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A note on composite operators in N=4 SYM

A note on composite operators in N=4 SYM
A note on composite operators in N=4 SYM

a r X i v :h e p -t h /0106238v 1 26 J u n 2001KCL-TH-01-??

February 1,2008

A note on composite operators in N =4SYM

P.J.Heslop and P.S.Howe Department of Mathematics King’s College,London Abstract We discuss composite operators in N =4super Yang-Mills theory and their realisations as super?elds on di?erent superspaces.The super?elds that realise various operators on analytic superspace may be di?erent in the free,interacting and quantum theories.In particular,in the quantum theory,there is a restricted class of operators that can be written as analytic tensor super?elds.This class includes all series B and C operators in the theory as well as some series A operators which saturate the unitarity bounds.Operators of this type are expected to be protected from renormalisation.

Over the past few years the Maldacena conjecture[1]has rekindled interest in four-dimensional superconformal?eld theories and this has led to the discovery of many new and interesting re-sults.Most of these results have concerned properties of short(series C)operators and their cor-relation functions derived both directly in?eld theory and from supergravity via the AdS/CFT correspondence.Some recent reviews and lists of references can be found in[2,3,4,5].A strik-ing feature of such operators is that their shortness protects them from renormalisation-they cannot develop anomalous dimensions because the representations under which they transform determine these dimensions uniquely.More recently,however,it has been found that certain series A operators,which are not short in the above sense and which had not been anticipated to be protected from renormalisation,turn out to also have vanishing anomalous dimensions. These results have been established using the OPE and AdS/CFT[6,7],from partial non-renormalisation of four-point functions[8,9],in perturbation theory[10]and,most recently, using the OPE in N=2harmonic superspace[11].

The representations of the superconformal group are well-known[12]and their realisations on super?elds have been studied by many authors,see for example[13,14,15,16,17,18].In particular,shortening conditions for series A representations which saturate unitarity bounds have been discussed in[14,17].In this note we point out that the series A operators fall into three distinct classes when looked at as explicit functions of the underlying N=4supersymmetric Yang-Mills?eld strength super?eld.There are3di?erent types of behaviour:(i)operators which do not saturate the unitarity bounds,even in the free theory,(ii)operators which saturate the unitarity bounds in the free theory but for which the number of components changes in the interacting theory and(iii)operators which saturate unitarity bounds in the interacting theory. This classi?cation holds in the classical theory where the dimensions are still(half)integral.In the quantum theory operators of types(i)and(ii)can develop anomalous dimensions because there are“nearby”representations with non-integral quantum numbers which have the same number of components.On the other hand,for operators of type(iii)this is not the case,and one therefore expects them to be protected in a similar fashion to the short representations of series B and C.All the operators which have been found to be non-renormalised in references [6,7,8,9,10,11]are of type(iii)as one might expect,but this classi?cation suggests that there are very many more of them.

Operators of type(i)take care of themselves in that there are no shortening conditions even in the free case.However,it is not so easy to distinguish between operators of types(ii)and (iii)merely by looking at the quantum numbers of the representations or at their realisation as (abstract)super?elds in Minkowski superspace.It turns out that the operators of type(iii)are those that can be written as products of chiral primary operators,possibly with spacetime or spinorial derivatives.Operators of type(ii)include single trace operators(with the exception of the chiral primaries)and more complicated operators which include such single trace functions as factors.The basic reason for this is that the constraints(on Minkowski superspace)which type (iii)super?elds must satisfy in order to saturate unitarity bounds follow from the constraints on the gauge-invariant factors whereas,for operators of type(ii),this is not the case,so that the corresponding interacting multiplets have more components than the free ones.One way of seeing this is to work on analytic superspace,this having the advantage that there are no further constraints to be imposed apart from analyticity.A general analytic superconformal

?eld will transform under the isotropy subgroup of the superconformal group which de?nes analytic superspace in a non-trivial manner,i.e.it will have superindices(whereas one can?nd analytic superspaces for the series C operators where no indices are required[17,18]).We shall work on the analytic superspace with the smallest number of odd coordinates and the smallest number of additional even coordinates compatible with these.All representations with(half) integral dimensions can be constructed from a set of free Maxwell?eld strength super?elds and derivatives with respect to the coordinates of this analytic superspace.1The di?erence between operators of types(ii)and(iii)can be stated very simply in this context:operators of type (ii)cannot be so represented in the interacting case because this would involve applying gauge-covariant derivatives to the non-Abelian SYM?eld strength super?eld and this is not allowed because the Yang-Mills potential is not itself a?eld on analytic superspace.Operators of type (iii)are therefore composite operators for which the analytic superspace derivatives only act on gauge-invariant factors.The claim,therefore,is that all such series A operators which satisfy a unitarity bound should be protected from renormalisation.In the quantum theory operators of types(i)and(ii)both cease to be realised as analytic tensor super?elds.They can still be viewed as analytic?elds but their transformation properties are not of the usual tensorial type. On the other hand,operators of type(iii)are analytic tensor super?elds even in the quantum theory.In this sense one can view protection from renormalisation as being due to analyticity even for series A operators.

Before discussing this in more detail we shall brie?y discuss an example of each type of operator in N=4super Minkowski space.The?eld strength super?eld W I transforms under the6of SO(6),and is subject to the constraint

?αi W I=(σI)ijΛjα(1) whereαis a2-component spinor index,i is an SU(4)index andσI is an SO(6)σ-matrix.The spinorial derivative includes a gauge?eld in the non-Abelian case.The leading component of W I is the set of six scalar?elds of N=4SYM while the leading component ofΛiαis the quartet of spin one-half?elds.The supercurrent is T IJ=tr(W I W J)?1/6tr(W K W K).From(1)it obeys the constraint that when Dαi is applied to it only the20-dimensional representation of SU(4) survives.The quantum numbers specifying a representation of the N=4superconformal group are(L,J1,J2,a1,a2,a3)where L is the dilation weight,J1and J2are spin labels and(a1,a2,a3) are SU(4)Dynkin labels.We thus see that T IJ has quantum numbers(2,0,0,0,2,0).The unitarity bounds are:

Series A:L≥2+2J1+2m1?m

2

Series B:L=m

;L≥1+m1+J2,J1=0

2

Series C:L=m1=m

1This is brie?y discussed in[19];a detailed account is in preparation.

where m is the total number of boxes in the Young tableau of the SU(4)representation and m1 the number of boxes in the?rst row.

An operator of type(i)is given by T IJ T IJ.This has quantum numbers(4,0,0,0,0,0).It is a series A operator which does not saturate either unitarity bound and is simply an unconstrained scalar super?eld on Minkowski superspace.In the quantum theory there is nothing to prevent this operator developing an anomalous dimension.

An example of a type(ii)operator is the N=4Konishi multiplet,K=tr(W I W I)[20,21].In the free theory this operator obeys the constraint

D ij K:=Dαi Dαj K=0(3) However,in the interacting theory one?nds

D ij K~tr([W ik,W jl]W kl):=S ij(4) so that K is now an unconstrained super?eld(W ij:=(σI)ij W I).This is similar in some respects to the behaviour of the Yang-Mills supercurrent in ten dimensions.In the free theory this consists of a quasi-superconformal multiplet(128+128)together with a constrained scalar super?eld

[22]whereas in the interacting theory the scalar super?eld is unconstrained[23].As in the type

(i)case,in the quantum theory,there is nothing to stop K developing an anomalous dimension and it is well-known that this indeed happens[24,25].

For an example of type(iii)we consider the operator O IJ:=T IK T JK?1/6δIJ T KL T KL,which transforms under the20′representation of SU(4).This has quantum numbers(4,0,0,0,2,0); it is a series A operator which saturates both unitarity bounds.This operator obeys the same constraints in the interacting theory as it does in the free theory because they can be derived from the gauge-invariant constraints that T IJ satis?es.There is a representation related to this one by changing L=4to L=4+2γwhereγis a real number,but it has many more components and so one expects O IJ to be protected from renormalisation.Indeed,this operator is one of those found to have vanishing anomalous dimensions in references[6,7,8,9,10,11].

To discuss these operators more generally we shall use super Dynkin diagrams.For the(com-plexi?ed)superconformal group SL(4|N)acting on C4|N,the Dynkin diagram depends on the choice of basis.If the basis is ordered in the standard fashion,4even-N odd,we have the distinguished basis with one odd root,but we shall use a di?erent basis,which we shall refer to as physical,in which the basis has the ordering,2even-N odd-2even.The physical basis has two odd roots so that the Dynkin diagram is

????···??

N?1??

(5)

Any representation can be speci?ed by giving labels associated to each node of the Dynkin diagram.The labels associated with the two external even(black)nodes are determined by the spin quantum numbers(J1,J2)and the(N?1)internal even labels are?xed by the Dynkin

labels of SL(N).The two odd(white)labels are then determined by the dilation(L)and the R-symmetry(R)quantum numbers.All the Dynkin labels should be non-negative integers except for the odd ones which can be positive real numbers.These continuous labels are directly related to anomalous dimensions of operators.

The super Dynkin diagram can also be used to represent coset spaces determined by parabolic subgroups.With respect to a given basis the Borel subalgebra consists of lower triangular matri-ces,and a parabolic subalgebra(which by de?nition is one which contains the Borel subalgebra) consists of lower block triangular matrices.The size of these blocks is determined by a set of at most N+3positive integers k1

????···????(6) Chiral superspaces have a single cross through one of the odd nodes,harmonic superspaces have crosses through both odd nodes and some internal nodes,and analytic superspaces have crosses only through internal nodes.Superspaces with crosses through the external nodes include projective super twistor space,but such spaces are inconvenient for representation theory and so will not be considered further here.

The crosses on a super Dynkin diagram factorise the diagram into sub-(super)-Dynkin diagrams corresponding to the semi-simple subalgebra of the Levi subalgebra(the diagonal blocks in the parabolic),while the Dynkin labels above the crosses correspond to charges under internal U(1)’s or dilation and R weights.In general the Levi subalgebra will be a superalgebra and so the?elds can carry superindices.Only in cases where both odd nodes have crosses through (such as for super Minkowski space and harmonic superspaces)does the Levi subalgebra contain no superalgebra.

In order to have unitary representations(of the real superconformal group SU(2,2|N))the Dynkin labels on the odd nodes must exceed those of the adjacent external nodes by at least one unless one or both pairs of these adjacent nodes are zero.This gives three series of unitarity bounds.We label the nodes from the left n1...n N+3so that the two odd nodes are n2and n N+2 and the adjacent external nodes are n1and n N+3respectively.For series A we have n2≥n1+1 and n N+3≥n N+2+1.For series B we have either n1=n2=0and n N+3≥n N+2+1or we have n2≥n1+1and n N+3=n N+2=0.Finally series C requires that n1=n2=n N+3=n N+2=0. For general N we have

n2=1

N

?m1

n N+2=1

N

(7)

where m is the total number of boxes in the internal Young tableau determined by the SU(N) Dynkin labels(a1,...a N?1)=(n3,...n N+1)and m1is the number of boxes in the?rst row. The external black labels are(n1,n N+3)=(2J1,2J2).For N=4we need to impose R=0in order to have representations of P SU(2,2|4).

The above discussion implies that all of the unitary representations can be represented in various ways on super?elds de?ned on superspaces,and that these?elds will transform linearly under representations of the Levi subalgebra.In particular,in N=4,all of the representations can be realised as(analytic)super?elds on(N,p,q)=(4,2,2)analytic superspace:

???×???

p.This means that A

p is a scalar under sl(2|2)⊕sl(2|2)and has charge p under the U(1)corresponding to the central node of the super Dynkin diagram.All other representations transform non-trivially under the sub-algebra sl(2|2)⊕sl(2|2).The series B super?elds must transform under the totally(generalised)antisymmetric tensor representation (or the trivial representation)of one of the sl(2|2)subgroups and the series C super?elds must transform under the totally antisymmetric representation of both sl(2|2)subgroups(trivially in the KK case).For a general representation the highest weight state is obtained from the tensor component which has the most number of internal(a or a′)indices.

We now describe how the three operators discussed earlier can be written as?elds on analytic superspace.The?rst one,T IJ T IJ in super Minkowski space,has super Dynkin labels(0200020). On analytic superspace its behaviour with respect to both of the sl(2|2)subalgebras is given by the super Dynkin labels(020).It can be constructed from two T’s and four derivatives with both sets of indices,primed and unprimed,in the representation corresponding to the super Young tableau with two boxes in the?rst and second rows.

The free Konishi multiplet on(4,2,2)analytic superspace is???×???

theory,the diagram is the same with the1replaced by1+γ,γ>0.Forγnon-integral the representation???

n

of sl(2|2) for n integral,n≥2,all have the same dimension as???

?(A′(A?B′)B)W2(10)

6

However,this expression cannot be generalised to the interacting case since there is no gauge covariant derivative?A′A on analytic superspace.Moreover(10)is misleading in the quantum theory.The quantum Konishi multiplet resembles more closely the operator???×???

121.Again this representation has an associated anomalous representation???×???

121can be expressed in terms of derivatives of the supercurrent T=???×???

?(A′(A?B′)B)T2.(11)

5

We next consider operators of the form?p T?q T on analytic superspace.Since all such operators are compatible with non-Abelian gauge invariance they are all either type(i)or type(iii).We shall consider these operators?rst in the classical theory where the Dynkin labels are all integers. Those that are type(i)can then develop anomalous dimensions in the quantum theory whereas the others will be protected.The result is simple:those operators which have vanishing internal Dynkin labels are type(i)and all the others are type(iii).This is in agreement with the results derived in[11]using the OPE in N=2harmonic superspace.

To study these operators we?rst de?ne Q=L?(J1+J2).Since Q(?)=0and Q(T)= 2Q(W)=2,it follows that Q=4for any of these operators.In terms of the Dynkin labels Q= i=6i=2n i?(n1+n7),so that we have

n′2+n′6+m1=4(12)

where n′2:=n2?n1≥1;n′6:=n6?n7≥1,the inequalities following from the unitarity bounds. The requirement that the R-charge be zero gives

n3+2n2?n1=n5+2n6?n7(13) Due to the bounds we need only consider the cases m1=0,1,2.For m1=2we have n′2=n′6=1. The possible internal Dynkin labels are[020],[110],[011],[101],[200]and[002].

m1=2;[020]

For[020]we?nd the super Dynkin labels are[k(k+1)020(k+1)k].These operators can be written in the form T?k+2T with the k+2A and A′indices completely symmetrised2.Clearly such operators saturate the bounds and so are type(iii).

m1=2;[101]

For this case the super Dynkin labels are[k(k+1)101(k+1)k].These operators can be written as T?k+3T where the(k+3)unprimed and primed indices are both in the representation with symmetrisation over(k+2)indices but not over all of them.Again these operators saturate the bounds.

m1=2;[110]

The super Dynkin labels are[k(k+1)110(k+2)(k+1)].In this case the left-hand sl(2|2) representation corresponding to the unprimed indices is the same as the previous case whereas the right-hand one is totally symmetric in k+3indices.These operators cannot be written with all the derivatives hitting one of the T’s,but can be written in the form?T?k+2T.Again these are saturated.The case[011]is conjugate to this one.Note that the leading component of this supermultiplet is fermionic.In super Minkowski space it will involve an odd derivative acting on one of the T’s.

m1=2;[200]

Here the super Dynkin labels are[k(k+1)200(k+3)(k+2)].The left-hand sl(2|2)representation has Young tableauwhile the right one iswhere the notation denotes the number of boxes in the?rst,second,third row,and so on.It is not possible to construct this representation from derivatives acting on two T’s by symmetry.The case[002]is conjugate to this one and also cannot be constructed.

m1=1;[010]

If we choose n′2=2,n′6=1we?nd the super Dynkin labels are[k(k+2)010(k+3)(k+2)].The corresponding tensor has left Young tableauand right Young tableau. These operators can be written in the form?2T?k+2T and they saturate only one of the unitarity bounds.Nevertheless,this is su?cient for them to be of type(iii).

m1=1;[100]

If we choose n′2=2,n′6=1,the super Dynkin labels are[k(k+2)100(k+4)(k+3)].The left

Young tableau iswhile the right one is.Such operators cannot be constructed from derivatives acting on two T’s by symmetry.

m1=1;[001]

For n′2=2,n′6=1the Dynkin labels are[k(k+2)001(k+2)(k+1)].The left Young tableau iswhile the right one is.These operators can be written in the form ?2T?k+2T and satisfy one unitarity bound.

m1=0

In this case we could in principle have n′2=3,n′6=1but these cannot be written in terms of derivatives acting on two T’s.So take n′2=n′6=2.The super Dynkin labels are[k(k+ 2)000(k+2)k],and the Young tableaux arefor both the primed and unprimed indices.So these operators can be written in the form T?k+4T and are unsaturated.Therefore these operators can acquire anomalous dimensions in the quantum theory.

Operators of the above form contain,as spacetime components,the operators constructed from spacetime derivatives acting on two factors of the leading scalars in T discussed in[6].The authors of[6]were not always able to specify which supermultiplet was involved when the component?eld under discussion was not the highest weight state.Here we brie?y indicate how these supermultiplets can be identi?ed using analytic superspace.Let T o be the leading component of T;it is a scalar?eld in the20′representation of SU(4).The operators of[6]are schematically of the form,

O[abc]

~(?˙αα)r r′(T o T o)[abc](14)

rL

where r′=1/2(L?(r+4)),L being the na¨?ve dimension.The indices on the spacetime derivatives are totally symmetrised and[abc]denotes the SU(4)representation.Since T o is in the20′representation,the possible representations that can arise are1,20′,84,105,15,175. To illustrate the procedure let us consider operators in the105=[040]representation.There were two series of non-renormalised105operators mentioned in[6],r=2k,L=4+2k and r=2k,L=6+2k(where k is a positive integer),the?rst non-renormalised operator being r=0,L=8.Now,as an operator on analytic superspace,the leading component of T2is a scalar in the105representation.To obtain the desired component we therefore need only include the right spacetime derivatives.To?nd the full multiplet we then replace the spacetime derivatives by analytic superspace derivatives.

~(?˙αα)2k(T2o)[040],so the desired supermultiplet is(schematically)(?A′A)2k T2 We have O[040]

2k4+2k

with the primed and unprimed indices symmetrised.The super Dynkin labels are[(2k?2)(2k?1)020(2k?1)(2k?2)],so this operator is protected.The second operator is O[040]

2k6+2k (?˙αα)2k(T2o)[040].For this case we have2k+2derivatives and the primed(unprimed)in-dices are symmetrised with respect to2k+1of them.In other words the associated super Young tableaux are<2k+1,1>for both sets of indices.The super Dynkin labels are [(2k?1)(2k)101(2k)(2k?1)],and the operator is protected.In the third case O[040]

~ 2(T2o)[040].

08

In this case the four derivatives fall into the representation<2,2>for both primed and un-primed indices so the super Dynkin labels are[0200020].This operator is unprotected.

For k=0,one can have no d’Alembertians,in which case the operator is simply T2,which is series C,or one can have one d’Alembertian in which case the operator has super Dynkin labels [0020200]and is again series C.

A slightly more complicated situation arises when one needs to add further internal deriva-tives in order to obtain the right SU(4)representation.For example,consider the operator O[101]

~(?˙αα)2k+1(T2o)[101].Here it is necessary to add three further derivatives.There are 2k+12k+5

three possibilities corresponding to the super Dynkin labels[2k(2k+2)000(2k+2)2k](renor-malised)and[(2k+1)(2k+2)101(2k+2)(2k+1)]or[2k(2k+2)001(2k+2)(2k+1)](protected). Presumably the precise spacetime components of the three cases will not be identical because there will be di?erent contributions from the other?elds in the SYM multiplet(and from terms required to make the operators primary).

As well as the operators discussed above one can construct many more which should be protected by the same argument.To build any such operator one begins(schematically)with a product of A p’s and analytic superspace derivatives,with the indices on the latter projected onto irreducible representations of the two sl(2|2)superalgebras.One then requires that the operator really is primary,i.e terms can be added in such a way to achieve this,and?nally that at least one of the unitarity bounds is satis?ed.

For example,representations with Dynkin labels[k(k+1)lml(k+1)k]can be obtained by applying derivatives to gauge invariant operators for all positive integers k and l and for all positive integers m such that m≥4?2l or m=2?2l.These have the form T?k+l+2T A2l+m?2, and since they saturate both unitarity bounds they should be protected.Another example is the representation[(k+1)(k+2)lm(l+1)(k+1)k]for positive integers k,l,m and m≥3?2l or m=1?2l.These are of the form?k+l+2T?T A2l+m?1,saturate both unitarity bounds and are therefore protected.There are also many more examples of protected operators that saturate just one unitarity bound.

Note that,as shown above,the only unprotected operator constructed from two T’s is in the singlet representation of the internal SU(4)in agreement with[11].Furthermore we cannot construct any protected operators that are singlets by using more T’s or A p’s.There are, however,plenty of examples of unprotected operators that are not singlets.A simple example can be obtained by multiplying the m1=0example above by T.This has the form T2?k+4T and has Dynkin labels[k(k+2)020(k+2)k]and is thus in the20′representation of SU(4).

To summarise,we have seen that there are many series A composite operators in N=4SYM which should be protected from renormalisation by virtue of the fact that they are short and re-main short in the interacting theory,whereas the corresponding representations with anomalous dimensions are not shortened.These protected multiplets are all multi-trace operators,since the single-trace series A operators which are short in the free theory do not remain short in the presence of interactions.If we write the composite operators as?elds on(4,2,2)analytic super-space,the protected operators(from any series)are analytic tensor?elds.The non-protected operators can still be interpreted as?elds on analytic superspace but they are not tensor?elds of the standard type.

Acknowledgement

This research was supported in part by PPARC SPG grant613.

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我的第一份工作是在酒店,我最青春的三年也是在酒店度过的。 在这里,我学会了很多东西,能够跟同事们在一起工作,我觉得很开心,这里的每一位都是我的大哥大姐,我的叔叔阿姨,是他们教给了 我在学校里面学不到的知识,如何为人、如何处事、如何工作……在 酒店里,领导们也对我十分的关心,从刚进入酒店开始,我就感受到 从上至下的温暖。因为我是酒店里年龄还一般,还不算小,也从来没 有在这么大的集体里生活过,自不过然的,心里面就会产生一种被呵 护的感觉。这是一种以前在集体里未曾有过的感觉,很温馨,很自豪,而且它一直陪伴着我,直到我离开…… 但这种感觉不会随着我的离开而走远,我想我永远也不会忘记, 毕竟我以前生活在一个温暖而又温馨的集体里。韩总,还记得第一次 跟您近距离接触和理解是在20xx.3.16号。随着时间的流逝,斗转星移,您多年积累的工作经验与个人才华也得到充分的施展。您是我们 酒店的经理。在我上班之前,制定了一系列的政策与方针,重新定位 了酒店的经营策略,持续地尝试新的机制与奖励、分配办法,力争让 酒店的经济效益持续迈上新高,也让酒店员工的福利待遇如芝麻开花 一般节节高樊。,这才是为员工谋利益的举动,这才是一位被员工在 心里面所认可的经理。 而我,作为这个集体的一份子,更加感觉到您对员工的关心与培养。您肯定想到,酒店要想在竞争激烈的社会中立于不败之地,人才 的培养与发展是不可忽视的环节之一。因为我自身水平的不足,近期 的工作让我觉得力不从心,所以想公司提出了辞呈,忘领导批准。 申请人:### 20xx年xx月xx日 【篇三】 尊敬的公司领导:

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