Implementation of conformal covariance by diffeomorphism symmetry

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Implementation of conformal covariance by di?eomorphism symmetry Claudio D’Antoni ?,Klaus Fredenhagen ?,S?ren K¨o ster ?Abstract Every locally normal representation of a local chiral conformal quantum theory is covariant with respect to global conformal trans-formations,if this theory is di?eomorphism covariant in its vacuum representation.The unitary,strongly continuous representation implementing con-formal symmetry is constructed;it consists of operators which are in-ner in a global sense for the representation of the quantum theory.The construction is independent of positivity of energy and applies to all locally normal representations irrespective of their statistical di-mensions (index).AMS Subject classi?cation (2000):81T40,81T05,81R101Introduction The conformal group in 1+1dimensions is an in?nite dimensional di?eomor-

phism group.Many interesting models exhibit this symmetry and typically these models factorise into their chiral parts,each of which depends on one light-cone coordinate only.In this short letter we prove that all locally nor-mal representations of chiral conformal nets which exhibit di?eomorphism symmetry in their vacuum representation admit an implementation of global conformal transformations.

This automatic implementability is interesting since for a class of confor-mal models with less symmetry the existence of non-covariant,locally normal representations has been established by Guido,Longo,Wiesbrock[GLW98]. Moreover,our method applies in a very general setting:neither the represen-tation theory of the di?eomorphism group,the index of the respective locally normal representation or even positivity of energy concern our approach at all.

Di?eomorphism covariance of a chiral conformal net A(see e.g.[GL96]for general properties)means that there is a strongly continuous map U0from the group of orientation preserving di?eomorphisms of the circle,Di?+(S1),into the unitaries on H0,the representation space of the vacuum representation of A,implementing a geometric automorphic actionαof Di?+(S1):

U0(φ)A(I)U0(φ)?≡αφ(A(I))=A(φ(I)),I?S1,φ∈Di?+(S1).

The localisation regions are proper,i.e.open and non-dense,intervals I, denoted I?S1,whose causal complements are their open complements,I′, in S1.The subgroup Di?I(S1)of di?eomorphisms localised in I consists, by de?nition,of elementsφ∈Di?+(S1)which act trivially on I′.Forφ∈Di?I(S1)the adjoint action of U0(φ)is to implement the trivial automorphism of A(I′)and hence it is,by Haag duality of A,a local observable,i.e.U0(φ)∈A(I).

U0de?nes a ray representation,as the cocycles U0(φ1)U0(φ2)U0(φ1φ2)?commute with A and A is irreducible.With results of Carpi[Car03]this shows that U0corresponds to a de?nite value of the central charge of the Virasoro algebra.We require U0(id)=1l andα?PSL(2,R)to be identical to the global conformal covariance of A.In models having a stress-energy tensor,the restricted representation U0?PSL(2,R)is in fact a representation of PSL(2,R).The further analysis does not require the answer to the coho-mological question whether this may be achieved always by a proper choice of phases for U0.

We deal with a locally normal representationπof A,i.e.a family of normal representationsπI of the local algebras A(I)by bounded operators on a Hilbert space Hπ,which is required to be consistent with isotony:I?J?πJ?A(I)=πI;given this condition is ful?lled,we say that the local representationsπI are compatible.

By local normality,the mapsπ?U0?Di?I(S1)de?ne unitary,strongly continuous projective representations of the respective local di?eomorphism

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subgroup with cocycles which are phases,since the local algebras are factors. We will use the presence of these representations in order to construct an implementing ray representation of the subgroup of global conformal trans-formations(Moebius group PSL(2,R)).We begin with a clari?cation on the relation of general di?eomorphisms to the representationπinduced by U0.

We introduce the universal C?-algebra A uni generated by the local alge-bras of A.The properties of A uni are summarised in

Proposition1[FRS92,GL92]There is a unique C?-algebra A uni such that

1.For all I?S1there exist injective,compatible embeddingsιI:A(I)→

A uni and A uni is generated by its subalgebrasιI(A(I)).

2.For any compatible family of representations{πI}there exists a unique

representationπof A uni by bounded operators on Hπsuch thatπ?ιI=πI.

Moreover,every representationπof A uni restricts to a representation of A.The vacuum representationπ0of A uni corresponds to the identity (de?ning)representation of A on H0:π0?A(I)=id?A(I).

The actionαof Di?+(S1)on the net A can be extended to an action by

automorphisms of A uni throughαφ?ιI:=ιφI?Ad U

0(φ)?A(I).Our next goal

is to establish that this action is inner.For this purpose we prove Proposition2Letφ∈Di?I(S1),then for each J?S1we haveαφ?

ιJ(A(J))=Adι

I(U0(φ))?ιJ(A(J)).Eachαφ,φ∈Di?+(S1),possesses an

implementation by unitary elements of A uni.

Proof:If there is a proper interval?J?I∪J,the statement is obvious.If

I3?J′and I1,2∪J both are contained in some proper interval.By Lemma6we

?nd a factorisationφ= 3i=1φi,φi∈Di?I i(S1),ifφis contained in the neighbourhood Uε?Di?I(S1)de?ned in the Lemma,forεsu?ciently small

.

The two operatorsιI(U0(φ))and 3i=1ιI i(U0(φi))coincide up to a scalar multiple of1l=ιI(1l A(I)).Thus we have:

Adι

I(U0(φ))?ιJ(A(J))=

3

i=1AdιI i(U0(φi))?ιJ(A(J)).

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This proves the statement forφ∈Uε∩Di?I(S1).

Now letφbe an arbitrary di?eomorphism localised in I.We will factorise φinto a product of di?eomorphisms such that the above applies.Let?denote the periodic di?eomorphism of R which corresponds toφ(cf proof of lemma 6).De?ne?λ(x)=x+λ(?(x)?x),x∈R,and denote the corresponding element in Di?I(S1)byφλ.

With the covering{I i}i=1,2,3above there is aδ>0such that for all

λ∈[0,1]the di?eomorphismφλ+δ?φ?1

λ∈Uε.Then we can representφas a

productφ= n?1k=0(φk+1/nφ?1k/n),

for n large enough,where each factor and henceφsatis?es the Proposition.

Since Di?+(S1)is a simple group(theorem of Epstein,Herman,Thurston, cf[Mil84])eachφ∈Di?+(S1)has a presentation by a?nite product of localised di?eomorphisms.One may take this presentation and the results proved so far in order to obtain the desired implementation,which completes the proof.

Remark:The implementation of a di?eomorphismφby an element of A uni is not unique,in general:two implementers ofαφmay di?er by an element from the centre of A uni.

In a representationπof A uni the implementers form a projective,unitary representation of Di?+(S1)with cocycle in the centre ofπ(A uni)and which implements the automorphic actionα;ifπis a factorial representation,the cocycles are automatically phases.One would like to derive a genuine ray representation of Di?+(S1)with cocycle which is given by phases for any general representationπ,but we only know a way to do this for the subgroup PSL(2,R)of global conformal transformations.

To this end we analyse the cocycle of the implementation ofαand there-fore we look for a de?nite choice of the implementing unitaries for di?eomor-phisms close to the identity:For elementsφof a suitable neighbourhood Uεof the identity the results of lemma6(appendix)allow us to choose a?xed covering{I i}i∈Z m of S1by proper intervals and a?xed set of localisation maps Ξi:Uε→Di?I

i

(S1)such that uπ(φ):= m i=1π?U0?Ξi(φ)de?nes a strongly continuous,unitary and unital map.The adjoint action of uπinduces an implementation ofαinπ,which we will use in the following section.

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2Obtaining the implementation

We will now restrict our attention to the subgroup of global conformal trans-formations,PSL(2,R),and construct a unitary,strongly continuous repre-sentation of its universal covering group PSL(2,R)～from uπ?Uε∩PSL(2,R) as de?ned at the end of the previous section.This representation will im-plement the automorphic actionαof PSL(2,R)on A in the representation πand will be inner in the global sense,i.e.it will be contained in the von Neumann algebra of global observables,π(A):= I?S1πI(A(I)).

Let us begin with a closer look at the group PSL(2,R)itself.We use the symbol T for the one-parameter group of translations,S for the special conformal transformations,D for the scale transformations(dilatations)and R for rotations.We choose parameters for the rotations such that the rotation group R is naturally isomorphic to R/2πZ.

We can write every g∈PSL(2,R)in the form g=T(p g)D(τg)R(t g), where each term depends continuously on g(Iwasawa decomposition,[GF93], appendix I).In fact,any g∈PSL(2,R)may be written as a product of four translations and four special conformal transformations,each single of them depending continuously on g,if one uses the identities:

D(τ)=S(?(eτ2)T(1)S(eτ2),(1) R(2t)=S((?1+cos t)(sin t)?1)T(sin t)S((?1+cos t)(sin t)?1).(2) According to lemma6(appendix),there are continuous,identity pre-serving localisation mapsΞj,j=1,..,m,which map a neighbourhood of the identity,Uε?Di?+(S1),into groups of localised di?eomorphisms such that we have m j=1Ξj(φ)=φ,φ∈Uε.If we specialise to translations,this means that there is an open interval Iεcontaining0for which the mapping t→ j uπ(Ξj(T(t)))is unital and strongly continuous.We extend this map to all of R through a choice of aτ∈Iε,τ>0,de?ning n t∈Z by its properties t=n tτ+(t?n tτ),t?n tτ∈[0,τ[,and setting

πA(T(t)):= j uπ(Ξj(T(τ))) n t j uπ(Ξj(T(t?n tτ))).

One can easily check that this is indeed a strongly continuous map into the unitaries of B(Hπ)by recognising that the mappings involved are continuous and unital(π(1l)=1l,Ξi(id)=id).

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This procedure applies to the special conformal transformations as well, and we may use the result,the Iwasawa decomposition and(1),(2)to de?ne for each g∈PSL(2,R):

πA(g):=

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i=1Tπ(A)(t(i)g)Sπ(A)(n(i)g),g∈PSL(2,R).(3)

We haveπA(id)=1l.The following Lemma asserts that theπA(g)de?ne an

inner-implementing representation up to a cocycle in the centre ofπ(A).To this end we de?ne operators sensitive to the violation of the group multipli-

cation law:z A(g,h):=πA(g)πA(h)πA(gh)?,g,h∈PSL(2,R).

Lemma3πA:g→πA(g)de?nes a strongly continuous mapping with uni-tary values inπ(A).The adjoint action ofπA(g),g∈PSL(2,R),onπ(A)

implements the automorphismαg.z A:(g,h)→z A(g,h)de?nes a strongly continuous2-cocycle with unitary values inπ(A)′∩π(A).

Proof:Unitarity is obvious.Strong continuity follows since we multiply

continuous functions.The implementing property of theπA(g)follows im-

mediately by the decomposition g= 4i=1T(t i g)S(s i g),the subsequent decom-position of these into products of localised di?eomorphisms,the de?nition of

πA(g)and the implementation property of the(generalised)ray representa-tion uπof Di?+(S1).At this point all properties of z A follow immediately from its de?nition.

We write the abelian von Neumann algebra generated by the cocycle

operators z A(g,h)as follows:Z A≡{z A(g,h),z A(g,h)?|g,h∈PSL(2,R)}′′.

Obviously Z A is contained in the centre ofπ(A).Now we are prepared to realise the construction itself:

Lemma4For every?g∈PSL(2,R)～there exists a unitary operator z A(?g)∈Z A such that

Uπ(?g):=z A(?g)πA(p(?g))(4) de?nes a unitary,strongly continuous representation,whose adjoint action implements the automorphic actionα?p onπ(A);p denotes the covering projection from PSL(2,R)～onto PSL(2,R).

Proof:As Z A?π(A)∩π(A)′we may apply the direct integral decompo-

sition(cf e.g.[KR86],chapter14).This yields a decomposition of Hπas a

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direct integral of Hilbert spaces H x and it implies:the action of z A(g,h)on H x,denoted by z A(g,h)(x),is a multiple of the identity1l x and thereby de-?nes for almost every x a continuous2-cocycleω(f,g)x∈S1?C.The action of the operatorsπA(g)on H x,denoted byπA(g)(x),de?nes for almost every x a unitary,strongly continuous,projective representation of PSL(2,R),cf [Moo76].

For Lie groups with a simple Lie algebra the lifting criterion is valid [Sim68].This ensures for almost every x the existence of continuous phases ω(?g)(x),?g∈PSL(2,R)～,such thatω(?g)(x)πA(p(?g))(x)de?nes a represen-tation of PSL(2,R)～.Integratingω(?g)(x)over all x yields a unitary z A(?g)∈Z A,depending strongly continuously on?g.Integrating theω(?g)(x)πA(p(?g))(x) yields a unitary,strongly continuous representation Uπsatisfying equation (4).Uπ(?g)is an element ofπ(A)for every?g and implementsαp(?g)by its adjoint action due to Lemma3.

The outcome of the construction presented above proves our main result; its uniqueness statement is a simple consequence of the fact that PSL(2,R)～is a perfect group(cf[K¨o s02],prop.2):

Theorem5Let A be a chiral conformal,di?eomorphism covariant theory. Then any locally normal representationπof A is covariant with respect to the automorphic action of PSL(2,R).The implementing representation may be chosen to be the unique globallyπ(A)-inner,implementing representation Uπof PSL(2,R)～.

If there are di?eomorphism covariant theories which possess locally nor-mal representations violating positivity of energy,the construction of the inner-implementing representation given here applies even in cases in which the Borchers-Sugawara construction[K¨o s02]cannot be used,as the latter depends on the existence of implementations of translations and special con-formal transformations which have positive energy.For representations with ?nite statistical dimension the spectrum condition is always ful?lled because of the theorem we have just derived and results of[BCL98].For in?nite in-dex representations,of which examples are known(cf[Car02])there exists a criterion for strongly additive theories,which was given in[BCL98],too.In presence of the spectrum condition both constructions agree by uniqueness.

Yngvason[Yng94]discussed conformally covariant derivatives of the U(1)-current as interesting examples of chiral conformal theories.It is straight-

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forward to see that the?rst conformally covariant derivative had to exhibit a di?eomorphism symmetry if it contained a stress-energy tensor(details in[K¨o s03b]).Guido,Longo,Wiesbrock studied locally normal representa-tions of this model[GLW98]and found representations which manifestly do not admit an implementation of global conformal symmetry.As stated in [GLW98]this excludes the presence of a stress-energy tensor and di?eomor-phism symmetry for this model as a consequence of theorem5.Presence of a stress-energy tensor may be excluded directly as shown in[K¨o s03a]. Acknowledgements

C.D.is grateful to MIUR and the Alexander von Humboldt Stiftung for?-nancial support.S.K.acknowledges gratefully?nancial support from the Ev. Studienwerk Villigst and helpful discussions with K.-H.Rehren(G¨o ttingen). Appendix

The following technical lemma is crucial for the construction in section2. Lemma6Let Uε?Di?+(S1),ε>0denote the neighbourhood of the iden-tity

{φ∈Di?+(S1);sup

z∈S1|φ(z)?z|<εinf

z∈S1

|φ(z)′|}(5)

Let{I i}i∈Z m be a?nite covering of the circle by proper intervals.Then forεsu?ciently small there exist continuous localising mapsΞi:Uε?→Di?I

i

(S1) with the following features:

φ=

m

i=1Ξi(φ),Ξi(id)=id.

Proof:We look at the equivalent formulation in terms of periodic di?eo-morphisms of the real axis:?∈C∞(R),?′(x)>0,?(x+2π)=?(x)+2π. The analogue of?in Di?+(S1)is denoted byˇ?.The preimage of an interval I?S1under the covering projection p will be called?I.We choose a smooth partitionμof unity on S1satisfying1≥μi≥0,supp(μi)?I i.On the covering space we de?neλi(x):=μi(p(x)).

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De?ning ψk [?](x ):=x + k

i =1

λi (x )(?(x )?x ),k =0,...,m ,we have:

Ψk [?]′

(x )=(1?

k i =1

λi (x ))+

k i =1

λi (x )?′

(x )+

k i =1

λ′i (x )(?(x )?x )

inf ξ∈R

(min {1,?′(x )})?sup ξ∈R (

m

i =1

|λ′1(ξ)|)·sup ξ∈R

|?(ξ)?ξ|(6)

For a periodic di?eomorphism ?we have inf ?′(x )≤1.With

ε

?1

>sup

ξ∈R

m k =1

|λ′k (ξ)|,

(5),(6),imply

Ψk [?(x )]′≥

inf ?′(x ) 1?εsup ξ∈R k

i =1

|λ′k (ξ)| >0,

hence Ψk [?]is a periodic di?eomorphism.

Moreover Ψk [?]satis?es the estimate |Ψk [?](x )?x |≤|?(x )?x |<ε.We can now choose εsmall enough ,such that |χ(x )?x |<ε,for all x ,

χ∈Di?+(S 1),implies χ(suppλk )?I k .Then Ψk ?1[?]?1(I ′

k

)?(suppλk )′.Moreover,Ψk ?1[?]and Ψk [?]coincide on (suppλk )′,hence Ξk (?):=Ψk [?]?Ψk ?1[?]?1de?nes a periodic di?eomorphism,whose counterpart in Di?+(S 1),

namely ˇΞ

k (?),is localised in I k .We de?ne the localising maps on U εas follows:Ξk (ˇ?):=ˇΞ

k (?).Continuous dependence of Ξk (?)on ?is obvious,Ψk [id ]=id yields Ξk (id )=id .Finally,with Ψm [?]=?and Ψ0[?]=id :

m

k =1

Ξk (?)=Ψm [?]?Ψm ?1[?]?1?Ψm ?1[?]?...?Ψ0[?]?1=??id

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