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R. Brunetti1∗•, D. Guido2∗and R. Longo2∗

arXiv:funct-an/9302008v1 25 Feb 1993December 1992MODULAR STRUCTURE ANDDUALITY IN CONFORMALQUANTUM FIELD THEORYR. Brunetti1∗•, D. Guido2∗and R. Longo2∗(1) Dipartimento di Fisica, Universit`a di Napoli “Federico II”Mostra d’Oltremare, Pad.

19I–80125, Napoli, ItalyE-mail BRUNETTI@NAPOLI.INFN.IT(2) Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”Via della Ricerca ScientificaI–00133, Roma, Italy.E-mail GUIDO@MAT.UTOVRM.IT,LONGO@MAT.UTOVRM.ITABSTRACT. Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e.

the Tomita-Takesakimodular group associated with the von Neumann algebra of a wedge region and the vacuumvector concides with the evolution given by the rescaled pure Lorentz transformations pre-serving the wedge. A similar geometric description is valid for the algebras associated withdouble cones.

Moreover essential duality holds on the Minkowski space M, and Haag dualityfor double cones holds provided the net of local algebras is extended to a pre-cosheaf on thesuperworld ˜M, i.e.the universal covering of the Dirac-Weyl compactification of M. As aconsequence a PCT symmetry exists for any algebraic conformal field theory in even space-time dimension. Analogous results hold for a Poincar´e covariant theory provided the modulargroups corresponding to wedge algebras have the expected geometrical meaning and the splitproperty is satisfied.

In particular the Poincar´e representation is unique in this case.∗Supported in part by Ministero della Ricerca Scientifica and CNR-GNAFA.• Supported in part by INFN, sez. Napoli.1

IntroductionHaag duality in Quantum Field Theory is the property that local observable alge-bras maximally obeys the causality principle: if R(O) is the von Neumann algebra ofthe observables localized in the double cone O of the Minkowski space M, then R(O)is the commutant of the von Neumann algebra R(O′) of the observables localized inthe space-like complement O′ of OR(O′) = R(O)′Duality plays an important role in the structural analysis of algebraic QuantumField Theory [9] and has long been verified in free field models [1]. If the local alge-bras are generated by Wightman fields, Bisognano and Wichmann [2] have shown thegeneral result that duality holds for the von Neumann algebras associated with wedgeshaped regions W, namelyR(W ′) = R(W)′where W is any Poincar´e transformed of the region {x ∈M | x1 > |x0|}.

This propertyis called essential duality since it allows to enlarge the original observable algebras ofdouble cones so that Haag duality holds true. Their basic result is obtained by thecomputation of the Tomita-Takesaki modular operator ∆W associated to R(W) withrespect to the vacuum vector Ω[20], the latter being cyclic and separating becauseof the Reeh-Schlieder theorem.

In this case the modular group is the (rescaled) one-parameter group of pure Lorentz transformations leaving W invariant and the modularconjugation JW is the product of the PCT symmetry and a rotation; the essentialduality then follows at once by Tomita’s commutation theoremR(W)′ = JW R(W)JW = R(W ′).This identification of the modular group has several interesting consequences; besideduality, we mention here, as an aside, the relation with the Hawking effect, see [9], dueto the KMS (temperature) condition characterizing this evolution, and the Poincar´ecovariance of the superselection sectors with finite statistics in [14].One should not expect a sharp geometrical description for the modular group ofthe algebra R(O) of a double cone O, since in general there are not enough space-timesymmetries that preserve O; however, as a consequence of the Bisognano-Wichmanntheorem, the modular group of R(O) has a geometrical meaning in a conformallyinvariant theory [15].The purpose of the present work is to provide an intrinsic, Wightman field in-dependent, algebraic derivation of the Bisognano-Wichmann theorem in the case of aconformally invariant theory.2

We were motivated by a recent general result of Borchers [3] showing that part ofthe geometric behavior of ∆W follows automatically from the positivity of the energy-momentum operators: ∆W has the expected commutation relations with the trans-lation operators.This result however does not furnish the commutation relationsbetween modular operators associated with different wedges and, in space-time dimen-sion greater than two, does not provide the Bisognano-Wichmann identification of themodular automorphism group of R(W), indeed simple counter-examples illustrate howit may be violated in general.We shall show that essential duality holds automatically in a conformal theory ofany space-time dimension. But Haag duality for double cones fails in general becausethe local algebras actually live in a superworld˜M [19,17], a ∞-sheeted cover of acompactified Minkowski space.

However there is a natural procedure to extend theoriginal net of local algebras to a causal pre-cosheaf of von Neumann algebras (i.e. aninclusion preserving map O →R(O)) on ˜M, and Haag duality holds there.As a further consequence, we shall show that an algebraic conformal theory admitsautomatically a PCT symmetry [21].In particular for a M¨obius covariant pre-cosheaf of local algebras on S1, dualityholds on S1 without any assumption other than positivity of the energy.

Neverthelessduality fails in general on the cut circle ≃R. This phenomenon, discussed in [6] inthis specialization, is already present in [15] concerning the time-like duality; we shallreformulate the examples in [15] to get models of conformal theories on S1 with thedesired properties.Our results parallel an independent work of Fredenhagen and J¨orss [11] wheresimilar results are obtained by a different route: they construct Wightman fields as-sociated with an algebraic conformal field theory on S1, under a finite multiplicitycondition for the M¨obius representation.Part of our analysis extends to the general case of Poincar´e covariant nets.

As-suming that the modular group of the von Neumann algebra of any given wedge regionhas a geometrical meaning, we can show that the Bisognano-Wichmann interpreta-tion holds, provided the net fulfills the split property. The latter condition is indeednecessary and guarantees the uniqueness of the Poincar´e covariant action.After our work was completed we received a preprint of Gabbiani and Fr¨ohlich[12] that contains similar analysis for algebraic conformal field theories on S1.3

1.Algebraic Conformal Quantum Field Theory, general set-ting.In this section we describe a Conformal Quantum Field Theory in the algebraic ap-proach.Our aim is to give a self-contained introduction in our setting of knownfeatures.Geometrical preliminaries.In the following we consider a Lie group G acting bylocal diffeomorphisms on a manifold M, i.e. there exists an open set W ⊂G × M anda C∞mapT :W→M(g, x)7→Tgx(1.1)with the following properties:(i) ∀x ∈M, Vx ≡{g ∈G : (g, x) ∈W} is an open connected neighborhood of theidentity e ∈G(ii) Tex = x, ∀x ∈M(iii) If (g, x) ∈W, then VTgx = Vxg−1 and moreover for any h ∈G such that hg ∈VxThTgx = ThgxWe say that a local action of a Lie group G on a manifold M is quasi-global if theopen set{x ∈M : (g, x) ∈W}is the complement of a meager set Sg, and the following equation holds:limx→x0 Tgx = +∞,g ∈G,x0 ∈Sg(1.2)where x approaches x0 out of Sg and a point goes to infinity when it is eventually outof any compact subset of M.1.1 Proposition.If T is a transitive quasi-global action of G on M, then there existsa unique “T-completion” of M, i.e.

a manifold M such that M is a dense open subsetof M and the action T extends to a transitive global action on M.ProofLet E be the space of the continuous bounded functions ϕ on M such that,for any given g ∈G, the function T ∗g ϕ defined byT ∗g ϕ(x) ≡ϕ(Tgx),x ̸∈Sghas a continuous extension to M.4

Since a countable union of meager sets is meager, it is easy to see that E is a C∗-algebra,and g →T ∗g is an action of G by isomorphisms of E.Moreover E contains the algebra C0(M) of continuous functions on M vanishing atinfinity because, by condition (1.2),T ∗g ϕ(x) ≡ ϕ(Tgx)if x ̸∈Sg0if x ∈Sgis a continuous function, for any ϕ ∈C0(M) and for any fixed g ∈G. We denote byCG the minimal C∗-subalgebra of E containing C0(M) and globally invariant underthe action T ∗.We show that the spectrum M of CG is the requested completion, where the (global)action of G on M is given byTgp = p · T ∗g ,p ∈M.We notice that CG does not necessarily contain the identity, therefore M is not compactin general.

Moreover, by well known arguments [13], the spectrum of the C∗-algebraobtained adding the identity to CG is a compactification of M and is the one-pointcompactification of M, hence the natural embedding M ֒→M is dense.The transitivity of the action on M follows by the minimal choice of CG. In fact, letus consider the orbit in M containing M,M 0 ≡{Tgx : g ∈G}where x is a point of M as a subspace of M. By the transitivity on M, M 0 does notdepend on x and the action of G is transitive on M0.

We have natural embeddingsC0(M) ⊆C0(M0) ⊆CGand C0(M 0) is globally invariant under T ∗. Since CG is, by construction, the minimalC∗-algebra with this properties, then M 0 = M.By minimality, any other T-completion of M contains M, and therefore the transitivityrequirement implies uniqueness.Finally, M is a homogeneous space for the Lie group G, hence it is a C∞manifold.1.2 Proposition.In the hypotheses of the previous proposition, we consider theuniversal covering ˜G of G and the universal covering ˜M of M. Then the action T liftsto a transitive global action ˜T of ˜G on ˜M.ProofThe group ˜G has a canonical global transitive action on M, i.e.

T · π whereπ : ˜G →G is the covering map. The identity component of the isotropy group of apoint ( ˜Gx)0 is a connected closed subgroup of ˜G, therefore [18], the manifold˜M ≡˜G/( ˜Gx)05

is simply connected and is a covering of M, i.e. it is the universal covering of M. Then˜M is a homogeneous space for ˜G, and the thesis follows.Conformal action on the Minkowski space.In the following we specialize thepreceding analysis to the action of the conformal group on the Minkowski space M.The action of the conformal group C on a point of M (resp.

of the universal covering ˜Con ˜M) will be denoted by x →gx. As is known, M is the manifold Rd with constantpseudo-metric tensor q with signature+1,−1, .

. .

, −1|{z}(d −1) −times. (1.3)The corresponding Minkowski norm of a vector x ∈M isx2 = x20 −x21 −.

. .

−x2d−1.When d > 2 the conformal group consists of the local diffeomorphisms ϕ of M whichpreserve the pseudo-metric tensor q up to a non-vanishing function µ:ϕ∗q = µqThis group is a (d+2)(d+1)2dimensional Lie group, and the Lie algebra of its identitycomponent C is generated (as a vector space) by the following objects:translationsd generators(1.4a)boostsd −1 generators(1.4b)rotations(d−1)(d−2)2generators(1.4c)dilations1 generator(1.4d)special transformationsd generators(1.4e)The special transformations are the elements of the formρτaρwhere τa, a ∈Rd, is a translation and ρ is the relativistic ray inversion:ρx = −xx2x2 ̸= 0.When d ≤2 the conformal group C is, by definition, the one generated by thetransformations described in (1.4).The Conformal Universe.The conformal group C acts quasi-globally on M, andtherefore the completion M is defined.6

For convenience of the reader, we follow [22] and give the more explicit constructionof M by Dirac and Weyl [7,23]. Given Rd+2 with signature+1,−1, .

. .

, −1+1|{z}d −timeswe consider the manifold N whose points are the isotropic rays of the light cone, i.e.N = {(ξ0, . .

. ξn+1) ∈Rd+2\{0} : +ξ20 −ξ21 −.

. .

−ξ2d + ξ2d+1 = 0}/R∗where R∗= R\{0} acts by multiplication on Rd+2. The Lie group PSO(d, 2) actstransitively on this manifold by global diffeomorphisms, and it is easy to check thatthe map M →N given byξi = xii < dξd = 1 −x22ξd+1 = 1 + x22(1.5)is a dense embedding such that the restriction of the action of PSO(d, 2) to M cor-responds to the conformal transformations.

The uniqueness proven in Proposition 1.2shows that N = M.The Dirac-Weyl description shows M to be a compact manifold diffeomorphic to(Sd−1 × S1)/Z2. When d > 2, the universal covering˜M is infinite sheeted and isdiffeomorphic to (Sd−1 × R).

The map described in equation (1.5) lifts to a naturalembedding of M into ˜M, and the covering map from ˜M to M shows that ˜M containsinfinitely many copies of M as submanifolds.One of the main advantages of dealing with the manifold ˜M instead of M is thata global causal structure is naturally defined on it, i.e. the time ordering gives riseto a global ordering relation which extends the ordering on M, and a notion of (non-positive definite) geodesic distance is also well-defined, and is locally equivalent to theone in M. As a consequence each point divides ˜M in three parts, the relative future,i.e.

the points at time-like distance which follow the point, the relative past, i.e. thepoints at time-like distance which precede the point, and the relative present, i.e.

thepoints at space-like (or light-like) distance from the point. As we shall see, conformalquantum field theories live naturally on ˜M.When d ≤2, M is diffeomorphic to (S1)d, and therefore its universal covering isRd, but this manifold is rather unphysical.

In fact we may find two space-like separatedembeddings of M in Rd. Therefore we use the convention ˜M ≡S1 × R when d = 2,and ˜M ≡S1 when d = 1.7

We also mention that when d is odd (d ̸= 1) the manifold M is not orientable,and physical theories live on orientable coverings of M (cf. Proposition 1.7 and theodd-dimensional examples at the end of Section 2).In the following we shall consider the family ˜K of the subregions of ˜M which areimages of double cones in M under conformal transformations in ˜C.

We notice that alldouble cones, wedges and light-cones of M belong to the family ˜K, see e.g. [15].

Nowwe list some properties of ˜K:1.3 Proposition. (i) All elements of ˜K are open contractible precompact submanifolds of ˜M.

They area fundamental set of neighborhoods for ˜M. (ii) ˜C acts “transitively” on ˜K, i.e.∀O1, O2 ∈˜K∃g ∈˜C : gO1 = O2(iii) The identity component ˜C(O)0 of the group of the conformal diffeomorphisms thatpreserve O,˜C(O) = {g ∈˜C : gO = O}acts transitively on O, O ∈˜K.

(iv) The space-like complement O′ of a region O ∈˜K belong to ˜K. (v) The family ˜K is not a net, in fact the union of a region and of its causal complementis not contained in any region of ˜K.ProofImmediate.With each region O ∈˜K, we shall associate a one-parameter group ΛOt of confor-mal transformations which preserve O and commute with all O-preserving conformaltransformations:O ∈˜K →{ΛOt ,t ∈R} ⊂˜CThese groups will have the following coherence property:ΛO2t= g−1ΛO1t g,O2 = gO1, Oi ∈Kg ∈˜CTherefore, by Proposition 1.3(ii), they will be completely determined if we assignΛOtfor one region O.

For reader’s convenience, we describe explicitly the conformaltransformations for three particular regions in M (cf. [15,4]).The wedge W1:W1 = {x ∈M : x1 > |x0|}.8

The group ΛW1tis the one-parameter group of pure Lorentz transformations (boosts)along the x1 axis; its action on (x0, x1) is given by the matricescosh 2πt−sinh 2πt−sinh 2πtcosh 2πtThe double cone O1:O1 = {x ∈M : |x0| + |⃗x| < 1}The group ΛO1tcommutes with the rotations, hence is determined by its action on the(x0, x1)-plane:ΛO1t x± = (1 + x±) −e−2πt(1 −x±)(1 + x±) −e−2πt(1 + x±)where we posed x± = x0 ± x1.The future cone V+:V+ = {x ∈M : x0 > 0, x2 > 0}The one-parameter group ΛV+tis the dilation subgroup:ΛV+t= D(et)The universal covering ˜C.The universal covering ˜C of the conformal group Cturns out to be a central extension of C with fiber Z × Z2. The Z2 component actstrivially on ˜M and ˜C/Z2 acts effectively on it (see e.g.

[22]).Now we prove some simple properties on the conformal group we shall need inSection 2.1.4 Proposition.The groups C and ˜C are perfect groups, i.e. they coincide withtheir commutator subgroups.ProofSince both C and ˜C are semi-simple Lie groups, the result follows by theobservations in [18], p. 345.1.5 Proposition.The conformal transformations Ri, i = 1, .

. ., d −1 given byRix = −1x2 (x0, .

. .xi−1, −xi, xi+1, .

. ., xd−1)are in the identity component of C. Each Ri has only two liftings ±Ri of order 4 in ˜C.Proof(Sketch) We observe thatU(α) ≡τi(−cot α)D((sin α)−2)Riτi(−cot α)9

is a one-parameter subgroup of C, where τi(·) are the translations along the i−th axis.Moreover U(α) satisfiesU(0) = U(π) = eU(π2 ) = Ri,hence the first assertion follows. Lifting this subgroup to ˜U(α) ∈˜C we get a group ofperiod 2π, therefore ±Ri ≡˜U(± π2 ) are the requested liftings.

Since the fiber of thecovering ˜C →C is Z × Z2, any other lifting has infinite order.1.6 Proposition.For each i = 1, . .

., d−1, the translation subgroup and Ri generatethe conformal group C. The same result holds in ˜C when Ri is replaced by any of itsliftings ±Ri (cf. [22] for an analogous statement in the 4-dimensional case).Proof(Sketch) A straightforward calculation shows that the equationτi(a)Riτi(1/a)Riτi(a)Ri = D(a2)holds in C. Then, since dilations are the transformations ΛV+ for the future light coneand such a cone can be mapped into any wedge by a suitable product of translationsand Ri, boosts are in the group generated by τ and Ri [15].

The first statement followsbecause boosts and translations generate the Poincar´e group and, together with Ri,the conformal group C (cf. formulas (1.4)).

The statement for ˜C is proven in a similarway.1.7 Proposition.When d is odd, the change of sign of a space coordinate Pi is inthe identity component of C. As a consequence M is not orientable.ProofThe first part of the Proposition follows from straightforward calculationssimilar to those in Propositions 1.5 and 1.6. The rest follows because the Jacobian ofPi is negative.Conformal Quantum Field Theories.A local Conformal Quantum Field Theoryin dimension d is described by a causal additive pre-cosheaf of von Neumann algebrason the double cones of M = Rd with Minkowski structure, i.e.

a mapA : O →A(O),O ∈K10

where K is the family of the double cones in M, such thatO1 ⊂O2 ⇒A(O1) ⊂A(O2)A(O) ⊂A(O′)′(causality)A(∪nOn) = ∨nA(On)(additivity)where O′ is the space-like complement of O and the regions O, On and ∪On belong toK. The A(O) are supposed to act on a common Hilbert space H. Since the family Kis a direct set, the map O →A(O) is indeed a net and the quasilocal C∗-algebra A0 isdefined as the direct limit of the local algebras.

The algebras associated with generalopen regions in M are defined by additivity.Now we describe the conformal covariance assumption. As already explained, thegroup C acts locally on M. We observe that, since double cones are precompact, usingthe notation of formula (1.1) there exists an open neighborhood VO of of the identity in˜C such that VO×O ⊂W, and therefore the elements in VO give rise to diffeomorphismsof O into M.We assume that C acts locally by covariant automorphisms of the pre-cosheaf A,namely, for any O ∈K there is a weakly continuous map from the open set VO to theset Iso(A(O), A0) of isomorphisms of A(O) into A0,g →αOgwith the following properties.If O1 ⊂O2 are double cones,αO2gA(O1) = αO1gBecause of the preceding property, and since double cones form a net, we may dropthe superscript which specify the region, αg ≡αOg .The map g →αg is a local action, i.e.αhg = αh · αgwhen it makes sense.The local action is covariant, i.e.αgA(O) = A(gO).Finally we assume the existence of a local unitary representation U of C and of aU-invariant vector Ω, cyclic for ∪A(O), such thatU(g)AΩ≡αg(A)ΩA ∈A(O),g ∈VO(1.6)11

The generators of the (local) one-parameter subgroups are well-defined selfadjoint op-erators on H. The energy-momentum is assumed to be positive.While the usual energy H corresopnds to the Lie algebra generator h of the timetranslations, the conformal energy K corresponds to the Lie algebra elementk ≡h + ρhρ(1.7)where ρhρ is the adjoint action of ρ on h. It is clear that if H is positive, K is thesum of positive operators, and therefore is positive too. The converse is also true,and can be checked on explicit realizations of the irreducible positive-energy unitaryrepresentation of C [19].By the Reeh-Schlieder theorem Ωis cyclic and separating for the algebras A(O)and U can be defined directly from (1.6).

Now we show how the pre-cosheaf A maybe canonically extended to the superworld ˜M.We observe that the map g →U(g) may be considered as a local unitary rep-resentation of the universal covering ˜C, where g ∈˜VO, the identity component of thepre-image of VO under the covering map. Then, since ˜C is simply connected, U extendsto a global unitary representations of ˜C on H.1.9 Lemma.If O ∈K, g ∈˜C and gO = O then U(g)A(O)U(g)∗= A(O).ProofWe take x ∈O, then gx ∈O and since the connected component of thestabilizer of O acts transitively on O (see Proposition 1.3) we find h ∈˜C(O)0 suchthat h−1gx = x.

Then h−1g ∈˜Cx, which is connected by construction (cf. the proofof Proposition 1.2).

Therefore we may find a one-parameter family l(t) ∈˜Cx suchthat l(0) = id and l(1) = h−1g, and there exists a neighborhood Bx ∈˜K of x suchthat l(t)Bx ⊂O, t ∈[0, 1], hence the local covariance of α implies covariance for allt ∈[0, 1], i.e.αh−1gA(Bx) = A(h−1gBx).Since h ∈˜C(O)0, we may connect it with id staying inside ˜C(O)0, and the sameargument used before implies αhA(O) = A(O). As a consequence,αgA(Bx) = αhαh−1gA(Bx) = A(gBx) ⊂A(O)By additivity A(O) = ∨x∈OA(Bx), thereforeαg(A(O)) =_x∈O(A(gBx)) = A(gO).12

Since ˜C acts transitively on ˜K, we may define˜A(gO) = U(g)A(O)U(g)∗,g ∈˜G, O ∈K. (1.7)By Lemma 1.9, the map ˜A is well defined on ˜K, and defines a pre-cosheaf of C∗-algebrason ˜K.

By definition ˜C acts globally covariantly on the pre-cosheaf ˜A. We observe thatif W ⊂M is a wedge, then ˜A(W) turns out to be weakly closed, while A(W) is not.However ˜A(W) = A(W)′′, therefore, since A(W) is defined by additivity and A iscausal, ˜A(W) ⊂˜A(W ′)′.

Then, by covariance and transitivity on ˜K, causality holdsfor ˜A. The additivity property for ˜A follows by the analogous property of A.

Thus wehave proven the following:1.10 Proposition.The pre-cosheaf A extends to a unique, causal, additive pre-cosheaf ˜A on˜M and the local action α canonically extends to a globally covariantaction of the group ˜CWe have therefore shown the equivalence between the locally covariant picture, onthe manifold M, and the globally covariant picture, on ˜M, for conformally covariantfield theories.As it is explained in ([10], see also [14]) we may define a universal C∗-algebra ˜A0,and the isomorphisms adU(g) extend to automorphisms ˜αg of ˜A0. The C∗-algebra ˜A0is larger than A0 since it contains the von Neumann algebras associated with wedgeregions, and it is not necessarily faithfully represented in the vacuum representation[10].Finally we mention that explicit models may live on finite coverings of M or evenon M itself (see examples in Section 2).

In the last part of the following section weshall illustrate with some examples this phenomenon.2. Duality property and the Bisognano-Wichman theorem forconformal theories.In this Section we prove that essential duality holds for an algebraic conformal fieldtheory on the Minkowski space M, and duality for double cones (and conformally equiv-alent regions) holds for the corresponding pre-cosheaf extension on ˜M.

Moreover themodular unitary group of a region O ∈˜K coincides with a one-parameter subgroup ofthe conformal group, namely we have an algebraic derivation of Bisognano-Wichmanntheorem in this case.As was explained in the preceding section, the Minkowski space M is embeddedin a canonical way in the superworld˜M, and a (locally) conformally covariant pre-cosheaf of von Neumann algebras on M extends uniquely to a pre-cosheaf on ˜M whichis globally covariant with respect to the universal covering ˜C of the conformal group C.13

Therefore, in the following, M will always be thought as a submanifold of ˜M, and A asa sub-pre-cosheaf of ˜A. We shall denote by R(O) the von Neumann algebra associatedwith O, both when O ∈K and when O ∈˜KR(O) = ˜A(O).In two-dimensional space-time theories we make the further assumption that parityis implemented, i.e.

there exists a selfadjoint unitary U(P) such thatU(P)R(O)U(P) = R(PO)where P is the change of sign of the space coordinate in M, or the corresponding trans-formation in ˜M. In particular, for chiral theories, namely pre-cosheaves which splitinto a product of two pre-cosheaves on S1, parity correspond to the flip automorphismof the tensor product.We recall that an internal symmetry of the theory is an automorphism of thepre-cosheaf, i.e.

a consistent family of automorphismsγO ∈Aut(R(O)),O ∈˜K2.1 Lemma.Let O be a region in ˜K, ∆O the modular operator for (R(O), Ω) andUO(t) the unitaries associated with the transformations ΛOt defined in (1.6). Thenz(t) = ∆itOUO(−t)is a one-parameter unitary group that commutes with the unitary representation of ˜C,implements internal symmetries, and does not depend on the region O.Proofd > 2.Let us consider the wedge W1 and the equation∆itW1U(g)∆−itW1 = UW1(t)U(g)UW1(−t).

(2.1)If g is a conformal diffeomorphism which preserves W1, then g commutes with ΛW1t.Moreover U(g) implements an automorphism of R(W1), hence, by Tomita-Takesakitheory (see e.g. [20]), it commute with ∆itW1 and (2.1) holds.

This is also the case wheng is a translation along the axes x2, . .

., xd−1 or the trasformations ±R2 introducedin Lemma 1.5.By Borchers theorem [3], (2.1) holds if g is a translation along x0or x1, and therefore, by Lemma 1.6, for all elements of the conformal group. As aconsequence, z(t) = ∆itW1U(−t) commutes with U( ˜C) and therefore, since it preservesR(W1), it preserves R(O) for any region O ∈˜K.

Hence z(t) is an internal symmetry14

and commutes with the modular groups of all regions in ˜K. The independence fromthe region follows by the mentioned commutation relations.If d = 2, the proof goes on as before provided we substitute R2 with R1P.If d = 1 (i.e.

on S1), we consider the tensor product of ˜A0 with itself, and we get achiral two-dimensional theory. Then formula (2.1) holds for this theory, hence for theoriginal one-dimensional theory.2.2 Corollary.The given unitary representation {U(g) : g ∈˜C} is the uniquerepresentation of the conformal group which implements a covariant action on thepre-cosheaf ˜A and preserves the vacuum vector.ProofBy Lemma 2.1 we get the equality of the multiplicative commutators[∆itO1, ∆isO2] = [U(ΛO1t ), U(ΛO2s )].Therefore the representation U is intrinsic for g ∈[ ˜C, ˜C], since it it determined by themodular group.

Since ˜C is perfect (Proposition 1.4), U is completely determined bythe modular operators, hence it is unique.2.3 Theorem.Let A be a conformally covariant pre-cosheaf on M, ˜A the corre-sponding pre-cosheaf on ˜M. Then:(i) Essential duality holds for the pre-cosheaf A, and duality holds for the pre-cosheaf˜A.

(ii) If O is a region in ˜K, ∆O the modular operator for (R(O), Ω) and UO(t) theunitary associated with ΛOt , then∆itO = UO(t).Proof(i) Let us consider the wedge W1. By causality, R(W ′1) is a subalgebra ofR(W1)′, and is globally stable under the action of the modular group of R(W1) becausethe latter acts geometrically by Proposition 2.1.

Therefore, by Tomita-Takesaki theoryand the cyclicity of the vacuum, it coincides with R(W1)′. Since the Poincar´e grouphas a global causally-preserving action on M the relationR(W ′) = R(W)′holds for each wedge W, i.e.

essential duality holds for the pre-cosheaf A. Since ˜C hasa global causally-preserving action on ˜M the relationR(O′) = R(O)′15

holds for any region O ∈K, i.e. duality holds for the pre-cosheaf ˜A.

(ii) We have to show that z(t) in Lemma 2.1 vanishes. Let us consider one of theconformal transformations ±R1 introduced in Proposition 1.5.

By definition, R1W1 =W ′1. Then, by essential duality,U(R1)∆W1U(R1) = ∆−1W1.Moreover,U(R1)UW1(t)U(R1) = U(R1ΛW1tR1) = UW1(−t).Therefore,z(t) = U(R1)z(t)U(R1) = U(R1)∆itW1UW1(−t)U(R1) = z(−t),hence z(t) = I.Now we show that in even dimensions there exists a canonical antiunitary Θ whichimplements the PCT transformation on ˜M, i.e.

implements a conformal transformationof ˜M that restricts to change of sign of all coordinates on M:ΘR(O)Θ = R(βO),O ∈Kwhere βx = −x, x ∈M. Such an antiunitary is not unique, since ΘV still implementsa PCT whenever V is a self-adjoint unitary which implements a internal symmetry.

Asuitable (positivity) condition will fix a canonical choice for Θ. We choose the wedgeW1 as in formula (2.1) and consider the unitary SW1 which corresponds to the changeof sign of the coordinates x2, .

. .

xd−1 (which is an element in the identity componentof ˜C when d is even). Then, with each wedge W in the family W0 of the Lorentztransformed regions of W1, we associate the unitary SW ≡U(g)SW1U(g−1) wheregW1 = W, g ∈L↑+.2.5 Theorem.The anti-unitary Θ = JW SW , W ∈W0, implements a PCT trans-formation and does not depend on the choice of the wedge W ∈W0.

It is the uniqueanti-unitary such thatΘR(O)Θ = R(βO)O ∈Kand(Ω, AΘSW AΩ) ≥0∀A ∈R(W),W ∈W0(2.2)ProofThe same arguments used in the proof of Theorem 2.1 imply thatJW1U(g)JW1 = U(r1gr1),g ∈C16

where JW1 is the modular conjugation for the algebra R(W1) and r1 is the reflectionwith respect to the edge of the wedge. Since JW1R(W1)JW1 = R(W ′1) = R(r1W1) wegetJW1R(O)JW1 = R(r1O),∀O ∈K.Hence JW1SW1 implements the transformation β defined by βx = −x.

Therefore themapg →ΘU(βgβ)Θg ∈˜Cis a covariant unitary representation of ˜C. By the uniqueness proved in Corollary 2.2,and since β commutes with the Lorentz group L↑+ we getΘU(g)Θ = U(g)g ∈L↑+.The previous commutation relation implies that Θ does not depend on W:in fact given two wedges W1, W2 we can find an element g ∈L↑+ such that gW1 = W2,therefore we haveJW1SW1 = U(g)JW2U(g)U(g)∗SW2U(g)∗= JW2SW2.Since Θ implements the transformation β, ΘSW is an antilinear conjugation whichmaps W in W ′, hence, by essential duality, ΘSW R(W)ΘSW = R(W)′.

Thereforecondition (2.2) is the characterization of the modular conjugation JW [20], and theuniqueness follows.We observe that in odd dimensions we may implement a PCT transformation upto a change of sign of one space coordinate. Therefore a complete PCT is implementedif and only if such a change of sign is unitarily implemented.2.6 Remark.The split property [8] holds automatically for a conformal pre-cosheafon (Sd−1 × S1)/Z2, where it is equivalent to the distal split property [8] by conformalinvariance, provided the conformal Hamiltonian K (that has spectrum equal to N)has the multiplicity of its eigenvalues growing at most exponentially.

Indeed e−βK isa trace-class operator (for β large enough) and nuclearity holds [5].2.7 Corollary.With the assumptions of the previous remark, any local algebraR(O),O ∈˜K, is the unique injective factor of type III1.ProofImmediate, see [16].Examples derived from free fields.We illustrate the present setting by some ex-amples given by the net A of local algebras associated with the free massless scalar17

field on the d-dimensional Minkowski space discussed in [15], with a reinterpretationof the duality behavior.Recall that the group of conformal time translations (i.e. the subgroup generatedby the conformal energy K) has period 2π, i.e.

the operator e2πiK acts as the identityon ˜A. The Z-component of the center of ˜C acts as an “helicoidal shift” on˜M andthat the 2πZ time translations correspond to the even part of the Z-component of thecenter of ˜C.

Therefore all even copies of M may be identified, and all odd copies of Mmay be identified, hence free massless scalar fields lives on the compactification M oron its 2-fold covering M 2. In the following, if O is a region in M, we shall indicatewith O′ its space-like complement in M, with Ot its time-like complement in M, andwith Oc its causal complement in M, resp.

M 2.• d even.Since the relativistic ray inversion ρ is unitarily implemented, the theorylives on M. The space-like complement O′ and the time-like complement Ot of adouble cone O ∈M form a connected region Oc in M, the causal complement ofO in M. The duality property in M meansR(O)′ = R(Oc).Since R(O′) and R(Ot) are subalgebras of R(Oc), space-like and time-like com-mutativity hold for the pre-cosheaf A. Moreover R(O′) = R(Oc), therefore Haagduality holds in M. If d > 2, R(Ot) ̸= R(Oc), i.e.

time-like duality does not holdin M [15]. If d = 2, R(Ot) = R(Oc), i.e.

time-like duality holds in M.• d odd (d ̸= 1).Since two different liftings of ρ in ˜C are unitarily implemented,the theory lives on the 2-fold covering M 2. The product of the two liftings of therelativistic ray inversion is the transformation O →˜O which maps a region in oneof the copies of M in M2 onto the corresponding region in the other copy.

Thecausal complement in M 2 of a double cone O ∈M is the regionOc = O′ ∪˜Ot.SinceR(O′) = R(Oc)then Haag duality follows by the duality property for the pre-cosheaf˜A.The“twisted time-like commutativity” in [15] is the inclusionR( ˜Ot) ⊂R(O)′i.e. the algebra associated to the twisted time-complement ˜Ot commutes withR(O).18

• d = 1.Fix n > 2 and let R(O) be the net on the n-dimensional Minkowski spacedescribed above. Then setAn(I) = R(O)where I ⊂R is an interval of the time axis and O is the double cone obtained bycausal completion from I.If n is even then the An(I) give a conformal net on R that extends to a M¨obiuscovariant positive energy pre-cosheaf on S1.

An satisfies duality on S1. However Ansatisfies duality on R iffn = 2.If n is odd, An extends to a M¨obius covariant positive energy pre-cosheaf on thedisjoint union S1 ⊔S1.

If I is an interval in one of the copies of S1, we denote by ˜Ithe corresponding interval in the other copy of S1. Then duality holds in the formAn(I)′ = An( ˜I′)where I′ is the interior of the complement of I.Finally we notice that, with the previous notations, the tensor productA4 ⊗A4is a conformally covariant pre-cosheaf on R2 such that Haag duality does not hold.3.

Further results for Poincar´e covariant theoriesIn this section we consider Poincar´e covariant Quantum Field Theories in d > 2 di-mensions, namely a pre-cosheaf of von Neumann-algebras O →R(O) where O is inthe family K of double cones in the Minkowski space M with the usual causality andadditivity properties (see Section 1).The quasi-local C∗-algebra A0 generated by the local algebras is supposed to actin the vacuum representation as usual [9].The Poincar´e group P↑+ acts by covariant automorphisms on A0 and positivity ofthe energy-momentum is required.The pre-cosheaf is extended by additivity to general open regions in M. The weakclosure of the algebra associated with an unbounded region O will be denoted by R(O).We shall make two main assumptions in this Section:(a) Given any wedge region W, the modular unitaries ∆itW of R(W) act geometricallyon the pre-cosheaf A,∆itW R(O)∆−itW = R(ΛWt O),O ∈K,∀W ∈W(3.1)where ΛWtis the one-parameter group of diffeomorphisms defined in 1.6, and Wis the family of all wedge regions in M.19

(b) Distal split property holds, i.e there exist two regions O1 ⊂O′2 in M such thatR(O1) and R(O2) generates a W ∗-tensor product.3.1 Theorem.If distal split property holds, there is only one covariant unitaryrepresentation of the Poincar´e group on H leaving the vacuum vector invariant.ProofDistal split property implies that the group of internal symmetries G is com-pact and commutes with any action of the Poincar´e group by automorphisms [8]. Thenif two unitary covariant representations U, V of P↑+ exist, adU(g)V (g−1) belong to G,and therefore gives rise to an action of P↑+ in G. Since P↑+ has no non-trivial finite-dimensional representations, and G is compact, such a representation is trivial, i.e.adU(g)V (g−1) = id.

Then U(g)V (g−1) is a one-dimensional representation of P↑+,and, repeating the preceding argument, U(g)V (g−1) = 1.3.2 Lemma.Let W be a wedge in W, ∆W the modular operator for (R(W), Ω) andUW (t) the unitary associated with ΛWt. Then, if assumptions (a) and (b) holdz(t) = ∆itW UW (−t)is a one-parameter group which commutes with the unitary representation of P↑+,implements internal symmetries and does not depend on the wedge W.ProofBy assumption (a), the unitarieszWt= ∆−itW UW (t),W ∈W,t ∈Rimplement internal symmetries, and therefore commute with the modular groups ofany local algebra.Moreover, if we fix W0 ∈W and W1 is any other wedge, we can find an element g ∈P↑+which maps W1 onto W0, and thereforezW1(t) = U(g−1)zW0(t)U(g)t ∈RSince assumption (b) implies that the internal symmetries commute with any actionof P↑+ by automorphisms [8], the unitary zW1(t)zW0(−t) ∈C.

Since it preserves thevacuum vector, then zt ≡zW0tis independent from W0.Finally, the group property of {z(t), t ∈R} is easily checked using the mentionedcommutation relations.3.3 Theorem.Let A be a local pre-cosheaf on M satisfying assumption (a). Thenthe following holds:20

(i) The pre-cosheaf A satisfies essential duality, i.e.R(W)′ = R(W ′)for each wedge region W.(ii) If distal split property holds and d > 2, then∆itW = UW (t)where ∆W is the modular operator for (R(W), Ω) and UW (t) is the unitary represen-tation of the boosts which preserve W. The same result holds when d = 2, providedthat the parity transformation is unitarily implemented.ProofThe proof of part (i) is identical to the proof of Theorem 2.3, part (i).Since Lemma 3.4 holds, the proof of (ii) is analogous to the proof of Theorem 2.3,part (ii), provided that Ri is replaced by a Poincar´e transformation mapping a wedgeW onto W ′. If d > 2 we may choose a rotation, if d = 2 we use the parity transforma-tion.We conclude this section with an example of a non-split net where (ii) of Theorem3.3 does not hold, namely z(t) is non trivial.Such a construction was mentioned to us by D. Buchholz in a different context.Let A be a Poincar´e covariant net on the d + 1-dimensional Minkowski space Md+1which satisfies the Bisognano-Wichmann theorem.

Define a net B on the d-dimensionalMinkowski space Md byB(O) = A(π−1O)O ⊂Mdwhere π : Md+1 →Md is the projection parallel to the (d + 1)-th coordinate.Let W1 be a wedge in Md, z(t) the translation along the (d + 1) axis. Then z(t)implements a one-parameter group of internal symmetries of B which commutes withthe action of the Poincar´e group on Md, henceU ′W (t) ≡z(t)UW (t)determines a new d-dimensional representation of P↑+ that violates the Bisognano-Wichmann theorem.21

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