results. Although our main objects are not C ∗ -dynamical systems, we recall here a standard result.All notions defined for nets, namely compact (gauge) group action, asymptotic γ -abelianness, γ -abelianness, weakly γ -clustering of states and inclusion of systems, andcorresponding results in Section 3, except Corollary 3.6, have variations for C ∗ -dynamicalsystem ([19], see also [3]). Among them, important is the theorem of Araki-Haag-Kastler-Takesaki [1]: for a C ∗ -dynamical system with the fixed point subalgebra with respect to agauge group, any KMS state on the smaller algebra extends to a KMS state with respectto a slightly different one-parameter group. In fact, to obtain the full extension, it is nec-essary to assume that the state ϕ is faithful and that there is the net structure. A detaileddiscussion is collected in Appendix A. 8 .5 Regularization To classify the KMS states on Vir , we need to extend a KMS state on Vir to A SU (2) (explained below). Since Vir is the fixed point subnet of A SU (2) with respect to the actionof SU (2) [34], one would like to apply Theorem A.1. The trouble is, however, that thetheorem applies only to C ∗ -dynamical systems where the actions of the translation groupand the gauge group are pointwise continuous in norm. The pointwise norm-continuityseems essential in the proof and it is not straightforward to modify it for locally normalsystems; we instead aim to reduce our cases to C ∗ -dynamical systems.More precisely, we assume that the net A has a locally *-strongly continuous action τ of translations (covariance, in subsection 2.1.1) and α of a gauge group G (subsection2.3), has an automorphism γ (subsection 2.1.1) and they commute, then we construct a C ∗ -dynamical system ( A r , τ ) with the regular subalgebra A r *-strongly dense in A . Proposition 2.2.
For any net A with locally *-strongly continuous action τ × α of R × G and an automorphism γ commuting with τ × α , the set A r of elements of the quasilocalalgebra A on which τ × α act pointwise continuously in norm is a ( τ × α, γ )-globally invariant*-strongly dense C ∗ -subalgebra. Any local element x ∈ A ( I ) can be approximated *-stronglyby a bounded sequence from A r ∩ A ( I C ) for some I C ≻ I .If we consider a continuous action γ s , then we can take A r such that A r is { γ s } -invariantand the action of γ is pointwise continuous in norm.Proof. Let A r be the set of elements of A on which R × G acts pointwise continuously innorm; A r is clearly a ∗ -algebra and is norm-closed, hence is a C ∗ -subalgebra of A . Globalinvariance follows since τ , γ and α commute.Let x be an element of some local algebra A ( I ). We consider the smearing of x with asmooth function f on R × G with compact support x f := Z R × G f ( t, g ) α g ( τ t ( x )) dtdg. By the definition of net and the compactness of the support of f , the integrand belongs toanother local algebra A ( I C ) and the actions α and τ are normal on A ( I C ), hence the weakintegral can be defined. Smoothness of the actions on x f is easily seen from the smoothnessof f , thus x f ∈ A r .Take a sequence of functions approximating the Dirac distribution, i.e. a sequenceof f n with R R × G f n ( x, g ) dxdg = 1 and whose supports shrink to the unit element in thegroup R × G , then x f n converges *-strongly to x , since group actions α and τ are *-strongly continuous by assumption. Thus, any element x in a local algebra A ( I ) can beapproximated by a bounded sequence of smeared elements in a slightly larger local algebra A ( I C ). As any element in A can be approximated in norm (and a fortiori *-strongly) bylocal elements, A r is *-strongly dense in A .Moreover, as the actions are norm continuous on A r , if x ∈ A r then x f n converges innorm to x . This means that the norm closure of the linear space generated by the smearedelements { x f } is an algebra and coincides with A r .9or a continuous action γ s , it is enough to consider a smearing on R × G × R withrespect to the action of τ × α × γ . Remark . If A is the fixed point subnet of B in the sense of Section 2.3 ( A = B G ), then A r = B r G . Indeed, from A r ⊂ B r ⊂ B it follows that A r ⊂ B r G ⊂ B G = A , since theelements of A are G -invariant; on the other side, from B r G ⊂ A it follows that B r G ⊂ A r ,since the elements of B r G are regular. Thus we obtain an inclusion of C ∗ -dynamical systems A r ⊂ B r . Lemma 2.4.
If a state ϕ on the net A is weakly γ -clustering, then the restriction of ϕ tothe regular system ( A r , τ ) is again weakly γ -clustering.Proof. The definition of weakly γ -clustering of a smaller algebra A r refers only to elementsin A r , hence it is weaker than the counterpart for A . Lemma 2.5.
Let ϕ be a locally normal state on A which is a KMS state on A r . Then ϕ is a KMS state on A .Proof. We only have to confirm the KMS condition for A . Let x, y ∈ A and take boundedsequences { x n } , { y n } from A r which approximate x, y *-strongly. Since ϕ is a KMS stateon A r , there is an analytic function f n such that f n ( t ) = ϕ ( x n τ t ( y n )) ,f n ( t + i ) = ϕ ( τ t ( y n ) x n ) . In terms of GNS representation with respect to ϕ , these functions can be written as ϕ ( x n τ t ( y n )) = h π ϕ ( x ∗ n )Ω ϕ , U ϕ ( t ) π ϕ ( y n )Ω ϕ i ,ϕ ( τ t ( y n ) x n ) = h U ϕ ( t ) π ϕ ( y ∗ n )Ω ϕ , π ξ ( x n )Ω ϕ i . Note that π ϕ ( x n ) (respectively π ϕ ( y n )) is *-strongly convergent to π ϕ ( x ) (resp. π ϕ ( y )) sincethe sequence { x n } (resp. { y n } ) is bounded. Let us denote a common bound of norms by M . We can estimate the difference as follows: | ϕ ( xτ t ( y )) − ϕ ( x n τ t ( y n )) | = |h π ϕ ( x ∗ )Ω ϕ , U ϕ ( t ) π ϕ ( y )Ω ϕ i − h π ϕ ( x ∗ n )Ω ϕ , U ϕ ( t ) π ϕ ( y n )Ω ϕ i|≤ M k π ϕ ( x ∗ ) − π ϕ ( x ∗ n )Ω ϕ k + M k π ϕ ( y ) − π ϕ ( y n )Ω ϕ k and this converges to 0 uniformly with respect to t . Analogously we see that ϕ ( τ t ( y n ) x n )converges to ϕ ( τ t ( y n ) x n ) uniformly. Then by the three-line theorem (which can be appliedbecause f n are bounded: see [3, Prop. 5.3.7]) f n ( z ) is uniformly convergent on the strip0 ≤ ℑ z ≤ f is an analytic function. Obviously f connects ϕ ( xτ t ( y )) and ϕ ( t t ( y ) x ), hence ϕ satisfies the KMS condition for A . Lemma 2.6.
Let ϕ be a locally normal state on A which is a KMS state on A r . If eachlocal algebra A ( I ) is a factor, then ϕ is faithful on A .Proof. By Lemma 2.5, ϕ is a KMS state on A . The GNS representation π ϕ is locallynormal and hence locally faithful since each local algebra is a factor, then it is faithfulalso on the norm closure A . On the other hand, [3, Corollary 5.3.9] (which applies alsoto locally normal systems) tells us that the GNS vector Ω ϕ is separating for π ϕ ( A ) ′′ , thus ϕ ( · ) = h Ω ϕ , π ϕ ( · )Ω ϕ i is faithful. 10 Extension results
In this section we provide variations of standard results on C ∗ -dynamical systems. Parts ofthe proofs of Lemma 3.7 and Proposition 3.8 are adaptations of [19] for the locally normalcase, as we shall see. In particular, when we consider the one-parameter group { γ s } , weneed local normality to assure the weak-continuity of the GNS implementation { U ϕ ( γ s ) } .For some propositions we need the split property in connection with local normality. Remark . If we treat one-parameter group { γ s } , in the following propositions (exceptfor Proposition 3.5, where the corresponding modification shall be explicitly indicated)it is enough to take just the von Neumann algebra ( π ϕ ( A ) ∪ { U ϕ ( γ s ) } ) ′′ and to considerinvariance under { γ s } or { U ϕ ( γ s ) } and the corresponding notion of γ -clustering propertyof states. Since { U ϕ ( γ s ) } is weakly continuous, we can utilize the mean ergodic theoremin this case as well.The following proposition is known (see e.g. [3, 19]). Proposition 3.2.
A state ϕ is extremal γ -invariant if and only if ( π ϕ ( A ) ∪ { U ϕ ( γ ) } ) ′′ = B ( H ϕ ) . Note that any finite convex decomposition of a locally normal state consists of locallynormal states, because a state dominated by a normal state is normal, too.The following is essential to our argument of extension for locally normal systems.
Theorem 3.3 ([13], A 86) . Let H = R ⊕ X H λ dµ ( λ ) be a direct integral Hilbert space, T i = R ⊕ X T i,λ µ ( λ ) be a sequence of decomposable operators, M be the von Neumann al-gebra generated by { T i } , and M λ be the von Neumann algebra generated by { T i,λ } . Thenthe algebra Z of diagonalizable operators is maximally commutative in M ′ if and only if M λ = B ( H λ ) for almost all λ . Since we assume the split property of the net A , there is a sequence of indices I i andtype I factors F i . Let K i be the ideal of compact operators of F i , and K be the C ∗ -algebragenerated by { K i } . With a slight modification about the index set, the following appliesto our situation. Theorem 3.4 ([21], Proposition 56) . Let π be a locally normal representation of a splitnet A on a separable Hilbert space and denote by π K the restriction to the algebra K . If wehave a disintegration π K = Z ⊕ X π λ dµ ( λ ) , then π λ extends to a locally normal representation e π λ of A for almost all λ . We need further a variation of a standard result. The next Proposition would followfrom a general decomposition of an invariant state into extremal invariant states and [38,Corollary 5.3] which affirms that any decomposition is locally normal. In the present articlewe take another way through decomposition of representation.11 roposition 3.5.
Let ϕ be a locally normal γ -invariant state of the C ∗ -algebra A and π ϕ be the corresponding GNS representation, then ϕ decomposes into an integral of locallynormal extremal γ -invariant states.Proof. We take a separable subalgebra K as above analogously as in [21]. We fix a max-imally abelian subalgebra m in the commutant ( π K ( K ) ∪ { U ϕ ( γ ) } ) ′ . Since K is separable,we can apply [13, Theorem 8.4.2] to obtain a measurable space X , a standard measure µ on X , a field of Hilbert spaces H λ and a field of representations π λ such that the originalrestricted representation π K is unitarily equivalent to the integral representation: π K = Z ⊕ X π λ dµ ( λ )and m = L ∞ ( X, µ ). Now, by Theorem 3.4 (note that the representation space H ϕ of theGNS representation with respect to a locally normal state ϕ is separable since we assumethat the original net A is represented on a separable Hilbert space H ), we may assume that π λ is locally normal for almost all λ , hence it extends to a locally normal representation e π λ and the original representation π ϕ decomposes into π ϕ = Z ⊕ X e π λ dµ ( λ ) . Furthermore, the GNS vector Ω ϕ decomposes into a direct integralΩ ϕ = Z ⊕ X Ω λ dµ ( λ ) . The representative U ϕ ( γ ) decomposes into direct integrals as well, since m commutes with U ϕ ( γ ): U ϕ ( γ ) = Z ⊕ X U λ ( γ ) dµ ( λ ) . From this it holds that Ω λ is invariant under U λ ( γ ), thus the state ϕ λ ( · ) := h Ω λ , π λ ( · )Ω λ i is invariant under the action of γ , for almost all λ . By the definition of the direct integralit holds that ϕ = Z ⊕ X ϕ λ dµ ( λ ) . It is obvious that ϕ λ is locally normal.It remains to show that each ϕ λ is extremal γ -invariant. By assumption, m is maximallycommutative in the commutant of ( π K ( K ) ∪ { U ϕ ( γ ) } ) ′′ . This von Neumann algebra isgenerated by a countable dense subset { π K ( x i ) } and a representative U ϕ ( γ ). Then, byTheorem 3.3, this is equivalent to the condition that ( { π λ ( x i ) } ∪ { U λ ( γ ) } ) ′′ = B ( H λ ),namely ϕ λ is extremal γ -invariant.If we consider a continuous family { γ s } , we only have to take a countable family ofoperators { π K ( x i ) } ∪ { U ϕ ( γ s ) } s ∈ Q . 12 orollary 3.6. Let A ⊂ B be an inclusion of split nets with a locally normal conditionalexpectation which commutes with γ . If ϕ is an extremal γ -invariant state on A , then ϕ extends to an extremal γ -invariant state on the quasilocal algebra B of the net B .Proof. The composition ϕ ◦ E is a γ -invariant state on B . By Proposition 3.5, ϕ ◦ E canbe written as an integral of extremal γ -invariant states: ϕ ◦ E = Z ⊕ X ψ λ dµ ( λ ) . By assumption, the restriction of ϕ ◦ E to A is equal to ϕ , which is extremal γ -invariant,hence the restriction ψ λ | A coincides with ϕ for almost all λ . Hence, each of ψ λ is anextremal γ -invariant extension of ϕ . Lemma 3.7.
If the net A is asymptotically γ -abelian, then it is γ -abelian.Proof. Let ϕ be a locally normal γ -invariant state on A . The action of γ is canonicallyunitarily implemented by U ϕ ( γ ). Let E be the projection onto the space of U ϕ ( γ )-invariantvectors in H ϕ and Ψ , Ψ ∈ E H ϕ . Let us put ψ ( x ) = h Ψ , π ϕ ( x )Ψ i .By the assumption of asymptotically γ -abelianness, it is easy to see thatlim N →∞ N N X i =1 ψ ( γ n ( x ) y ) = lim N →∞ N N X i =1 ψ ( yγ n ( x )) . On the other hand, by the mean ergodic theorem we havelim N →∞ N N X i =1 ψ ( γ n ( x ) y ) = lim N →∞ N N X i =1 h Ψ , U ϕ ( γ ) n π ϕ ( x )( U ϕ ( γ ) ∗ ) n π ϕ ( y )Ψ i = lim N →∞ N N X i =1 h Ψ , π ϕ ( x )( U ϕ ( γ ) ∗ ) n π ϕ ( y )Ψ i = h Ψ , π ϕ ( x ) E π ϕ ( y )Ψ i = h Ψ , E π ϕ ( x ) E π ϕ ( y ) E Ψ i . Similarly we have lim N →∞ N P Ni =1 ψ ( yγ n ( x )) = h Ψ , E π ϕ ( y ) E π ϕ ( x ) E Ψ i . Together withthe above equality we see that h Ψ , E π ϕ ( x ) E π ϕ ( y ) E Ψ i = h Ψ , E π ϕ ( x ) E π ϕ ( y ) E Ψ i ,which means that E π ϕ ( x ) E and E π ϕ ( y ) E commute. Proposition 3.8. If ϕ is a locally normal γ -invariant state on the asymptotically γ -abeliannet A , then the following are equivalent:(a) in the GNS representation π ϕ , the space of invariant vectors under U ϕ ( γ ) is onedimensional.(b) ϕ is weakly γ -clustering. c) ϕ is extremal γ -invariant.Proof. First we show the equivalence (a) ⇔ (b). By the asymptotic γ -abelianness we havelim N →∞ N N X i =1 ϕ ( γ n ( x ) y ) = lim N →∞ N N X i =1 ϕ ( yγ n ( x )) , and it holds by the mean ergodic theorem thatlim N →∞ N N X i =1 ϕ ( γ n ( x ) y ) = h Ω ϕ , E π ϕ ( x ) E π ϕ ( y ) E Ω ϕ i , lim N →∞ N N X i =1 ϕ ( yγ n ( x )) = h Ω ϕ , E π ϕ ( y ) E π ϕ ( x ) E Ω ϕ i . Now if E is one dimensional, then it holds that h Ω ϕ , E π ϕ ( y ) E π ϕ ( x ) E Ω ϕ i = h Ω ϕ , π ϕ ( y )Ω ϕ ih Ω ϕ , π ϕ ( x )Ω ϕ i = h Ω ϕ , E π ϕ ( x ) E π ϕ ( y ) E Ω ϕ i , and this is weakly γ -clustering.Conversely, if A is weakly γ -clustering, the above equality holds and it implies that E is one dimensional, since Ω ϕ is cyclic for π ϕ ( A ).Next we see the implication (a) ⇒ (c). Let us take a projection P in the commutant( π ϕ ( A ) ∪ { U ϕ ( γ ) } ) ′ . Since P commutes with U ϕ ( γ ), P Ω ϕ is again an invariant vector. Byassumption the space of invariant vector is one dimensional, it holds that P Ω ϕ = Ω ϕ orthat P Ω ϕ = 0. We may assume that P Ω ϕ = Ω ϕ (otherwise consider − P ). By thecyclicity of Ω ϕ for π ϕ ( A ), it is separating for π ϕ ( A ) ′ , thus P = .Finally, we prove the implication (c) ⇒ (a). By Lemma 3.7, the algebra E π ϕ ( A ) E isabelian, but by assumption (c), π ϕ ( A ) ∪ { U ϕ ( γ ) } act irreducibly and U ϕ ( γ ) acts trivially on E . Hence E π ϕ ( A ) E acts irreducibly on E . This is possible only if E is one dimensional. In this section we partly follow the steps in [1]. We give an overview of the proof ofTheorem II.4 of [1] in Appendix A, where some notations are introduced.Let A ⊂ B be an inclusion of asymptotically γ -abelian split nets of factors, and supposethat A is the fixed point subnet of a locally normal action α by a separable compact group G which commutes with γ and τ . We take a weakly γ -clustering primary τ -KMS state ϕ on A and fix a γ -clustering extension ψ to B (whose existence is assured by Corollary 3.6). Lemma 3.9.
There is a one-parameter group ε t ∈ Z ( G ψ , G ) such that the restriction of ψ to B G ψ is a faithful KMS state with respect to τ ′ t := τ t ◦ α ε t . roof. We consider the inclusion of C ∗ -algebras A r = B r G ⊂ B r and the restriction of τ .This is an inclusion of C ∗ -systems. The restriction of ψ to the regular subalgebra B r isstill γ -clustering by Lemma 2.4. We claim that the restriction of ψ (hence of ϕ ) to B r G (see Remark 2.3) is still a primary KMS state. Indeed, the GNS representation of ϕ | B r G can be identified with a subspace of the representation π ϕ of A . By the local normality,this subspace for B r G (which coincides with A r ) includes the subspace generated by A ( I )for each fixed index set. The whole representation space of π ϕ is the closed union of suchsubspaces, hence these spaces coincide. Furthermore, by the local normality, π ϕ ( B r G ) ′′ contains π ϕ ( A ( I )) ′′ for each I . Hence the von Neumann algebras generated by π ϕ ( A ) and π ϕ ( B r G ) coincide and ϕ | B r G is primary.Now we can apply Lemma A.2 to obtain a one-parameter group ε t ∈ Z ( G ψ , G ) suchthat ψ restricted to B r G ψ is a KMS state with respect to τ ′ t . Then by Lemmas 2.5, 2.6 wesee that ψ is a KMS state on the net B G ψ and it is faithful. Theorem 3.10.
Let A ⊂ B be an inclusion of asymptotically γ -abelian split nets of factors,and suppose that A is the fixed point subnet of a locally normal action α by a separablecompact group G which commutes with γ and τ . Then, for any weakly γ -clustering ex-tension ψ to B of a primary τ -KMS state ϕ on A (such an extension always exists byCorollary 3.6), there is a one-parameter subgroup ( ε ◦ ζ ) in G such that ψ is a primary e τ -KMS state where e τ t = τ t ◦ α ε t ◦ ζ t . The state ψ is automatically faithful.Proof. The restriction of ψ to B r is primary as we saw in Lemma 3.9.This time we consider the inclusion B r G ψ ⊂ B r . Any locally normal representation ofthe regularized algebra extends to a representation of the net on the same Hilbert space,hence it is faithful on the quasilocal algebra since we treat a net of factors. Then wecan apply Lemma A.3 together with Theorem A.5 to see that there is a one-parametersubgroup ζ t ∈ G ψ such that ψ | B r is a e τ -KMS state where e τ t = τ t ◦ α ε t ◦ ζ t and ε t is takenfrom Lemma 3.9. By Lemma 2.5 ψ is a KMS state on the net B . Again by local normality,the primarity of ψ | B r and ψ are equivalent. The faithfulness is proved as in Proposition3.9.We have the following corollary. Note that the gauge group of a split net is separablebecause the underlying Hilbert space is automatically separable. Corollary 3.11.
Let A ⊂ B be an inclusion of split conformal nets on the real line R ,and suppose that A is the fixed point subnet of a locally normal action α by a compactgroup G which commutes with translations τ . Then, for every weakly τ -clustering primary τ -KMS state ϕ on A , there exists a weakly τ -clustering extension ψ to B . For any suchan extension ψ there is a one-parameter subgroup ( ε ◦ ζ ) in G such that ψ is a primary e τ -KMS state where e τ t = τ t ◦ α ε t ◦ ζ t . The state ψ is automatically faithful. U (1) -current model From now on, we discuss concrete examples from one-dimensional Conformal Field Theory.We recall some constructions regarding the U (1)-current and discuss its KMS states for two15easons: being a free field model, it is simple enough to allow a complete classification ofthe KMS states, showing an example of non completely rational model with multiple KMSstates; it is useful in the classification of states for the Virasoro nets, whose restrictions to R are translation-covariant subnets of the U (1)-current net. U (1) -current model The U (1)-current model is the chiral component of the derivative of massless scalar freefield in the 2-dimensional Minkowski space time. See [5, 23] for detail.In the R picture, the space C ∞ c ( R , R ) can be completed to a complex Hilbert space (theone-particle space) with the complex scalar product ( f, g ) := R p> p b f ( p ) b g ( p ), where b f isthe Fourier transform of f , and the imaginary unit is given by b I f ( p ) := − i sgn( p ) b f ( p ). Theimaginary part of the scalar product is a symplectic form σ ( f, g ) := R R f g ′ dx . The U (1)-current algebra A U (1) is the Weyl algebra constructed on this symplectic space, generatedby Weyl operators W ( f ) = e iJ ( f ) acting on the corresponding Fock space (if f is a realfunction, J ( f ) is essentially self-adjoint on the finite particle-number subspace). The netstructure is given by A U (1) ( I ) := { W ( f ) : supp( f ) ⊂ I } ′′ . This defines a conformal neton S in the sense of Part I. The current operators satisfy [ J ( f ) , J ( g )] = iσ ( f, g ) and theWeyl operators satisfy W ( f ) W ( g ) = W ( f + g ) exp (cid:18) − i σ ( f, g ) (cid:19) . Let us briefly discuss the split property of the U (1)-current net. A sufficient conditionfor the split property for a conformal net on S is the trace class condition , namely thecondition that the operator e − sL , where L is the generator of the rotation automorphism,is a trace class operator for each s > J ( e − n ) J ( e − n ) · · · J ( e − n k )Ω, where e n ( θ ) = e i πnθ , 0 ≤ n ≤ n ≤· · · ≤ n k , k ∈ N , and all these vectors are linearly independent and eigenvectors of L witheigenvalue P ki =1 n i . Hence the dimension of the eigenspace with eigenvalue N is p ( N ), thepartition number of N . There is an asymptotic estimate of the partition function [18]: p ( n ) ∼ n √ e π √ n/ . Hence with some constants C s , D s , we haveTr( e − sL ) = ∞ X n =0 p ( n ) e − sn ≤ ∞ X n =0 C s e − D s n , which is finite for a fixed s >
0. Namely we have the trace class condition, and the splitproperty.The Sugawara construction T := : J :, using normal ordering, gives the stress-energytensor, satisfying the commutation relations:[ T ( f ) , T ( g )] = iT ([ f, g ]) + i c Z R f ′′′ g dx (2)16ith c = 1 and [ f, g ] = f g ′ − gf ′ . This is the relation of Vect( S ), which is the Lie algebraof Diff( S ). This (projective) representation T of Vect( S ) integrates to a (projective)representation U of Diff( S ). Furthermore, T and J satisfy the following commutationrelations [ T ( f ) , J ( g )] = iJ ( f g ′ ) . (3)Accordingly, U acts on J covariantly: if γ is a diffeomorphism of R , then U ( γ ) J ( f ) U ( γ ) ∗ = J ( f ◦ γ − ) (see [6, 32] for details). U (1) -current model We give here the complete classification of the KMS states of the U (1)-current model, firstappeared in [39, Theorem 3.4.11]. Proposition 4.1.
There is a one-parameter group q γ q of automorphisms of A U (1) | R commuting with translations, locally unitarily implementable, such that γ q ( W ( f )) = e iq R R fdx W ( f ) . (4) Proof.
For any I ⋐ R , let s I be a function in C ∞ c ( R , R ) such that ∀ x ∈ I s I ( x ) = x ; then σ ( s I , f ) := R R f dx if supp f ⊂ I and thereforeAd W ( qs I ) W ( f ) = e − iσ ( qs I ,f ) W ( f ) = e iq R R fdx W ( f ) . Set γ q | A ( I ) = Ad W ( qs I ), this is a well-defined automorphism, since Ad W ( qs I ) | A ( I ) =Ad W ( qs J ) | A ( I ) when I ⊂ J , which can be extended to the norm closure A U (1) satisfying(4) and commuting with translations because so is the integral. Lemma 4.2.
A state ϕ is a primary KMS state of the U (1) -current model if and only ifso is ϕ ◦ γ q for one value (and hence all) of q ∈ R .Proof. By a direct application of the KMS condition and the fact that γ q is an automor-phism commuting with translations. Theorem 4.3.
The primary (locally normal) KMS states of the U (1) -current model atinverse temperature β are in one-to-one correspondence with real numbers q ∈ R ; eachstate ϕ q is uniquely determined by its value on the Weyl operators ϕ q ( W ( f )) = e iq R f dx · e − k f k Sβ (5) where k f k S β = ( f, S β f ) and the operator S β is defined by d S β f ( p ) := coth βp b f ( p ) . The geo-metric KMS state is ϕ geo = ϕ and any other primary KMS state is obtained by compositionof the geometric one with the automorphisms (4) : ϕ q = ϕ geo ◦ γ q . roof. The algebra of the U (1)-current model is a Weyl CCR algebra, for which the generalstructure of KMS states w.r.t. a Bogoliubov automorphism is essentially known: seee.g. [35, Theorem 4.1] or [3, Example 5.3.2]. It is however easier to do an explicit andstraightforward calculation for the present case.Let ϕ be a KMS state and f, g ∈ C ∞ c ( R , R ). Recall that a product of Weyl oper-ators is again a (scalar multiple of) Weyl operator, so that the quasilocal C ∗ -algebra islinearly generated by Weyl operators. Hence the state ϕ is uniquely determined by itsvalues on { W ( f ) } . Furthermore, under the KMS condition, the function t F ( t ) = ϕ ( W ( f ) W ( g t )), where g t ( x ) := g ( x − t ), has analytic continuation in the interior of D β := { ≤ ℑ z ≤ β } , continuous on D β , satisfying F ( t + iβ ) = e − iσ ( g t ,f ) F ( t ) = e iσ ( f,g t ) F ( t ) . (6)We search for a solution F of the form F ( z ) = exp K ( z ), where K is analytic in the interiorof D β and has to satisfy the logarithm of (6), K ( t + iβ ) = iσ ( f, g t ) + K ( t ). The Fouriertransform of t iσ ( f, g t ) is p p b f ( p ) b g ( p ), thus we have a simple equation for the Fouriertransform w.r.t. t : exp( − βp ) b K ( p ) = b K ( p ) + p b f ( p ) b g ( p ), from which b K ( p ) = p b f ( p ) b g ( p )exp( − βp ) − .It can be explicitly checked that F is a solution of (6); any other solution, divided bythe never vanishing function F , has to be constant (w.r.t. t ) by analyticity. The generalsolution can therefore be written as F ( t ) = c ( f, g t ) · F ( t ), with c ( f, g t ) independent of t .To obtain (5), notice that ϕ ( W ( f + g t )) = F ( t ) e i σ ( f,g t ) = c ( f, g t ) · exp (cid:20) K ( t ) + i σ ( f, g t ) (cid:21) , and K ( t ) + i σ ( f t , g ) is the Fourier antitransform of p b f ( p ) (cid:18) e − βp − (cid:19) b g ( p ) = − p b f ( p ) coth βp b g ( p ) = − p b f ( p ) d S β g ( p )which is given by − Z e itp p b f ( p ) d S β g ( p ) dp = −
12 ( f, S β g t ) = − (cid:16) k f + g t k S β − k f k S β − k g t k S β (cid:17) , since ( f, S β g t ) is a real form. Note that k g t k S β is independent of t . We finally have thegeneral solution in the form ϕ ( W ( f + g t )) = c ( f, g t ) · e (cid:16) k f k Sβ + k g t k Sβ (cid:17) · e − k f + g t k Sβ . Note that factors ϕ ( W ( f + g t )) and e − k f + g t k depend only on the sum f + g t , hence sodoes the remaining factor: we define c ( f + g t ) := c ( f, g t ) · e (cid:16) k f k Sβ + k g t k Sβ (cid:17) . Since c ( f, g t )and k g t k S β are independent of t , so is c ( f + g t ). As ϕ ( W ( f )) = ϕ ( W ( − f )), c ( f ) = c ( − f ).Now we have ϕ ( W ( f + g t )) = c ( f + g t ) · e − k f + g t k Sβ , c ( f + g t ).Concerning the continuity, we notice that k f k S β ≥ k f k , because coth p ≥ p ∈ R + ; the map f W ( f ) is weakly continuous when C ∞ c ( R , R ) is given the topologyof the (one-particle space) norm k·k and a fortiori of the norm k·k S β ; being ϕ a KMSstate and locally normal, f ϕ ( W ( f )) is continuous w.r.t. both norms and f c ( f ) = ϕ ( W ( f )) · exp( − k f k S β ) is continuous w.r.t. the norm k·k S β ; finally, both λ λf and t f t are continuous w.r.t. the k·k S β norm, thus in particular λ c ( λf ) (and triviallythe constant function t c ( f + g t )) is continuous.If we require ϕ to be primary, it satisfies the clustering property: for t → ∞ ϕ ( W ( f + g t )) = ϕ ( W ( f ) W ( g t )) exp (cid:18) i σ ( f, g t ) (cid:19) → ϕ ( W ( f )) ϕ ( W ( g ))and thus c ( f + g ) = c ( f ) · c ( g ) , (7)because both σ ( f, g t ) and ( f, S β g t ) go to 0. It follows that c (0) = 1, c ( − f ) = c ( f ) − = c ( f )and | c ( f ) | = 1. As R ∋ λ c ( λf ) is a continuous curve in { z ∈ C : | z | = 1 } , there is aunique functional ρ : C ∞ c ( R , R ) → R s.t. c ( f ) = exp( iρ ( f )), ρ (0) = 0 and λ ρ ( λf ) iscontinuous.Clearly, (7) implies ρ ( f + g ) − ρ ( f ) − ρ ( g ) ∈ π Z ; by continuity of λ ρ ( λf + λg ) − ρ ( λf ) − ρ ( λg ) and ρ (0) = 0, we get ρ ( f + g ) = ρ ( f ) + ρ ( g ). Similarly, from [28, Proposition6.1.2] we know that ρ has the same continuity property of c , i.e. w.r.t. the k·k S β norm; c ( f t ) = c ( f ) implies ρ ( f t ) − ρ ( f ) ∈ π Z , but this difference vanishes because t ρ ( f t ) iscontinuous. Therefore, ρ is a real, translation invariant and linear functional. Accordingto [31], any translation invariant linear functional (even without requiring continuity) ρ on C ∞ c ( R , R ) is of the form ρ ( f ) = q R f ( x ) dx . So, if ϕ is a primary KMS state, it has to beof the form (5). Conversely, Lemma 4.2 implies that all these states are KMS.These are regular states (i.e. λ ϕ ( W ( λf )) is a C ∞ function ∀ f ) and the one pointand two points functions are given by ϕ q ( J ( f )) = q Z f dx (8) ϕ q ( J ( f ) J ( g )) = 12 ℜ ( f, S β g ) + i σ ( f, g ) + q Z f dx Z g dx, (9)where ℜ means the real part. The geometric KMS state has to coincide with one of those:it is ϕ . This can be proved by noticing that, if supp f ⊂ I , ϕ geo ( W ( λf )) = (Ω , Ad U ( γ I,β ) W ( λf )Ω) = (cid:0) Ω , W (cid:0) λf ◦ γ − I,β (cid:1) Ω (cid:1) = e − λ k f ◦ γ − I,β k where the exponent is a quadratic form in f , therefore the state is regular and taking thederivative w.r.t. λ we get ϕ geo ( J ( f )) = 0, which implies q = 0 by comparison with (8). Remark . The gauge automorphism γ z defined by the map J ( f )
7→ − J ( f ) acts as achange in the sign of q : ϕ q ◦ γ z = ϕ − q . 19he ’energy density’ of a state can be read from the expectation value of the stress-energy tensor as the constant c in the formula ϕ ( T ( f )) = c R f dx . Beside its physicalinterpretation, this formula is also useful to classify the states on the Virasoro net (seeSections 5.2 and 5.3). In order to evaluate ϕ q ( T ( f )), we need two technical lemmas.In the following, D fin := span { ψ = J ( f ) ...J ( f n )Ω : n ∈ N , f , . . . , f n ∈ C ∞ ( S , R ) } isthe space of finite number of particles and D ∞ := ∩ n ∈ N D ( L n ) is the common domain ofthe powers of L ; D ( L n ) ⊃ D ∞ and D fin are all dense in the vacuum Hilbert space, containthe space of finite energy vectors and are cores for L n (the following Lemma implies alsothat D ∞ ⊃ D fin ). Lemma 4.5 (Energy bounds) . Let P n ( J, T, L ) be a (noncommutative) polynomial in L and some J ( f i ) and T ( f j ) of total degree n , with f i , f j ∈ C ∞ ( S , R ) , then ∀ ψ ∈ D ∞ k P n ( J, T, L ) ψ k ≤ r n k ( + L ) n ψ k , (10) with an appropriate r n (depending on { f k } and on n but not on ψ ).Proof. The operators J ( f ) and T ( f ) satisfy similar bounds k J ( f ) ψ k ≤ c f k ( + L ) ψ k k T ( f ) ψ k ≤ c f k ( + L ) ψ k (11)for any ψ ∈ D fin with c f independent of ψ [6, ineqalities (2.21) and (2.23)], and similarcommutation relations on D fin : [ L , J ( f )] = iJ ( ∂ θ f ), [ L , T ( f )] = iT ( ∂ θ f ). Since D fin isa core for L , ∀ ψ ∈ D ( L ), using a sequence ψ n ∈ D fin , s.t. ψ n → ψ and L ψ n → L ψ ,and the closedness of J ( f ) and T ( f ), the bounds (11) hold on D ( L ) ⊃ D ∞ ; hence thecommutators hold also on D ∞ , using ∀ ψ ∈ D ∞ a sequence ψ n ∈ D fin , s.t. ψ n → ψ and L ψ n → L ψ (from which L ψ n → L ψ ). One sees also that D ∞ is invariant under J ( f )and T ( f ).We can generalize the inequalities (11), which are equivalent to (10) for n = 1, to any n . Indeed, induction and commutation relations show that on D ∞ ( + L ) n J ( f ) = X ≤ k ≤ n (cid:18) nk (cid:19) i k J ( ∂ kθ f )( + L ) n − k . (12)Then, we use induction in the degree of the polynomial to prove (10). Suppose (10) holdsfor degree n . Any polynomial of degree n + 1 is a linear combination of polynomials ofdegree n multiplied from the right by J ( f ) or T ( f ) or L .First, let us consider J ( f ). k P n ( J, T, L ) J ( f ) ψ k ≤ r n k (1 + L ) n J ( f ) ψ k by inductionhypothesis and, applying (12) (notice that + L ≥ L , ), the last norm is smaller than P ≤ k ≤ n c k k J ( ∂ kθ f n )( + L ) n − k ψ k where each term is estimated, using (11), by constantstimes k ( + L ) n +1 ψ k .Secondly, we consider T ( f ). With T in place of J , equation (12) still holds and thesame argument as above applies.Finally, k P n ( J, T, L ) L ψ k ≤ r n k (1 + L ) n L ψ k ≤ r n k (1 + L ) n +1 ψ k and thus (10)holds for degree n + 1. 20 emma 4.6. D ∞ is invariant for the Weyl operator W ( f ) = e iJ ( f ) , ∀ f ∈ C ∞ ( S ) , andthe unitary U ( g ) , ∀ g ∈ Diff( S ) , implementing the conformal symmetry.Proof. The subspace D fin is included in D ∞ by (10), and it is invariant under L , as[ L , J ( f )] = iJ ( ∂ θ f ). Using also the commutator [ J ( f ) , J ( g ) k ] = ikσ ( f, g ) J ( g ) k − (easyconsequence of [ J ( f ) , J ( g )] = iσ ( f, g )), we compute ∀ ψ ∈ D fin [ L , J ( f ) n ] ψ = (cid:18) inJ ( f ) n − J ( ∂ θ f ) − n ( n − J ( f ) n − σ ( ∂ θ f, f ) (cid:19) ψ. (13)We apply it to the expansion of Weyl operators W ( f ) = P k i k k ! J ( f ) k , which is abso-lutely convergent on D fin (it is well known that finite particle vectors are analytic forthe free field, see e.g. the proof of [33, Theorem X.41], with the estimate (cid:13)(cid:13) J ( f ) k ψ (cid:13)(cid:13) ≤ k/ p ( n + k )! k f k k k ψ k , where n is the number of particles of ψ ). By the closedness of L and the absolute convergence of L P k i k k ! J ( f ) k ψ , thanks to (13), we conclude that W ( f ) D fin is in the domain of L . We then easily compute, using the convergent series, thecommutation relations W ( f ) ∗ L W ( f ) = L − J ( ∂ θ f ) + σ ( ∂ θ f, f ) and their powers W ( f ) ∗ L n W ( f ) ψ = (cid:18) L − J ( ∂ θ f ) + 12 σ ( ∂ θ f, f ) (cid:19) n ψ. (14)Finally, (10) applied to the r.h.s., which is a polynomial of degree n in J ( ∂ θ f ) and L , gives k L n W ( f ) ψ k ≤ r k ( + L ) n ψ k (15) ∀ ψ ∈ D fin . As D fin contains the space of finite energy vectors (the vectors of D fin where f , · · · , f n are trigonometric polynomials), it is dense in D ∞ and is a core for L n ; any ψ ∈ D ∞ is the limit of a sequence { ψ i : i ∈ N } such that ( + L ) n ψ i is convergent, thus,by (15) and the closedness of L n W ( f ), W ( f ) D ∞ is in the domain of L n . We have provedthat W ( f ) D ∞ ⊂ D ∞ ; the same is true for W ( f ) − = W ( − f ), thus W ( f ) D ∞ = D ∞ .A similar argument apply to U ( g ). First one consider the case where g = exp T ( f )is contained in a one-parameter group. We replace (14) with the known transformationproperty of the stress-energy tensor ( L = T (1), where 1 has to be understood as thegenerator of rotations, the constant vector field on the circle; in the real line picture, itwould be the smooth vector field x x ) [15]: U ( g ) L n U ( g ) ∗ = ( T ( g ∗
1) + r g ) n (16)and then apply (10). For a general diffeomorphism g , it is possible to write g as a finiteproduct of diffeomorphisms contained in one-parameter groups, since Diff( S ) is alge-braically simple [14, 30] and the subgroup generated by one-parameter groups is normal,hence Diff( S ) itself. Thus we obtained the claimed invariance for any element g .21 heorem 4.7. For any primary KMS state ϕ q (cf. (5) ) the map t ϕ q (cid:0) e itT ( f ) (cid:1) is C ∞ , ∀ f ∈ C ∞ c ( R , R ) , and the expectation value of the stress-energy tensor is given by ϕ q ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx. (17) Moreover, in the GNS representation ( π ϕ q , H ϕ q , Ω ϕ q ) , Ω ϕ q is in the domain of any (noncommutative) polynomial of the stress-energy tensors π ϕ q ( T ( f k )) := − i ddt π ϕ q ( e iT ( f k ) ) , with f k ∈ C ∞ c ( R , R ) , k = 1 , . . . , n .Proof. Fix f ∈ C ∞ c ( R , R ) with supp f ⊂ I ⋐ R .We first consider the case q = 0. According to the proof of Proposition 2.4 and Theorem2.5 of Part I [7], the GNS representation of ϕ geo is the triple (cid:0) π ϕ geo = Exp β , H Ω , Ω (cid:1) andthere is a g β,I ∈ C ∞ c ( R , R ) s.t. Exp β | A ( I ) = Ad U ( g β,I ). It follows that the one parametergroup t π ϕ geo (cid:0) e itT ( f ) (cid:1) = Ad U ( g β,I ) (cid:0) e itT ( f ) (cid:1) has a generator Ad U ( g β,I ) ( T ( f )) whichcan be computed: indeed, [15, Proposition 3.1] proves that, in general diffeomorphismcovariant nets, if g ∈ Diff( S ) fixes the point ∞ , Ad U ( g ) T ( f ) = T ( g ∗ f ) + r R ( g, f ),with g ∗ f ( x ) = g ′ · f ( g − ( x )) and r R ( g, f ) = c π R p g ′ ( x ) d dx f ( x ) √ g ′ ( x ) dx with the centralcharge c set equal to 1 for the U (1) case. Therefore, with g β,I in place of g , recalling that g β,I ( t ) = e πtβ on the support of f , we get π ϕ geo ( T ( f )) = Ad U ( g I ) T ( f ) = T ( g I ∗ f ) + πc β Z f dx. (18)The vacuum vector Ω is in the domain of the operator (18) and any product of suchoperators; from (Ω , T ( h )Ω) = 0 for any h ∈ C ∞ c ( R , R ), we easily compute (17). The case q = 0 is proved.We now consider the general case for q . In this case the GNS representation is( π ϕ q = Exp ◦ γ q , H Ω , Ω) with γ q | A ( I ) = Ad W ( qs I ) defined in Proposition 4.1. The one pa-rameter group t Ad U ( g I ) ◦ Ad W ( qs I ) (cid:0) e itT ( f ) (cid:1) has a self-adjoint generator Ad U ( g I ) ◦ Ad W ( qs I ) ( T ( f )), which has to be computed. According to Lemma 4.6, for any ψ ∈ D fin ⊂ D ∞ with a finite number of particles, Ad W ( qs I ) ( T ( f )) ψ is well-defined because D ∞ is in the domain of T ( f ). Using, as for equation (13), [ J ( f ) , J ( g ) k ] = ikσ ( f, g ) J ( g ) k − and [ T ( f ) , J ( g )] = iJ ( f g ′ ), we compute ∀ ψ ∈ D fin a generalization of (13):[ T ( f ) , J ( g ) n ] ψ = (cid:18) inJ ( g ) n − J ( f g ′ ) − n ( n − J ( g ) n − σ ( f g ′ , g ) (cid:19) ψ. We use a similar argument to that following equation (13). The expansion of Weyl op-erators W ( g ) = P k i k k ! J ( g ) k is absolutely convergent on D fin ; using the absolute con-vergence of L P k i k k ! J ( g ) k ψ and the estimate k T ( f ) ψ k ≤ c f k ( + L ) ψ k , we concludethat also T ( f ) P k i k k ! J ( g ) k ψ is absolutely convergent and therefore, by the closedness of T ( f ), W ( g ) D fin is in the domain of T ( f ). The convergent series lets us compute (cf.2214)) W ( g ) ∗ T ( f ) W ( g ) ψ = (cid:0) T ( f ) − J ( f g ′ ) + σ ( f g ′ , g ) (cid:1) ψ . In the particular case in which g = − qs I , and thus f g ′ = − qf (recall that supp f ⊂ I ), we obtainAd W ( qs I ) ( T ( f )) = T ( f ) + qJ ( f ) + q Z f dx on the dense set D fin and also on D ∞ , where both sides are defined. We can apply Ad U ( g I )to this operator, as D ∞ is invariant for U ( g I ), and taking into account its action on J ( f )and T ( f ), we get π ϕ q ( T ( f )) = T ( g I ∗ f ) + π β Z f dx + qJ (cid:0) f ◦ g − I (cid:1) + q Z f dx. (19)Ω is in the domain of the operator (19) and any power of such operators; as before, usingalso (Ω , J ( h )Ω) = 0 for any h ∈ C ∞ c ( R , R ), we easily compute (17).We finally observe that the thermal completion (defined in Part I [7]), in the case ofthe U (1)-current model, does not give any new net. Theorem 4.8.
The thermal completion of the U (1) -current net w.r.t. any of its primary(locally normal) KMS states is unitarily equivalent to the original net.Proof. In the case of the geometric KMS state, this is the content of Theorem 2.5 in Part I[7]. The general case follows from the fact that any other primary KMS state of the U (1)-current model is obtained by composition of the geometric one with an automorphism,so that the local algebras ˆ A ϕ q ( e πt , e πs ) := A ϕ q ( t, ∞ ) ∩ A ϕ q ( s, ∞ ) ′ do not depend on thevalue of q . Vir c The Virasoro nets Vir c with c < ϕ geo . Thisis not the case for c ≥
1. Before going to the classification of the KMS states of Vir anda (possibly incomplete) list of KMS states for the Virasoro net with central charge c > c [39, Theorem 3.6.2]. Theorem 5.1.
The (primary locally normal) geometric KMS states of the
Vir c net w.r.t.translations assume the following value on the stress-energy tensor ϕ geo ( T ( f )) = (cid:18) πc β (cid:19) Z f dx. (20) Proof.
The evaluation of the state on the stress-energy tensor (20) follows from (18) usingthe same argument of the proof of Theorem 4.7.23 .2 KMS states of the Virasoro net
Vir Recall [7, Section 2.3] that the Virasoro net Vir is defined as the net generated by therepresentatives of diffeomorphisms. In fact, it holds that Vir ( I ) = { e iT ( f ) : supp( f ) ⊂ I } ′′ ,since the latter contains the representatives of one-parameter diffeomorphisms, which forma normal subgroup of Diff( I ) (the group of diffeomorphisms with support in I ), then thisturns out to be the full group because Diff( I ) is algebraically simple [14, 30]. The net Vir is realized as a subnet of the U (1)-current; we have seen that e − sL is trace class, henceVir is split as well.The primary (locally normal) KMS states of the U (1)-current, restricted to the Virasoronet, give primary (locally normal) KMS states. They are still primary because primarity forKMS states is equivalent to extremality in the set of τ -invariant states [3, Theorem 5.3.32],and this is in turn equivalent to the clustering property (Proposition 3.8) for asymptoticallyabelian nets; clustering property is obviously preserved under restriction. We denote thesestates ϕ | q | . We know their values on the stress-energy tensor (17). Notice that the twodifferent states ϕ q and ϕ − q coincide when restricted to Vir . We have thus a family ofprimary (locally normal) KMS states classified by a positive number | q | ∈ R + . We willshow that these exhaust the KMS states on Vir .An important observation for this purpose is that the U (1)-current net and Vir can beviewed as subnets of an even larger net. Namely, let B := A SU (2) be the net generated bythe vacuum representation of the loop group LSU (2) at level 1 [16], or by the SU (2)-chiralcurrent at level 1 [34], on which the compact group SU (2) acts as inner symmetry (anautomorphism of the net which preserves the vacuum state). This net satisfies the traceclass condition by an analogous estimate as for U (1)-current net in Section 4.1, hence itis split. It has been shown [34] that the Virasoro net Vir can be realized as the fixedpoint subnet of B with respect to this inner symmetry. Moreover, as shown in [8], all thesubnets of B are classified as fixed points w.r.t. the actions of closed subgroups of SU (2)(conjugate subgroups give rise to isomorphic fixed points); in particular, let A U (1) be the U (1)-current net, it is the fixed point B H of the net B w.r.t. the action of the subgroup H ≃ S of rotations around a fixed axis. Therefore, we have the double inclusionVir = B SU (2) ⊂ A U (1) = B H ⊂ A SU (2) =: B , and a complete classification of the KMS states of the intermediate net A U (1) . As we arenot able to directly extend a τ -KMS state on Vir to a τ -KMS state on A U (1) , we use anauxiliary extension to B exploiting the existence of the gauge group SU (2) and Corollary3.11. Theorem 5.2.
The primary (locally normal) KMS states of the
Vir net w.r.t. translationsare in one-to-one correspondence with positive real numbers | q | ∈ R + ; each state ϕ | q | canbe evaluated on the stress-energy tensor and it gives ϕ | q | ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx. (21)24 roof. For any q ∈ R , the restriction of the KMS state ϕ q to the Vir subnet gives a KMSstate. The evaluation of the state on the stress-energy tensor (21), depending only on | q | ,follows again from (18) using the same argument of the proof of Theorem 4.7.We have to prove that any primary KMS state of Vir arises in this way. Let ϕ be a primary KMS state of Vir = B SU (2) . By applying Corollary 3.11, we obtain alocally normal primary (i.e. extremal) τ -invariant extension ˜ ϕ on B , which is a KMS statew.r.t. the one parameter group t ˜ τ t = τ t ◦ α ε t ◦ ζ t , with a suitable one parameter group t ε t ◦ ζ t ∈ SU (2). The image of t ε t ◦ ζ t ∈ SU (2) is a closed subgroup H ≃ S since SU (2) has rank 1 and any one-parameter subgroup forms a maximal torus, therefore, if weconsider the subnet A = B H , it is ˜ τ invariant and, as ˜ τ t | A = τ t | A , the state ˜ ϕ is a primaryKMS state of A w.r.t. τ . It then follows that the KMS state ϕ of Vir is the restriction ofa KMS state ˜ ϕ | A of A , isomorphic to the U (1)-current net A U (1) . Remark . The geometric KMS state corresponds to q = 0, because it is the restrictionof the geometric KMS state on the U (1)-current net, and the corresponding value of the‘energy density’ π β + q is the lowest in the set of the KMS states. Remark . In contrast to the case of the U (1)-current net (Theorem 4.3), here the differ-ent primary KMS states are not obtained through composition of the geometric one withautomorphisms of the net.By contradiction, suppose that there were an automorphism α of the net such that ϕ | q | = ϕ ◦ α with q = 0. The KMS condition for ϕ ◦ α w.r.t. the one parameter group t τ t is equivalent to the KMS condition for ϕ w.r.t. the one parameter group t α ◦ τ t ◦ α − and, by the uniqueness of the modular group, τ t has to coincide with α ◦ τ t ◦ α − , i.e. theautomorphism of the net commutes with translations. By Proposition 4.2 of Part I [7], α cannot preserve the vacuum state and, by Lemma 4.5 of Part I, there is a continuous familyof pairwise non unitarily equivalent automorphisms of A | R commuting with translations.By Proposition 4.6 of Part I, there is a continuous family of automorphic sectors of A ,which contradicts the fact, proved in [9], that Vir can have at most countable sectors withfinite statistical dimension.Recall that in Part I the thermal completion net played a crucial role. Let A ϕ ( t, s ) := π ϕ ( A ( t, s )) and A dϕ ( t, s ) := A ϕ ( t, ∞ ) ∩ A ϕ ( s, ∞ ) ′ . Putting A ≡ Vir and ϕ ≡ ϕ | q | with q = 0, we have examples for which A ϕ ( t, s ) = A dϕ ( t, s ) . Indeed, if the inclusion A ϕ ( t, s ) ⊂ A dϕ ( t, s ) were an equality, as A = Vir has the splitproperty, Theorem 3.1 of Part I tells that ϕ would have to be ϕ geo ◦ α . The observation inthe previous paragraph would give a contradiction . Vir c with c > Here we show a (possibly incomplete) list of KMS states of the net Vir c with c > This shows that the formula (10) in [37] is incorrect and does not hold in general. to the real line R can be embedded as a subnet of the restrictionto R of the U (1)-current net. One can simply define a new stress-energy tensor [6, equation(4.6)], with k ∈ R and f ∈ C ∞ c ( R , R ) e T ( f ) := T ( f ) + kJ ( f ′ )and, using the commutation relations (2), calculate that h e T ( f ) , e T ( g ) i = i e T ([ f, g ]) + i k Z R f ′′′ g dx. It follows that the net generated by e T ( f ) as Vir c ( I ) := n e i e T ( f ) : supp f ⊂ I o ′′ with I ⋐ R ,is the restriction to R of the Virasoro net with c = 1 + k > c ( I ) ⊂ A U (1) ( I ) for I ⋐ R . Indeed, we know the locality of J and T , hence if supp( f ) ⊂ I , then e i e T ( f ) commutes with W ( g ) with supp( g ) ⊂ I ′ by the Trotter formula. By theHaag duality it holds that e i e T ( f ) ∈ A U (1) ( I ).The primary (locally normal) KMS states of the U (1)-current, restricted again tothis Virasoro net, give primary locally normal KMS states, noticing that ϕ q ( J ( f ′ )) = q R f ′ dx = 0: ϕ | q | (cid:16) e T ( f ) (cid:17) = ϕ | q | ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx ;as in the c = 1 case, the restrictions of ϕ q and ϕ − q are equal. We have thus the following Theorem 5.5.
There is a set of primary (locally normal) KMS states of the
Vir c net with c > w.r.t. translations in one-to-one correspondence with positive real numbers | q | ∈ R + ;each state ϕ | q | can be evaluated on polynomials of stress-energy tensor T ( f ) and on a single T ( f ) it gives: ϕ | q | ( T ( f )) = (cid:18) π β + q (cid:19) Z f dx. (22) The geometric KMS state corresponds to q = β q π ( c − and energy density πc β .Proof. As in the case of Vir , the restriction of a primary KMS state of the U (1)-currentnet is a primary KMS state and ϕ q = ϕ p if and only if q = ± p .The last statement on the geometric KMS state follows by comparison of (22) with(20). Remark . Unlike the Vir case, here the geometric KMS state does not correspond eitherto q = 0 or the lowest possible value π β of the energy density.26 n argument toward classification We give here an argument that could be useful in the classification of KMS states onVirasoro nets.Let ϕ be a primary (locally normal) KMS state on the Vir c net w.r.t. translations andsuppose that ϕ (( T ( f ) · · · T ( f n )) ∗ ( T ( f ) · · · T ( f n ))) < ∞ , f , · · · , f n ∈ C ∞ c ( R , R ). This isthe case for all the known KMS states, listed above, although we cannot prove it for ageneral KMS state. As the state is locally normal, the GNS representation π ϕ is locallynormal (thus a unitary equivalence of type III factors) and can be extended to the stress-energy tensors T ( f ) ( f ∈ C ∞ c ( R , R )), which are unbounded operators affiliated to localvon Neumann algebras. The above hypothesis is equivalent to the requirement that theGNS vector Ω ϕ is in the domain of any (noncommutative) polynomial of the representedstress-energy tensors π ϕ ( T ( f )). We show that the values of the state on polynomials of thestress-energy tensor ϕ ( T ( f ) · · · T ( f n )) are uniquely determined by the value of the stateon a single stress-energy tensor ϕ ( T ( f )), for f ∈ C ∞ c ( R , R ). This fact seems to determineuniquely the KMS state ϕ , as the net is in some sense generated by such polynomials,however this is not a rigorous statement.First of all, one can generalize the KMS condition in order to treat unbounded operators:it is shown in [39, Prop. 3.5.2] that equations (1) hold with x, y possibly unbounded oper-ators affiliated to a local algebra, such that Ω ϕ is in the domain of π ϕ ( x ) , π ϕ ( x ∗ ) , π ϕ ( y ) and π ϕ ( y ∗ ). Then we show, by induction in n , that ϕ ( T ( f )), together with the KMS conditions,uniquely determines the values ϕ ( T ( f ) ...T ( f n )). It is obvious for n = 1. It is supposedthat Ω ϕ is in the domain of the polynomials of T ( f ), the value of ϕ ([ T ( f ) ...T ( f n − ) , T ( f n )])can be computed from the values of ϕ on polynomials of degree n −
1, using the commuta-tion relations (2) which hold on Ω ϕ . According to the KMS condition, there is a function F ( t ) = ϕ ( T ( f ) ...T ( f n − ) τ t T ( f n )), continuous and bounded in D β := { ≤ ℑ z ≤ β } andanalytic in its interior, such that F ( t + iβ ) − F ( t ) = ϕ ([ T ( f ) ...T ( f n − ) , τ t T ( f n )]). If G has the same properties, then F − G is continuous in D β , analytic in its interior and( F − G )( t + iβ ) − ( F − G )( t ) = 0, thus F − G can be continued to an analytic boundedfunctions on C , which has to be a constant. As ϕ is primary, the clustering property impliesthat the constant is 0: lim t →∞ F ( t ) = ϕ ( T ( f ) ...T ( f n − )) ϕ ( T ( f n )) = lim t →∞ G ( t ). Thus F is uniquely determined and, in particular, ϕ ( T ( f ) ...T ( f n − ) T ( f n )).If the above argument can be made rigorous, one would get that a KMS state on theVirasoro net is uniquely determined by the value of the ’energy density’, the constant k appearing in ϕ ( T ( f )) = k R f ( x ) dx (this in the only possible expression with translationinvariance). In order to prove that the list in (22) is complete, it would be enough to provethat the set of possible energy density has π β as greatest lower bound. In this section we consider the free fermion net and the KMS states on its quasilocal C ∗ -algebra. For an algebraic treatment of this model, see [2, 29]. In contrast to the U (1)-27urrent model, the free boson model, it turns out to admit a unique KMS state (for eachtemperature). The model is not local, but rather graded local. It is still possible to definea (fermionic) net [11].The free fermion field ψ defined on S satisfies the following Canonical AnticommutationRelation (CAR): { ψ ( z ) , ψ ( w ) } = 2 πi · δ ( z − w ) , and the Hermitian condition ψ ( z ) ∗ = zψ ( z ), or, if we consider the smeared field, we have { ψ ( f ) , ψ ( g ) } = I S dz πiz f ( z ) g ( z ) . We put the Neveu-Schwarz boundary condition: ψ ( ze πi ) = ψ ( z ). Then it is possible toexpand ψ ( z ) in terms of Fourier modes as follows. ψ ( z ) = X r ∈ Z + b r z − r − . The Fourier components satisfy the commutation relation { b s , b r } = δ s, − r , s, r ∈ Z + .There is a faithful *-representation of this algebra which contains the lowest weightvector Ω, i.e., b s Ω = 0 for s > U be the unitaryrepresentation U of SL (2 , R ) ∼ = SU (1 ,
1) which makes ψ covariant. It holds that U ( g )Ω =Ω. Let P be the orthogonal projection onto the space generated by even polynomials of { b s } . It commutes with U ( g ) and the unitary operator Γ = 2 P − defines an innersymmetry (an automorphism which preserves the vacuum state h Ω , · Ω i ).For an interval I , we put A ( I ) := { ψ ( f ) : supp( f ) ⊂ I } ′′ . Then A is a M¨obius covariantfermi net in the sense of [11], and graded locality is implemented by Z , where Z := − i Γ − i .As a consequence, we have twisted Haag duality: It holds that A ( I ′ ) = Z A ( I ) ′ Z ∗ . Inaddition, we have Bisognano-Wichmann property: ∆ it = U (Λ( − πt )), where ∆ it is themodular group of A ( R + ) with respect to Ω under the identification of S and R ∪ {∞} , andΛ is the unique one-parameter group of SL (2 , R ) which projects to the dilation subgroupin P SL (2 , R ) under the quotient by { , − } [12].With { b s } we can construct a representation of the Virasoro algebra with c = asfollows (see [29]): L n := 12 X s> n (cid:16) s − n (cid:17) b − s b n + s , for n ≥ , and L − n = L ∗ n . For a smooth function f on S , we can define the smeared stress-energytensor T ( f ) := P n f n L n , where f n = H S dz πi z − n − f ( z ). The two fields ψ and T arerelatively local, namely if f and g have disjoint supports, then [ ψ ( f ) , T ( g )] = 0 ( ψ ( f ) is abounded operator and this holds on a core of T ( g )).By the twisted Haag duality, we have e iT ( g ) ∈ A ( I ) if supp( g ) ⊂ I (since ψ ( f ) is boundedfor a smooth function f , there is no problem of domains). Let us define Vir ( I ) := { e iT ( g ) :28upp( g ) ⊂ I } . This Virasoro net Vir has been studied in [20] and it has been shown thatVir admits a unique nonlocal, relatively local extension with index 2. Hence the fermi net A is the extension. Furthermore, by the relative locality, A is diffeomorphism covariant byan analogous argument as in [10, Theorem 3.7].We consider the restricted net A | R on R as in Section 2.1.1, the quasilocal C ∗ -algebra A and translation. Theorem 6.1.
The free fermion net A admits one and only KMS state at each temperature.Proof. By the diffeomorphism covariance and Bisognano-Wichmann property, we can con-struct the geometric KMS state as in Part I [7, Section 2.8] (locality is not necessary). Onthe other hand, Vir is completely rational [20], hence it admits a unique KMS state. Inthis case, we have proved without using locality [7, Theorem 4.11] that also the finite indexextension A admits only the geometric KMS state. Acknowledgement
We would like to thank the referee for pointing out imprecise statements in Appendix A.
Appendix A On the full extension of a KMS state
In this Appendix we discuss the theorem of Araki-Haag-Kastler-Takesaki [1]. Let B be a C ∗ -algebra, G a compact group acting on B and A = B G the fixed point with respect tothe action of G . We take a KMS state ϕ on A and a weakly γ -clustering extension ψ . Ifone looks at the statement carefully, it splits into two parts. The first part (Theorem II.4)claims that there is a distinguished subgroup N ψ (depending on ψ ) of G such that ψ isa KMS state on B N ψ with respect to an appropriate one-parameter automorphism group e τ . Then the second part (Remark II.4) says that N ψ is trivial when ψ is faithful on A ,so ψ is a KMS state on the whole algebra B . We believe that the first part is correct,but the proof of the second part is missing in the paper and we provide a counterexampleat the end of this Appendix. Hence the extension to the full algebra B is not clear ingeneral . Here we prove this complete extension with an additional assumption, which canbe applied to the case of nets.Let G ψ be the group of the stabilizers of ψ : G ψ := { g ∈ G : ψ ( α g ( a )) = ψ ( a ) for all a ∈ B } . The actions α of G , τ of R and ρ of Z are assumed to be norm-continuous. We alwaysassume that G is compact, the action of γ on B is asymptotically abelian. We preciselycite (the relevant part of)[1, Theorem II.4] (Note that we changed the notation. In theoriginal literature they use A , F for algebras, α for the time-translation, γ for the compactgroup action, τ for the space-translation and ϕ for the state). The same statement of full extension is found, for example, in [3, Theorem 5.4.25]. But we think thatat least the argument is flawed. We will give later a counterexample to the argument in [3] heorem A.1 (Araki-Haag-Kastler-Takesaki) . Assume that G is separable. Let ψ be aweakly γ -clustering state of B , whose restriction to A is an extremal ( τ t , β ) -KMS state.Then there exists a closed normal subgroup N ψ of G ψ , a continuous one-parameter subgroup ε t of Z ( G ψ , G ) and a continuous one-parameter subgroup ζ t of G ψ such that the restrictionof ψ to the fixed point algebra under N ψ B N ψ = { a ∈ B : α g ( a ) = a for all g ∈ N ψ } is a ( e τ t , β ) -KMS state where e τ t = τ t ◦ α ε t ◦ ζ t . We recall that the proof of this Theorem is further split into two parts ([1, TheoremII.2, Section II.5 and Section II.6]).
Lemma A.2.
Under the hypothesis of Theorem A.1, there is a one-parameter subgroup R ∋ t ε t ∈ Z ( G ψ , G ) such that the restriction of ψ to B G ψ is an ( τ ′ t , β ) -KMS state where τ ′ t := τ t ◦ α ε t . Lemma A.3.
Under the hypothesis of Theorem A.1, there is a continuous one-parametersubgroup ζ t of G ψ such that the restriction of ϕ to B N ψ is an ( e τ t , β ) -KMS state where e τ t := τ t ◦ α ε t ◦ ζ t . We think both of Lemmas are correct, hence the only task is to show that N ψ triviallyacts on B under certain conditions. Then let us recall how N ψ is defined.Consider the space of functions C ψ ( G ψ ) := { f ψa,b ∈ C ( G ψ ) : f ψa,b ( g ) = ψ ( aα g ( b )) , a, b ∈ B } . It has been shown that the norm closure C ψ ( G ψ ) is a Banach subalgebra of C ( G ψ ) [1,Lemma II.3], thus the intersection C ψ ( G ψ ) ∩ C ψ ( G ψ ) ∗ is a C ∗ -subalgebra of C ( G ψ ). It iseasy to see that this intersection is globally invariant under left and right translation by G ψ since by definition ψ is invariant under G ψ , hence there is a closed normal subgroup N ψ such that C ψ ( G ψ ) ∩ C ψ ( G ψ ) ∗ ∼ = C ( G ψ /N ψ ), where the isomorphism intertwines the naturalactions of G ψ [1, Lemma A.1]. Explicitly, N ψ is defined as follows: N ψ := { g ∈ G ψ : f ( g · ) = f ( · ) for all f ∈ C ψ ( G ψ ) ∩ C ψ ( G ψ ) ∗ } On the other hand, we can define another normal subgroup N ′ ψ of G ψ : N ′ ψ := { g ∈ G ψ : f ψa,b ( g · ) = f ψa,b ( · ) for all a, b ∈ B } . It is easy to see, by uniform approximation, that N ′ ψ := { g ∈ G ψ : f ( g · ) = f ( · ) for all f ∈ C ψ ( G ψ ) } . Hence, N ′ ψ ⊂ N ψ . Under a general assumption, N ′ ψ has a simple interpretation.30 emma A.4. Suppose that the GNS representation π ψ of B is faithful. Then it holds that N ′ ψ = { g ∈ G ψ : α g ( a ) = a for all a ∈ B } , namely N ′ ψ is the subgroup of the elementsacting trivially on B .Proof. We show that N ′ ψ ⊂ { g ∈ G ψ : α g ( a ) = a for all a ∈ B } , since the other inclusionis obvious. In the GNS representation, the defining equation of N ′ ψ is equivalent to h π ψ ( a ∗ )Ω ψ , U ψ ( g ) π ψ ( b )Ω ψ i = h π ψ ( a ∗ )Ω ψ , π ψ ( b )Ω ψ i , for all a, b ∈ B , which implies that U ψ ( g ) = and Ad U ψ ( g ) = id. In particular, we have π ψ ( α g ( a )) = π ψ ( a )for all a ∈ B and, by the assumed faithfulness of π ψ , we obtain α g ( a ) = a . Theorem A.5.
If the GNS representation of B G ψ with respect to (the restriction of ) ψ isfaithful and if π ψ is faithful on B , then N ψ acts trivially on B .Proof. We only have to show that N ψ = N ′ ψ by Lemma A.4 and the latter hypothesis.Under the former assumption, we show that the intersection C ψ ( G ψ ) ∩ C ψ ( G ψ ) ∗ is equalto C ψ ( G ψ ), then the Theorem follows from the definitions of N ψ and N ′ ψ .We remark that the assumption implies that ψ is faithful on B . Indeed, first theassumption that the GNS representation of ψ restricted to B G ψ is faithful implies that ψ is faithful on B G ψ , since the GNS vector of a KMS state is separating [3, Corollary 5.3.9].Now let x ∈ B such that ψ ( x ∗ x ) = 0. Then, by the definition of G ψ , ψ is invariant under G ψ , thus we have 0 = Z G ψ ψ ( α g ( x ∗ x )) dg = ψ Z G ψ α g ( x ∗ x ) dg ! . But R G ψ α g ( x ∗ x ) dg is positive and belongs to B G ψ , hence must be zero by the faithfulnessof ψ on B G ψ . This is possible only if x ∗ x = 0 by the continuity of α .As recalled in Appendix B, for f ψa,b ∈ C ψ ( G ψ ), one can take its Fourier component f ψa,b χ and the original function f ψa,b is uniformly approximated by its components. Henceit is enough to consider irreducible representations. If f ψa,b contains χ -component for some a, b ∈ B , then this in particular means that b χ = 0. By the faithfulness of ψ on B provedabove, one sees that ψ ( b χ b ∗ χ ) = 0. Since b ∗ χ belongs to the irreducible representation χ , oneconcludes that the conjugate representation χ is contained in H ψ . Then any function in C χ ( G ψ ) (see Appendix B) belongs to C ψ ( G ψ ).Summing up, the adjoint of each component of f ψa,b ∈ C ψ ( G ψ ) belongs again to C ψ ( G ψ )and each function in C ψ ( G ψ ) is recovered from its components. This completes the proofof self-adjointness of C ψ ( G ψ ).The hypothesis of the Theorem are satisfied not only in our case of conformal nets, aswe see in Section 3.2, but also in a wide class of models of statistical mechanics where localalgebras are finite dimensional factors M n ( C ).31 n the proofs of full-extension in the literature As noted before, Theorem A.5 without the assumption of faithfulness of π ψ is claimed in [1]without proof. In [3, Theorem 5.4.25] the theorem of full extension (i.e. N ψ acts trivially)is stated with the assumption of faithfulness of ϕ = ψ | A on A . But we think that the proofis not complete. The argument in [3] goes as follows. At the first step, they assume that ψ is faithful on A and show that ψ is a KMS state on B G ψ . At the second step, they saythat one can assume that ψ is invariant under G and the rest follows. The point is that,in the first extension, the faithfulness of ψ on B G ψ is not automatic. The symmetry of thespectrum of π ψ is essential in the second extension and the faithfulness is used for it. Herewe provide an example which shows that this faithfulness does not hold in general. We donot know whether the theorem holds without these assumptions. The same constructiongives a counterexample to [1, Remark II.4].We take an auxiliary system ( B , A , τ, α, γ ), where A = B G and take a KMS state ϕ on A with respect to τ and a γ -clustering extension ψ . Suppose for simplicity that ψ isfaithful and has the whole group as the stabilizer: G ψ := { g ∈ G : ψ ◦ α g = ψ } = G . Wehave many such examples: one can take just the geometric KMS state on the regularizedquasilocal algebra of a conformal net with a compact group action and the inclusion of thefixed point subnet.Consider now the field system ( b B , b A , b τ , b α, b γ ) where b B := B ⊕ B , b G := ( G × G ) ⋊ Z with Z acting on G × G as the flip, b τ t := τ t ⊕ τ t , the action b α of ( G × G ) ⋊ Z on B ⊕ B being the action α of each copy of G on each copy of B and the action of Z as the flip.The fixed point B b G is the diagonal algebra b A ⊂ A ⊕ A , which is isomorphic to A . Thesystem ( b B , b τ ) is asymptotically abelian as so is ( B , τ ).Let π i : b B → B be the projections on a component and ψ i := ψ ◦ π i . The two states ψ i are the two γ -clustering extensions of ϕ on b B (other extensions are convex combinationsof ψ and ψ and are KMS states w.r.t. b τ ). The stabilizer is in both cases b G ψ i = ( G × G ),a normal subgroup of b G , while the flip exchanges the two states: ψ i ◦ α z = ψ zi for z ∈ Z .The intermediate algebra is b B b G ψi = A ⊕ A and ψ i is obviously not faithful on it; thefaithfulness was assumed implicitly in the second step of the proof in [3].Let π ψ : B → B ( H ψ ) be the GNS representation of ψ (faithful as ψ is faithful), then π ψ i = π ψ ◦ π i : b B → B ( H ψ ) is not faithful, although it is true that π ψ i ( b A ) ′′ = π ψ i ( b B G ψi ) ′′ .As π ψ i is not faithful, although ψ i is faithful on b A , we cannot deduce that N ′ ψ i := { g ∈ b G : ψ i ( a b α g ( b )) = ψ i ( ab ) for all a, b ∈ b B } is trivial nor that it acts trivially. Indeed, N ′ ψ = (1 , G,
1) and N ′ ψ = ( G, , N ′ ψ i = N ψ i , also in this case. By proceeding as in the third paragraph of theproof of Theorem A.5, let b χ belong to the irreducible representation χ and ψ ( b ∗ χ b χ ) = 0.Then, as b χ is of the form b χ = b ⊕ b , ψ ( b χ b ∗ χ ) = ψ ( b b ∗ ) = 0 by the faithfulness of ψ ,which implies that χ is contained in H ψ ; the rest follows as in Theorem A.5. One sees that N ψ i is a normal subgroup of b G ψ i , ψ i and π ψ i are faithful neither on b B nor on b B N ψ = B ⊕ A .Moreover, the one parameter group b τ w.r.t. which ψ i is KMS is not uniquely defined, as ψ is not faithful and is KMS w.r.t. b τ ◦ α (1 ,g t , for any t g t ∈ G .32 ppendix B Noncommutative harmonic analysis Here we briefly summarize elementary methods to treat actions of a compact group G ona C ∗ -algebra. For the classical facts from the representation theory of compact groups,we refer to the standard textbooks, for example, [22]. The classical Peter-Weyl theoremsays that any irreducible representation of G is finite dimensional. To a finite-dimensionalrepresentation one can associate a character χ in the space C ( G ) of continuous functionson G . On this space G acts by left and right translations. This becomes a pre-Hilbertspace by the inner product induced by the Haar measure and its completion is denoted by L ( G ). The action by translation is referred to as the left or right regular representation.Again the Peter-Weyl theorem states that the left or right regular representation containsany irreducible representation and the multiplicity is equal to its dimension. If a function f belongs to an irreducible representation χ of dimension n of the left (or right) regularrepresentation, then the images of f under right and left translation of G × G span thewhole n dimensional space. Here we call this subspace C χ ( G ). Two characters χ, χ ′ areorthogonal iff the corresponding representations are disjoint. Any unitary representation U can be written as the direct sum of irreducible representations. The decompositioninto classes of inequivalent representations is canonical: for a character χ associated to anirreducible representation, the map ξ ξ χ = Z G χ ( g ) U ( g ) ξdg. is the projection from the representation space onto the direct sum of irreducible subrep-resentations of U equivalent to the one corresponding to χ . It holds that ξ = P χ ξ χ and ξ χ ⊥ ξ χ ′ if χ and χ ′ are inequivalent. The above formula is an extension of the Fourierdecomposition.An action α of G on a C ∗ -algebra B is an infinite dimensional representation of G ona Banach space. It is still possible to define the Fourier components: for a ∈ B , we put a χ := Z G χ ( g ) α g ( a ) dg. In general the sum P χ a χ is not necessarily norm-convergent. Now let us assume thatthere is a G -invariant state ψ . Then in the GNS representation ( H ψ , π ψ , Ω ψ ) there is aunitary representation U ψ which implements the action α . The components defined for U ψ and α are compatible: we have π ψ ( a χ )Ω ψ = Z G χ ( g ) π ψ ( α g ( a ))Ω ψ = Z G χ ( g ) U ψ ( g ) π ψ ( a )Ω ψ = ( π ψ ( a )Ω ψ ) χ . From the orthogonality in the representation U ψ , one sees that if χ and χ ′ correspondto two disjoint representations, then ψ (( a χ ) ∗ α g ( b χ ′ )) = h π ψ ( a χ )Ω ψ , U ψ ( g ) π ψ ( b χ ′ ))Ω ψ i =0. It is immediate to see that the function g ψ ( aα g ( b χ )) = ψ (( a χ ) α g ( b )) belongs to C χ ( G ). The decomposition of the vector π ψ ( b )Ω ψ = P χ π ψ ( b χ )Ω ψ converges in norm, hencefor the function f ψa,b ( g ) := ψ ( aα g ( b )), the decomposition f ψa,b = P χ f χa,b where f χa,b ( g ) = h π ψ ( a ∗ )Ω ψ , U ψ ( g ) π ψ ( b χ )Ω ψ i converges uniformly in the norm of C ( G ).33 eferences [1] H. Araki, D. Kastler, M. Takesaki and R. Haag: Extension of KMS states and chemicalpotential. Comm. Math. Phys. (1977), no. 2, 97-134.[2] J. B¨ockenhauer: Localized endomorphisms of the chiral Ising model. Commun. Math.Phys. (1996), 265–304.[3] O. Bratteli and D. Robinson:
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