The Wess-Zumino Model and the AdS_4CFT_3 Correspondence

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The Wess-Zumino Model and the AdS 4/CFT 3Correspondence W.M¨u ck ?and K.S.Viswanathan ?Department of Physics,Simon Fraser University,Burnaby,British Columbia,V5A1S6Canada February 1,2008Abstract We consider the non-interacting massive Wess-Zumino model in four-dimensional anti-de Sitter space and show that the conformal dimensions of the corresponding boundary ?elds satisfy the relations expected from superconformal invariance.In some cases the irregular mode must be used for one of the scalar ?elds.1Introduction The AdS/CFT correspondence,originally conjectured by Maldacena [1]formulates a duality between a ?eld theory on anti-de Sitter space (AdS)and a conformal ?eld theory (CFT)on its boundary.The most noted example is the duality between AdS type IIB string theory and N =4super Yang-Mills theory [2].The precise form of the AdS/CFT correspondence [3,4]in classical approximation reads exp (?I AdS [φ])= exp d d xφ0(x )O (x ) .

On the AdS side,the function φ0represents the boundary value of the ?eld φ,whereas on the CFT side it couples as a current to the conformal ?eld operator O .There is a characteristic relation between the mass of the AdS ?eld φand the conformal dimension of the CFT ?eld O [5].This has been investigated for scalar [4,6,7],spinor [8,9],vector

[7,9,10],graviton [11,12,13]and Rarita-Schwinger ?elds [14,15,16].Supersymmetry and supergravity in the AdS/CFT context have been considered in [17,18,19,5,20,21,22,23,24,25].

For AdS supersymmetric ?eld theories,supersymmetry relates the masses of ?elds in the same multiplet with each other.Hence,the AdS/CFT correspondence predicts that the conformal dimensions of the corresponding boundary CFT operators satisfy speci?c rela-tions.On the other hand,superconformal symmetry imposes a condition on the conformal

dimensions of the primary?elds of a superconformal multiplet.One would expect that the AdS prediction coincides with the CFT condition,which would mean that the AdS/CFT correspondence couples an AdS super multiplet to a superconformal multiplet on the AdS boundary.However,to the best of our knowledge,no direct comparison has yet been made.

In this paper,we shall tackle this problem by looking at the non-interacting massive Wess-Zumino model in AdS4,?nding the relations between the conformal dimensions of the corresponding scalar and spinor boundary operators and comparing them with the relations expected from superconformal invariance.We shall?nd in agreement with the classic AdS papers[26,27]that in some cases one must use the irregular mode for one of the AdS scalar ?elds in order for the AdS/CFT correspondence to hold true.This modi?es the standard prescription[4],which uses only the regular modes.

Let us start with some preliminaries and use them to explain our notation.For simplicity, we shall consider AdS4with Euclidean signature.As is well known[28],it can be constructed as a hyperboloid embedded into a?ve-dimensional Minkowski space with a metric tensor ηAB(A,B=?1,0,1,2,3),where

η?1?1=?1,ημν=δμν,andη?1μ=0(1) (μ,ν=0,1,2,3).Then,AdS4can be de?ned by the embedding

y A y BηAB=?1,y?1>0,(2) where the“radius”of the hyperboloid has been chosen equal to1for simplicity.The metric

ds2=dy A dy BηAB(3) represents the AdS metric,if one takes the yμas AdS4coordinates and de?nes y?1via equation(2).

While the representation(2)proves useful for?nding the AdS symmetries,a change of variables will reveal the conformal symmetry of the AdS boundary.Introducing the variables xμby

x0=

1

(x0)2

dxμdxν.(5)

The use of the Minkowski?ve-space suggests the introduction of4×4gamma matrices ?γA satisfying{?γA,?γB}=2ηAB.The spin matrices of the corresponding Lorentz algebra in ?ve dimensions are?S AB=1

4[γa,γb].Covariant gamma matrices are de?ned byΓμ=e aμγa and covariant spin matrices

byΣμν=e aμe bνS ab.

Finally,let us give a short outline of the rest of the paper.The AdS4symmetry algebra and its N=1grading will be derived in section2.In section3we recast these algebras in the form of conformal and superconformal algebras and recall the relations between the conformal weights of the primary?elds in a superconformal multiplet.The AdS4superspace is constructed in section4.In section5we consider the Wess-Zumino model in AdS4and calculate the conformal dimensions of the corresponding boundary?elds.We refer our read-ers to the appendices A and B for information on Grassmann variables and the calculation of Killing spinors,respectively.

2Symmetry Algebra and its N=1Grading

The AdS Symmetries are easiest found considering the embedding(2).In fact,equation(2) is invariant under Lorentz transformations of the Minkowski?ve-space,which are of the form (y′)A=R A B y B,where the matrix R satis?es R TηR=ηand R?1?1>0.For the purposes of this paper we consider only the connected subgroup of such matrices,namely the Lie group SO(4,1).An in?nitesimal transformation,R=1+M,has the form

1

δy A=M A B y B=

2 ω0i+ω?1i ,b i=1

We would like to add two remarks at this point.First,the validity of equation(12)is conditional upon the fact that we grade the?ve-dimensional Minkowski algebra.For higher dimensions(e.g.AdS5)one would have to introduce additional bosonic operators to obtain closure of all Jacobi identities[29].Second,the equations(8),(11)and(12)de?ne the complex superalgebra B(0/2),whose real form is osp(1,4)[30].Unfortunately,osp(1,4) does not contain so(4,1)in its even part,which means in other words that no Majorana spinors exist for our Minkowski?ve-space.However,osp(1,4)contains so(3,2),which is the symmetry group of AdS4with Minkowski signature.Resorting to a Wick rotation at the end to make our results valid,we shall ignore this fact and formally carry out the analysis. 3Conformal and Superconformal Algebra

As mentioned in section2,the AdS symmetry group acts as the conformal group on the AdS boundary.In this section,we shall for completeness explicitely show the isomorphisms be-tween the d=3conformal algebra and so(4,1)as well as between their N=1superalgebras. Let us introduce the conformal basis of so(4,1)by de?ning

D=M?10,K i=M0i+M?1i,

L ij=M ij,P i=M0i?M?1i.(13) Then,an element M∈so(4,1)takes the form

M=1

2

ωij L ij,(14)

with the parameters a i,b i andλgiven by equation(10).One easily?nds from equation(13) the commutation relations of D,P i,K i and L ij,which are given by

[D,P i]=?P i,

[D,K i]=K i,

[L ij,P k]=δjk P i?δik P j,

[L ij,K k]=δjk K i?δik K j,

[P i,K j]=2(δij D?L ij),

[L ij,L kl]=δil L jk+δjk L il?δik L jl?δjl L ik,

[P i,P j]=[K i,K j]=[D,L ij]=0.

(15)

Equations(15)are the standard representation of the conformal algebra[31].

The N=1grading of the conformal algebra(15)is well known in the literature[29], but again,a direct comparison with the superalgebra given in section2seems useful.This is done by choosing a particular representation of the?ve-dimensional Cli?ord algebra of matrices?γA.Choosing

?γi= σi00?σi ,?γ0= 0110 ,?γ?1= 01?10 ,(16)

whereσi are the Pauli spin matrices and1is the2×2unit matrix,one easily?nds from the de?nition(13)the spinor representations of the conformal basis elements,which are

?S(L

ij )=

1

2 100?1

,

?S(P

i

)= 00σi0 ,?S(K i)= 0?σi00 .(17) Splitting the spinor operator Qαinto two2-component spinors,

Qα= q s ,

we?nd from equation(11)the commutators

[L ij,qα]=?1

2

(σij s)α,

[D,qα]=?1

2

sα,

[P i,qα]=0,[P i,sα]=?(σi q)α,

[K i,qα]=(σi s)α,[K i,sα]=0.

(18)

Furthermore,the charge conjugation matrix?C has the form

?C= 0c c0 ,

where c is the charge conjugation matrix in three dimensions.Hence,using the identity

?S AB M

AB

=?2?S(D)D+?S(K i)P i+?S(P i)K i+?S(L ij)L ij,(19) equations(12)and(17)yield the anticommutators

{qα,qβ}=2 σi c?1 αβP i, {sα,sβ}=?2 σi c?1 αβK i, {qα,sβ}= 2c?1D?σij L ij αβ.(20)

Equations(15),(18)and(20)form the N=1superconformal algebra in three dimensions [29].

Obviously,the operators L ij,P i and qαform the three-dimensional N=1Poincar′e superalgebra.Let us therefore consider a scalar super-Poincar′e multiplet consisting of the scalar?elds O and F and the spinor?eldχ,which satisfy the supersymmetry relation

qαO=χα,

qαχβ=(σi c?1)αβ?i O+(c?1)αβF,

qαF=?(σi?iχ)α.

(21)

Imposing conformal symmetry on the multiplet means that the scaling dimensions of the ?elds must satisfy

1

?O=?χ+

ωAB M AB.In the case of

2

g0∈H,i.e.an even transformation,equation(24)takes the form e M eΘh(x)=eΘ′e M′h(x), where M′andΘ′are determined by the Baker-Campbell-Hausdor?formula.For in?nitesimal M one?nds M′=M andΘ′=Θ+[M,Θ].By de?nition,the even part e M h(x)=h(x′) yields equation(9)and thus does not contain new information,whereas the odd part yields a linear transformation of the Grassmann coordinates,namely

1

δ?ξα=

Θ2?1

3

and

M= ?Θ24 ∧R,(28) where the notation

1∧Y=Y,X∧Y=[X,Y],X2∧Y=[X,[X,Y]],etc.

has been used.Equations(27)and(28)are evaluated explicitly using the anti-commutator (12)and various Fierz identities listed in appendix A,leading to

δ?ξα=?εα 1?59(?ξ?ξ)2 ?1

6

?ξ?ξ (?ξ?S AB?ε)M AB,(30) respectively.The transformation formula(29)can be simpli?ed by de?ning

?θ=?ξ 1?1

2

ωAB(?S AB)αβ?θβ,(32) the odd transformation laws,equations(29)and(30),become

δ?θα=?εα 1??θ?θ?1

2

?θ?θ (?θ?S AB?ε)M AB,(34) https://www.sodocs.net/doc/9b2275787.html,ing equations(14),(19)and(6)one?nds

λ=? 1+1

2 1+1

2 1+1

2?θ?θ ?θS ij?ε .

(35)

Thus,the supersymmetry transformationδxμis given by equation(9)using the parame-ters of equation(35).Although this solves the problem of?nding the superspace transforma-tions,a space-time covariant formulation would be much more desirable.Such a formulation involves the Killing spinors,which are calculated in appendix B.In fact,it is easy to show from equation(B.6)that the quantity?ηΛΓμΛ?1?εis a Killing vector.On the other hand, because alsoδxμis a Killing vector and is linear in?ε,it must have exactly this form with?ηbeing a function of?θonly.A direct comparison using equations(9),(35),(B.3),(B.4)and (B.6)shows that

δxμ= 1+1

g(x)=d4x(x0)?4 is in itself invariant under any variable transformation,i.e.also under the supersymmetry transformation(36).For the fermionic part of the integral measure,let us make the ansatz d4?θρ(?θ)and demand that it be invariant under the transformation?θ→?θ′=?θ+δ?θ,where δ?θis given by equation(33).From equation(33)follows that

d4?θ′=d4?θ 1?2 1+?θ?θ ?θ?ε .(37) Multiplying equation(37)withρ(?θ′)and expanding to terms linear in?εwe?nd the equation

1??θ?θ?1

ρ=2?θα(1+?θ?θ)ρ,

??θα

whose solution up to a multiplicative constant is

3

ρ(?θ)=1+?θ?θ+

g(x)d4?θ 1+?θ?θ+3

5The Wess-Zumino Model

Let us start this section with the expansion of a chiral super?eld in powers of the Grassmann variables?θin order to identify its scalar and spinor?eld contents.Because of the existence of previous work[33,34,26,27]only the results will be given.However,it should be noted that our derivation di?ers in some points from[33,34].Keck[33]coupled a spinor ?eld directly to the SO(4,1)spinor variable?ξof section4,thereby demanding that the spinor?eld too be an SO(4,1)instead of a Lorentz spinor.On the other hand,Ivanov and Sorin[34]considered the Killing spinorθ(see appendix B)as the independent Grassmann variable,which can directly be coupled to a Lorentz spinor?eld.However,the complicated transformation rule forθunder supersymmetry transformations is a minor drawback of their very complete formulation,which led us to consider the SO(4,1)spinor?θas the independent superspace variable and realize the coupling to Lorentz spinor?elds via a matrixΛ(x),which is calculated in appendix B.We feel that this treatment combines the nice features of both references,[33]and[34].In addition,it yields the Killing spinorθas a side product.

The Wess-Zumino multiplet is given by the scalar?elds A,B,F and G and the Dirac spinor?eldψ.Their supersymmtry transformations are easiest found by considering chiral super?elds.Therefore,let us de?ne

A=A L+A R,B=A L?A R,

F=F L+F R,G=F L?F R,(40) and let us introduce the chiral projection operators

L=1

2

(1+i?γ?1).(41)

Then,the left and right handed chiral super?elds are given by ΦL(x,?θ)=A L+?θΛLψ+(?θΛLΛ?1?θ)F L

?i2(?θ?θ)?θΛ/DLψ+1

2(?θΛ?ΓμΛ?1?θ)DμA R+

1

8

(?θ?θ)2DμDμA R,(43)

respectively.It is straightforward to show using the transformation rules(33)and(36)that the supersymmetry transformations of the left handed super?eld components are given by

δA L=?εLψ,

δ(Lψ)=?2L(/DA L+F L)ε,

δF L=εLψ?ε/DLψ,

(44)

where we introduced the Killing spinorεα=(?εΛ)α.It takes somewhat more e?ort to show that all terms in the expansion(42)transform correctly.To?nd the supersymmetry transformations of A R,F R and Rψ,simply replace L with R in equation(44).

For the Wess-Zumino model we also introduce“conjugate”super?elds by de?ning

ˉΦ

R

=ˉA L+θRˉψ+(θRθ)ˉF L+···,

ˉΦL =ˉA R+θLˉψ+(θLθ)ˉF R+···,

(45)

where we used the Killing spinorθα=(?θΛ)αand where the dots indicate terms similar to those in equations(42)and(43).

A manifestly supersymmetric action is then given by the expression

S= d4x

g 1

g 1

2.Therefore,let us

not exclude the irregular modes for the scalar?elds.Then the conformal dimensions of the

boundary operators are given by[4,6,9]

?A=3

2 +m ,

?B=3

2?m

,

?ψ=

3

2,the irregular mode must be used for B in order to make

this identi?cation.

In conclusion,we found for a simple example that the AdS/CFT correspondence relates ?elds of AdS supersymmetry multiplets to the primary?elds of superconformal multiplets. This fact was derived by explicitly constructing the AdS supersymmetric model and com-paring its predictions with the relations expected from super CFT.In some cases,irreg-ular modes must be considered for AdS?elds,changing the standard prescription of the AdS/CFT correspondence.This conclusion stems from both,a pure AdS point of view and the AdS/CFT correspondence.We feel that a more general treatment should be attempted in future work.

Acknowledgements

We are very grateful to D.Z.Freedman for pointing out to us the importance of using the irregular modes,without which we would have drawn the wrong conclusion.

This work was supported in part by a grant from NSERC.W.M.is grateful to Simon Fraser University for?nancial support.

A Spinor Grassmann Variables

This appendix summarizes our notations and useful formulae for Grassmannian spinor vari-ables in?ve dimensions.We shall concentrate on facts which do not depend on the signature of the?ve-dimensional metric,thus avoiding explicit matrix representations and the intro-duction of complex conjugate spinors.Most of the following is derived from information on Cli?ord algebras and their representations,which can be found in[30].

A spinor?θhas components?θα(α=1,2,3,4),which are Grassmannian variables,i.e.the components of any two spinors?θand?ηsatisfy

?θα,?ηβ =0.(A.1)

Spinor matrices usually carry an upper and a lower index,such asδαβ,(?γA)αβetc.How-ever,indices can be lowered and raised with the charge conjugation matrix and its inverse,

respectively:

?θ

α

=?Cαβ?θβ,?θα=(?C?1)αβ?θβ.(A.2) For D=5the charge conjugation matrix is anti-symmetric.One can now de?ne the scalar product of two spinors by

?η?θ=?ηα?θα=??ηα?θα=?θ?η.(A.3) The vector space of4×4matrices with only lower indices is spanned by16matrices, which can conveniently be chosen to be i)the anti-symmetric charge conjugation matrix?C, ii)the5anti-symmetric matrices(?C?γA)and iii)the10symmetric matrices(?C?S AB)[36].The symmetry properties of the latter two follow directly from

?C?γ

A

=?γT A?C,(A.4)

?C?S

AB

=??S T AB?C.(A.5) One can easily derive the matrix identity

δαγδβ

δ=?

1

4

(?C?γA)γδ(?γA?C?1)αβ?1

4

(?θ?θ)(?η?ε)?

1

2

DνδxμΣμνψ

As the parameters a i ,b i ,λand ωij in δx are independent,equation (B.1)yields the following system of equations for Λ:

?S (P i )Λ+?i Λ=0,

?S

(D )Λ+x μ?μΛ=

0,?S (K i )Λ?2x μΛS iμ+ 2x i x μ?μ?x 2?i Λ=0,?S (L ij )Λ?ΛS ij +(x i ?j ?x j ?i )Λ=0.

(B.2)

The solution of equations (B.2)is not unique,but any solution will su?ce.A solution of equations (B.2)is

Λ(x )=√2(1+γ0)?1x 0(1?γ0)+x i

x 0

γi (1?γ0).(B.3)

It is also useful to know the inverse Λ?1,which is easily found to be

Λ?1(x )=1x 0(1+γ0)?√2(1?γ0)+

x i

x 0

γi (1?γ0).(B.4)

Consider the spinor (?θΛ)α,which by construction is a Lorentz spinor.Since ?θ?η=?θΛΛ?1?η,one ?nds

(Λ?1?θ)α=(C ?1)αβ(?θΛ)β,(B.5)

where C ?1is the inverse of the charge conjugation matrix C for Lorentz spinors.

Equation

(B.5)yields C =ΛT ?C Λ,which,together with equation (B.3),leads to C =??C .

Finally,one can check explicitly from the expression (B.3)that

D μ(?θΛ)α=1

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