Scalar Field Theory in the AdS/CFT Correspondence Revisited

Transcription

1 Scalar Fiel Theory in the AS/CFT Corresponence Revisite arxiv:hep-th/ v4 Feb 000 Pablo Minces an Victor. Rivelles Universiae e São Paulo, Instituto e Física Caixa Postal CEP São Paulo - Brazil Abstract We consier the role of bounary conitions in the AS + /CFT corresponence for the scalar fiel theory. Also a careful analysis of some limiting cases is presente. We stuy three possible types of bounary conitions, Dirichlet, Neumann an mixe. We compute the two-point functions of the conformal operators on the bounary for each type of bounary conition. We show how particular choices of the mass require ifferent treatments. In the Dirichlet case we fin that there is no ouble zero in the two-point function of the operator with conformal imension. The Neumann case leas to new normalizations for the bounary two-point functions. In the massless case we show that the conformal imension of the bounary conformal operator is precisely the unitarity boun for scalar operators. We fin a one-parameter family of bounary conitions in the mixe case. There are again new normalizations for the bounary two-point functions. For a particular choice of the mixe bounary conition an with the mass square in the range /4 < m < /4 + the bounary operator has conformal imension comprise in the interval [, ]. For mass square m > /4 + the same choice of mixe bounary conition leas to a bounary operator whose conformal imension is the unitarity boun. PACS numbers:.0.kk.5.mf Keywors: AS/CFT Corresponence, Bounary Conitions, Holographic Principle pablo@fma.if.usp.br rivelles@fma.if.usp.br

2 Introuction Since the proposal of Malacena s conjecture, which gives a corresponence between a fiel theory on anti-e Sitter space (AS) an a conformal fiel theory (CFT) on its bounary [], an intensive work has been evote to get a eeper unerstaning of its implications. In particular, a precise form to the conjecture has been given in [][3]. It reas ( ) Z AS [φ 0 ] = Dφ exp ( I[φ]) Z CFT [φ 0 ] = exp xφ 0, () φ 0 Ω where φ 0 is the bounary value of the bulk fiel φ which couples to the bounary CFT operator. This allows us to obtain the correlation functions of the bounary CFT theory in imensions by calculating the partition function on the AS + sie. The AS/CFT corresponence has been stuie for the scalar fiel [3][4][5][6], the vector fiel [3][5][7][8], the spinor fiel [7][9][0], the Rarita-Schwinger fiel [][][3], the graviton fiel [4][5], the massive symmetric tensor fiel [6] an the antisymmetric p-form fiel [7][8]. In all cases Dirichlet bounary conitions were use. Several subtle points have been clarifie in these papers an all results len support to the conjecture. In a broaer sense Malacena s conjecture is a concrete realization of the holographic principle [9][0]. We expect that any fiel theory relationship in AS space must be reflecte in the borer CFT. An example of this is the well known equivalence between Maxwell-Chern-Simons theory an the self-ual moel in three imensional Minkowski space []. This equivalence hols also in AS 3 an using the AS/CFT corresponence we have shown that the corresponing bounary operators have the same conformal imensions [8]. Another situation involves massive scalar fiels in AS spaces. If the scalar fiel has mass-square in the range /4 < m < /4 + then there are two possible quantum fiel theories in the bulk []. The AS/CFT corresponence with Dirichlet bounary conition can easily account for one of the theories. The other one appears in a very subtle way by ientifying a conjugate fiel through a Legenre transform as the source of the bounary conformal operator [3]. The existence of two conjugate bounary operators has been first pointe out in [4]. Since a fiel theory is etermine not only by its Lagrangian but also by its bounary terms in the action we expect that the AS/CFT corresponence must be sensitive to these bounary terms. This is easily seen to be true by computing the left-han sie of Eq.() for a classical fiel configuration. All that is left is a bounary term. If we start with ifferent bounary terms in the action then we obtain ifferent correlation functions on the right-han sie.

3 The origin of bounary terms in the action is ue to the variational principle. In orer to have a stationary action bounary terms, which will epen on the choice of the bounary conitions, must be introuce. The importance of these bounary terms for the AS/CFT corresponence was recognize in the case of spinor fiels where the action is of first orer in erivatives an the classical action vanishes on-shell [5]. They also playe an important role in the case of Chern-Simons theory [8]. Therefore it is crucial to unerstan the implications of ifferent types of bounary conitions for the same theory since they in general imply ifferent bounary terms. In this work we will stuy the role of ifferent types of bounary conition for the scalar fiel theory. We will consier Dirichlet an Neumann bounary conitions, an a combination of both of them which we will call mixe bounary conition. Each type of bounary conition requires a ifferent bounary term. We will show that the mixe bounary conitions are parametrize by a real number so that there is a one-parameter family of bounary terms consistent with the variational principle. We will also show that ifferent types of bounary conition give rise to ifferent conformal fiel theories at the borer. For the scalar fiel this was somehow expecte. The two solutions foun in [] correspon to two ifferent choices of energy-momentum tensor. Both of them are conserve an their ifference gives a surface contribution to the isometry generators. Although these two solutions were foun in the Hamiltonian context by requiring finiteness of the energy they will reappear here by consiering ifferent types of bounary conition which amounts to ifferent bounary terms in the action. We can also look for the asymptotic behavior of the scalar fiel near the bounary accoring to the chosen type of bounary conition. For the Dirichlet bounary conition it is well known that the scalar fiel behaves as x / /4+m 0 near the borer at x 0 = 0. There is no upper restriction on the mass in this case. It correspons to one of the solutions foun in [] an gives rise to a bounary conformal operator with conformal imension / + /4 + m. We will show that for a particular choice of mixe bounary conition an when the mass square is in the range /4 < m < /4 + the scalar fiel behaves as x /+ /4+m 0 near the borer. It correspons precisely to the secon solution of [] an gives rise to a bounary conformal operator with conformal imension / /4 + m. Note that the upper limit for the mass square /4 + is consistent with the unitarity boun ( )/. Another important point that we will show is the existence of bounary conitions which give rise to bounary conformal operators for which the unitarity boun ( )/ is reache. They correspon to a massless scalar fiel with Neumann bounary

4 conition or to a massive scalar fiel with m > /4 + with a particular choice of the mixe bounary conition (the same choice which gives the bounary operator with conformal imension / /4 + m ). In this way, using ifferent bounary conitions, we obtain all scalar conformal fiel theories allowe by the unitarity boun. We will also analyze carefully two cases where the mass of the scalar fiel takes special values. In some cases the usual expansion of the moifie Bessel functions in powers of x 0 breaks own an we must use expansions involving logarithms. When m = /4 it gives rise to the asymptotic behavior x / 0 ln x 0 an the two-point function is obtaine without troubles. This is to be contraste with the usual limiting proceure where the mass goes to m = /4 but the two-point function has a ouble zero in the limit [3]. The other case correspons to /4 + m integer but non-zero. In this case we just reprouce the known results. We shoul stress the fact that the use of ifferent types of bounary conition (for given values of m an ) allows us in general to get bounary two-point functions with ifferent normalizations. This will affect the three-point an higher-point functions. Maybe this is relate to the fact that AS an fiel theory calculations agree up to some imension epenent normalization factors [6] but we will not iscuss this further. The paper is organize as follows. In section we fin the bounary terms corresponing to each type of bounary conition. In section 3 we consier the Dirichlet case while in Section 4 we treat Neumann bounary conitions. In section 5 we consier mixe bounary conitions. Finally section 6 presents our conclusions. In appenix A we list all bounary two-point functions that we compute an appenix B contains some useful formulae. The Variational Principle We take the usual Eucliean representation of the AS + in Poincaré coorinates escribe by the half space x 0 > 0, x i R with metric The action for the massive scalar fiel theory is given by I 0 = s = x µ x µ. () x 0 µ=0 + x g ( g µν µ φ ν φ + m φ ), (3) 3

5 an the corresponing equation of motion is ( m ) φ = 0. (4) The solution which is regular at x 0 reas [5] φ(x) = k k x (π) e i x 0 a( k) K ν (kx 0 ), (5) where x = (x,..., x ), k = k, K ν is the moifie Bessel function, an From Eq.(5) we also get 0 φ(x) = ν = k k x (π) e i x 0 a( k) 4 + m. (6) [( ) ] + ν K ν (kx 0 ) kx 0 K ν+ (kx 0 ). (7) In orer to have a stationary action we must supplement the action I 0 with a bounary term I S which cancels its variation. The appropriate action is then I = I 0 + I S. (8) In orer to capture the effect of the Minkowski bounary of the AS +, situate at x 0 = 0, we first consier a bounary value problem on the bounary surface x 0 = ǫ > 0 an then take the limit ǫ 0 at the very en. Then the variational principle applie to the action I gives δi = x ǫ + 0 φ ǫ δφ ǫ + δi S = 0, (9) where φ ǫ an 0 φ ǫ are the value of the fiel an its erivate at x 0 = ǫ respectively. This equation will be use below to fin out the appropriate bounary term I S for each type of bounary conition. For Dirichlet bounary conition the variation of the fiel at the borer vanishes so that the first term in Eq.(9) also vanishes an the usual action I 0 is alreay stationary. Making use of the fiel equation the action I takes the form I D = + x µ ( g φ µ φ) = x ǫ + φ ǫ 0 φ ǫ. (0) 4

6 It is to be unerstoo that 0 φ ǫ in Eq.(0) is evaluate in terms of the Dirichlet ata φ ǫ. To consier Neumann bounary conitions we first take a unitary vector which is inwar normal to the bounary n µ (x 0 ) = (x 0,0). The Neumann bounary conition then fixes the value of n µ (ǫ) µ φ ǫ n φ ǫ. The bounary term to be ae to the action reas I S = + x µ ( g g µν φ ν φ) = x ǫ + φ ǫ 0 φ ǫ, () so that we fin the following expression for the action at the bounary I N = x ǫ φ ǫ n φ ǫ. () Here φ ǫ is to be expresse in terms of the Neumann value n φ ǫ. Notice that the on-shell value of the action with Neumann bounary conition Eq.() iffers by a sign from the corresponing action with Dirichlet bounary conition Eq.(0). We now consier a bounary conition which fixes the value of a linear combination of the fiel an its normal erivative at the borer φ(x) + αn µ µ φ(x) ψ α (x). (3) We will call it as mixe bounary conition. Here α is an arbitrary real but non-zero coefficient. In this case the surface term to be ae to the action is IS α = α + x µ ( g g µν ν φ n ρ ρ φ) = α x ǫ + 0 φ ǫ 0 φ ǫ, (4) an we fin the following expression for the action at the bounary I α M = x ǫ + ψ α ǫ 0 φ ǫ. (5) Clearly 0 φ ǫ in the above expression must be written in terms of the bounary ata ψ α ǫ. We then have a one-parameter family of surface terms since the variational principle oes not impose any conition on α. In this way the value of the on-shell action Eq.(5) also epens on α. In the following sections we will consier each bounary conition separately. 5

7 3 Dirichlet Bounary Conition We begin by recalling the main results for the Dirichlet case [4][5]. Let φ ǫ ( k) be the Fourier transform of the Dirichlet bounary value of the fiel φ(x). From Eq.(5) we get a( k) = ǫ φ ǫ ( k), (6) K ν (kǫ) an inserting this into Eq.(7) we fin 0 φ ǫ ( x) = y φ ǫ ( y) k (π) e i k ( x y) ǫ [ + ν kǫ K ] ν+(kǫ). (7) K ν (kǫ) Then the action Eq.(0) reas I D = x y φ ǫ ( x) φ ǫ ( y) ǫ k (π) e i k ( x y) [ + ν kǫ K ] ν+(kǫ). (8) K ν (kǫ) The next step is to keep the relevant terms in the series expansions of the Bessel functions an to integrate in k. We consier first the case ν 0 that is m. 4 For completeness we list the relevant moifie Bessel functions in Appenix B. For ν not integer we make use of Eqs.(90,94), whereas for ν integer but non-zero we use Eqs.(9,95). In both cases we get the same result I ν 0 D = ν π Γ( + ν) Γ(ν) x y φ ǫ ( x) φ ǫ ( y) ǫ (ν ) x y ( +ν) +, (9) where the ots stan for either contact terms or higher orer terms in ǫ. Taking the limit [5] lim ǫ 0 ǫν φǫ ( x) = φ 0 ( x), (0) to go to the borer an making use of the AS/CFT equivalence in the form exp ( I AS ) we fin the following two-point function ( exp ) x ( x) φ 0 ( x), () ν 0 D ( x) ν 0 D ( y) = ν π Γ( + ν) Γ(ν). () x y ( +ν) 6

8 Then the conformal operator ν 0 D on the bounary CFT has conformal imension +ν. From Eq.(0) we fin that near the borer φ behaves as x/ ν 0 φ 0 ( x) as expecte. In this way we have extene the results of [4][5] to the case ν integer but non-zero. For future reference we note that in the particular case m = 0, that is ν =, Eq.() reas ν= D ( x) ν= D ( y) = π Γ() Γ( ), (3) x y so that the operator ν= D has conformal imension. Now we consier the case ν = 0, that is m =. Since the two-point function 4 Eq.() has a ouble zero for ν = 0 it was argue [3] that the correct result can be foun by introucing a normalization on the bounary operator. Instea we will make use of the expansion for the Bessel function K 0. Using Eqs.(9,93) we get kǫ K (kǫ) K 0 (kǫ) = [ + (kǫ) ln ǫ = ln k ln ǫ ln ǫ + ( ǫ )] [ ln k ln + γ ln ǫ + ( ǫ )] +, (4) where the ots enote all other terms representing either contact terms in the twopoint function or terms of higher orer in ǫ. Notice that it is essential to separate the contributions of k an ǫ in the terms ln kǫ in orer to ientify the relevant contributions. Substituting in Eq.(8) we fin ID ν=0 = x y φ ǫ ( x) φ ǫ ( y) ǫ ln ǫ k (π) e i k ( x y) ln k +. (5) The integration in k is carrie out by making use of Eq.(95) yieling I ν=0 D = Γ ( ) 4π x y φ ǫ ( x) φ ǫ ( y) ǫ ln ǫ +. (6) x y To go to the borer we have to rescale φ ǫ using a factor of ln ǫ. This makes the rescaling somewhat arbitrary since any power of ǫ in ln ǫ woul o the job. So choosing the limit lim (ǫ ln ǫ) φ ǫ ( x) = φ 0 ( x), (7) ǫ 0 7

9 an making use of the AS/CFT equivalence Eq.() we fin the following two-point function ν=0 D ( x) D ν=0 ( y) = Γ ( ) π x y. (8) Then the conformal operator D ν=0 on the bounary CFT has conformal imension as expecte. As anticipate in [5] the scalar fiel approaches the bounary as x / 0 ln x 0 φ 0 ( x) ue to the logarithm appearing in the expansion of the Bessel function. 4 Neumann Bounary Conition Using the Neumann bounary conition we get from Eq.(7) a( k) = an substituting this in Eq.(5) we fin φ ǫ ( x) = y n φ ǫ ( y) Then the action Eq.() reas I N = x y n φ ǫ ( x) n φ ǫ ( y) ǫ ǫ n φ ǫ ( k) ( + ν)k ν(kǫ) kǫk ν+ (kǫ), (9) k k ( x y) (π) e i + ν kǫ K ν+(kǫ) K ν(kǫ) k k ( x y) (π) e i + ν kǫ K ν+(kǫ) K ν(kǫ). (30). (3) In orer to keep the relevant terms in the series expansions of the Bessel functions we must consier the massive an massless cases separately. In the massless case we have ν =. For o we make use of Eq.(90), whereas for even we use Eq.(9). In both cases we get for > + ν kǫ K ν+(kǫ) K ν(kǫ) = ( )(kǫ) +, (3) up to contact terms an higher orer terms in ǫ. Substituting this in Eq.(3) we fin I ν= N = x y n φ ǫ ( x) n φ ǫ ( y) ǫ 8 k (π) e i k ( x y) k +, (33)

10 an performing the integral in k we get I ν= N = Γ( ) 4π x y n φ ǫ ( x) n φ ǫ ( y) ǫ x y +, (34) where the ots stan for either contact terms or higher orer terms in ǫ. Taking the limit lim ǫ n φ ǫ ( x) = n φ 0 ( x), (35) ǫ 0 an making use of the AS/CFT equivalence of the form ( ) exp ( I AS ) exp x ( x) n φ 0 ( x), (36) we fin the following bounary two-point function ν= N ( x) ν= N ( y) = Γ() π x y. (37) Then for >, even or o, the conformal imension of the operator ν= N is precisely the unitarity boun. From Eq.(35) we fin that near the borer the scalar fiel goes as x /+ 0 n φ 0 ( x). Comparing Eqs.(3,37) we see that the conformal imensions of the bounary operators for the massless Dirichlet an Neumann cases are ifferent an for the later case the unitarity boun is reache. For the massive scalar fiel, that is ν, we first consier the case ν 0 i.e. m. We have again to consier separately the cases with ν not integer an ν 4 integer but non-zero. In both cases we fin I ν 0, N = ν π ( ν) Γ( + ν) Γ(ν) x y n φ ǫ ( x) n φ ǫ ( y) ǫ (ν ) x y ( +ν) +. (38) Taking the limit lim ǫ 0 ǫν n φ ǫ ( x) = n φ 0 ( x), (39) an making use of the AS/CFT equivalence Eq.(36) we fin the following bounary two-point function ν 0, N ( x) ν 0, N ( y) = ν π ( ν) Γ( + ν) Γ(ν). (40) x y ( +ν) 9

11 Then the operator ν 0, N has conformal imension + ν an the fiel φ goes to the borer as x / ν 0 n φ 0 ( x). Comparing Eqs.(,40) we notice that the normalizations of the bounary two-point functions corresponing to the massive ν 0 Dirichlet an Neumann cases are in general ifferent. Now we consier the case ν = 0 that is m =. Following the now usual steps 4 we get I ν=0 N Taking the limit = Γ ( ) π x y n φ ǫ ( x) n φ ǫ ( y) ǫ ln ǫ +. (4) x y lim ǫ 0 (ǫ ln ǫ) n φ ǫ ( x) = n φ 0 ( x), (4) an making use of the AS/CFT equivalence Eq.(36) we fin the following bounary two-point function ν=0 N ( x)ν=0 N ( y) = Γ ( ) π x y. (43) Then the conformal operator N ν=0 on the bounary CFT has conformal imension. Near the borer the scalar fiel has a logarithmic behavior x / 0 ln x 0 n φ 0 ( x). Again we fin that the normalizations of the bounary two-point functions corresponing to the ν = 0 Dirichlet an Neumann cases are in general ifferent. 5 Mixe Bounary Conition Using the mixe bounary conition Eq.(3) an again Eqs.(5,7) we get a( k) = ǫ ψ ǫ ( k) [β (α, ν) + αν]k ν (kǫ) αkǫk ν+ (kǫ), (44) where β(α, ν) is efine as β (α, ν) = + α ( ) ν. (45) 0

12 Substituting Eq.(44) into Eq.(7) we fin 0 φ ǫ ( x) = y ψ ǫ ( y) k (π) e i k ( x y) ǫ Using this we can write the action Eq.(5) as I M = x y ψ ǫ ( x) ψ ǫ ( y) ǫ k (π) e i k ( x y) + ν kǫ K ν+(kǫ) K ν(kǫ) β (α, ν) + αν αkǫ K ν+(kǫ) K ν(kǫ). (46) + ν kǫ K ν+(kǫ) K ν(kǫ) β (α, ν) + αν αkǫ K ν+(kǫ) K ν(kǫ) (47) As we shall see it is important to consier the cases β = 0 an β 0 separately in orer to fin out the relevant terms in the series expansions of the Bessel functions. Let us start with the case β = 0. For β = 0 we have α = /( ν) an m 0. We first consier the massive case with ν 0,. Again we have to stuy separately the cases with ν not integer an ν integer but non-zero. Let us first consier the case ν not integer. Making use of Eq.(90) with β = 0 we get an β (α, ν) + αν αkǫ K ν+(kǫ) K ν(kǫ) + ν kǫ K ν+(kǫ) K ν (kǫ) = = ν +, (48) ( ν) (kǫ) ν ν Γ( ν) Γ(ν) (kǫ) ν +. (49) Notice that for 0 < ν < the ominating term in the enominator of the r.h.s of Eq.(49) is (kǫ) ν. Substituting in Eq.(47) we get. I β=0,0<ν< M ( ) = ν ν Γ(ν) Γ( ν) x y ψ ǫ ( x) ψ ǫ ( y) ǫ ν k (π) e i k ( x y) k ν +. (50) Integration over k thus yiels ( ν I β=0,0<ν< M = 4π ) Γ( ν) Γ( ν) x y ψ ǫ ( x) ψ ǫ ( y) ǫ (ν+ ) x y ( ν) +. (5)

13 For ν > the ominating term in the enominator of the r.h.s of Eq.(49) is (kǫ) an Eq.(47) reas I β=0,ν> M = (ν ) ( ) ν x y ψ ǫ ( x) ψ ǫ ( y) ǫ k (π) e i k ( x y) k +. (5) Integration over k is carrie out for > thus giving I β=0,ν> M = (ν ) ( ) ν Γ( ) 4π x y ψ ǫ ( x) ψ ǫ ( y) ǫ x y +. (53) For the case ν integer an non-zero we make use of Eq.(9). The logarithmic terms vanish in the limit ǫ 0 an we fin that the same result Eq.(53) hols for ν integer an ν not integer. Now in the action Eq.(5) we take the limit whereas in the action Eq.(53) the limit to be taken is Using the AS/CFT equivalence exp ( I AS ) lim ǫ ν ψǫ ( x) = ψ 0 ( x), (54) ǫ 0 lim ǫ 0 ǫ ψ ǫ ( x) = ψ 0 ( x). (55) ( exp we get the following bounary two-point functions β=0,0<ν< M ( x) β=0,0<ν< M ( y) = π β=0,ν> M ( x) β=0,ν> M ( y) = (ν ) ) x ( x) ψ 0 ( x) ( ) ν Γ( ν) Γ( ν) ( ) ν Γ ( ) π, (56), (57) x y ( ν) x y. (58) Then the operators β=0,0<ν< M an β=0,ν> M have conformal imensions ν an respectively. For 0 < ν < the fiel φ approaches the bounary as x /+ν 0 ψ 0 ( x).

14 The erivation of the conformal imension ν for its associate bounary operator β=0,0<ν< M is a rather striking feature. It is worth noting that the upper constraint ν < in Eq.(57) is consistent with the unitarity boun. For ν > we foun a bounary operator whose conformal imension is the unitarity boun. Whereas we have alreay foun such a conformal imension in the massless Neumann case Eq.(37) we have here a ifferent normalization for the bounary twopoint function. We note that the behavior of the scalar fiel for small x 0 is as it shoul be. Now we consier the case ν = 0, that is m =, keeping still α = 4. We then fin ( I β=0,ν=0 M = Γ ) x ǫ y ψ 6π ǫ ( x) ψ ǫ ( y) +. (59) x y Taking the limit lim ǫ 0 ǫ ψǫ ( x) = ψ 0 ( x), (60) an making use of the AS/CFT equivalence Eq.(56) we get the following bounary two-point function β=0,ν=0 M ( x) β=0,ν=0 M ( ( y) = Γ ) 8π x y. (6) Then the conformal operator β=0,ν=0 M on the bounary CFT has conformal imension. Now the fiel φ goes to the borer as x/ 0 ψ 0 ( x) an no logarithm is present. We fin again that the normalization of the two-point function is ifferent from the corresponing ones of the Dirichlet an Neumann cases. Let us now consier the case when β 0. We stuy first the case ν 0. Again the cases ν not integer an ν integer but non-zero must be consiere separately. We first stuy the case ν not integer. Up to contact terms or higher orer terms in ǫ we fin an + ν kǫ K ν+(kǫ) K ν (kǫ) = β (α, ν) + αν αkǫ K ν+(kǫ) K ν(kǫ) ( ) [ ν ν Γ( ν) ν Γ(ν) = [ + ν α β(α, ν) β(α, ν) Γ( ν) Γ(ν) (kǫ) ν + ] (kǫ) ν +, (6) ]. (63) 3

15 Substituting in Eq.(47) we get I β 0,ν 0, M = Γ( ν) ν β (α, ν) Γ(ν) x y ψ ǫ ( x) ψ ǫ ( y) ǫ ν k (π) e i k ( x y) k ν +. (64) Integration over k yiels I β 0,ν 0, M = ν π β (α, ν) Γ( + ν) Γ(ν) x y ψ ǫ ( x) ψ ǫ ( y) ǫ (ν ) x y ( +ν) +. (65) Consier now the case ν integer an non-zero. We fin + ν kǫ K ( ) [ ] ν+(kǫ) = K ν (kǫ) ν ( ) ν ν ν Γ (ν) (kǫ)ν ln k +, (66) an β (α, ν) + αν αkǫ K ν+(kǫ) K ν(kǫ) Substituting in Eq.(47) we get = [ + ( ) ν ν α β(α, ν) β(α, ν) I β 0,ν 0, M = ( ) ν ν β (α, ν) Γ (ν) x y ψ ǫ ( x) ψ ǫ ( y) ǫ ν ] Γ (ν) (kǫ)ν ln k +. k (π) e i k ( x y) k ν ln k +. Making use of Eq.(95) we get Eq.(65) again. So both cases ν integer an ν not integer yiel the same result. Now taking the limit lim ǫ ν ψǫ ( x) = ψ 0 ( x), (69) ǫ 0 an making use of the AS/CFT corresponence Eq.(56) we fin the following bounary two-point function β 0,ν 0, M ( x) β 0,ν 0, M ( y) = ν π β (α, ν) 4 Γ( + ν) Γ(ν) (67) (68), (70) x y ( +ν)

16 so that the operator β 0,ν 0, M has conformal imension + ν. From Eq.(69) we fin that the behavior of φ for small x 0 is as expecte. Comparing Eqs.(,40,70) we conclue that the normalizations of the bounary two-point functions corresponing to the massive ν 0 Dirichlet, Neumann an β 0 mixe cases are ifferent. We now consier the case ν = 0 that is m =. We fin I β 0,ν=0 M = β (α, 0) Γ ( ) 4π x y ψ ǫ ( x) ψ ǫ ( y) ǫ ln ǫ 4 +. (7) x y Taking the limit lim (ǫ ln ǫ) ψ ǫ ( x) = ψ 0 ( x), (7) ǫ 0 an making use of the AS/CFT corresponence Eq.(56) we fin the following bounary two-point function β 0,ν=0 M ( x) β 0,ν=0 M ( y) Γ ( ) = β (α, 0) π x y, (73) so that the conformal operator β 0,ν=0 M on the bounary CFT has conformal imension. For small x 0 we fin a logarithmic behavior x / 0 ln x 0 ψ 0 ( x). Again the normalization of the two-point function is ifferent when compare to the corresponing ones of the Dirichlet, Neumann an β = 0 mixe cases. In the massless case we have ν =. For o we make use of Eqs.(90,94), whereas for even we use Eqs.(9,95). In both cases we get I ν= M = π Γ() Γ( ) x y ψ ǫ ( x) ψ ǫ ( y) +. (74) x y Taking the limit lim ψ ǫ ( x) = ψ 0 ( x), (75) ǫ 0 an making use of the AS/CFT equivalence Eq.(56) we get the following bounary two-point function ν= M ( x)ν= M ( y) = Γ() π Γ( ). (76) x y Then the operator ν= M has conformal imension. The scalar fiel goes to the borer as ψ 0 ( x) as expecte. Comparing Eqs.(3,76) we conclue that the bounary CFT s corresponing to the massless Dirichlet an mixe cases are equal. 5

17 6 Conclusions We have shown how ifferent bounary conitions in the AS/CFT corresponence allow us to erive bounary two-point functions which are consistent with the unitarity boun. We have also one a careful analysis of the particular cases when ν is an integer an we have shown that when ν = 0 there are no ouble zeroes in the two-point functions. In general the use of ifferent types of bounary conitions lea to bounary twopoint functions with ifferent normalizations. It is not clear at this point whether they are important or not. It is necessary to compute the three-point functions in orer to clarify if the ifferent normalizations are leaing to ifferent bounary CFT s. In the Neumann case the unitarity boun is obtaine for m = 0 while with mixe bounary conitions it is reache when β = 0 an m > /4+. The corresponing two-point functions have ifferent normalizations. The conformal imension / ν is obtaine in the case of mixe bounary conition with β = 0 an /4 < m < /4 +, an the normalization of the corresponing bounary two-point function iffers from the one foun in [3]. We have also trie to relate our formalism to the Legenre transform approach [3]. We coul think that both formalisms are relate through some fiel reefinition but this is not the case. It is not possible to reefine the scalar fiel in orer to turn a Dirichlet bounary conition into a Neumann or mixe one. If there is any relation between the two approaches it must be a very subtle one. Another important point is the interpretation of the new bounary conitions in the string theory context. Dirichlet bounary conitions are natural when thinking of the asymptotic behavior of the supergravity fiels reaching the borer of the AS space. Possibly Neumann an mixe bounary conitions are relate to more complex solutions involving strings an membranes reaching the borer in more subtle ways. This of course nees a more etaile stuy. 7 Acknowlegements P.M. acknowleges financial support by CAPES. V..R. is partially supporte by CNPq an acknowleges a grant by FAPESP. 6

18 8 Appenix A. Bounary Two-Point Functions The coefficients ν, α an β(α, ν) are efine in Eqs.(6,3,45) respectively. Let us also efine 8. Dirichlet Bounary Conition σ(ν) = ν. (77) ν 0 D ν= ν=0 D ( x) ν 0 D ( y) = ν π Γ( + ν) Γ(ν) x y ( +ν) (78) D ( x) ν= D ( y) = Γ() π Γ( ) (79) x y ( x)ν=0 D ( y) = Γ ( ) (80) x y π 8. Neumann Bounary Conition ν 0, N ( x) ν 0, N ( y) = σ (ν) ν= N ( x) ν= N ( y) = Γ() π ν=0 N ( x)ν=0 N ( y) = σ (0) 8.3 Mixe Bounary Conition ν 0 D ( x) ν 0 D ( y) (8) x y (8) ν=0 D ( x)ν=0 D ( y) (83) β=0,0<ν< M ( x) β=0,0<ν< M ( y) = σ (ν) Γ( ν) π Γ( ν) β=0,ν> M ( x) β=0,ν> M ( y) = σ (ν) (ν ) Γ ( ) π x y ( ν) (84) x y (85) 7

19 β 0,ν 0, M ( x) β 0,ν 0, M ν= M ( x) ν= M ( y) β 0,ν=0 M = ( x) β 0,ν=0 ( y) = ν= M ( y) = β (α, ν) D ( x) ν= D ( y) β (α, 0) ν 0 D ( x) ν 0 D ( y) (86) (87) ν=0 D ( x)ν=0 D ( y) (88) β=0,ν=0 M ( x) β=0,ν=0 M ( y) = σ (0) D ν=0 ( x) D ν=0 ( y) (89) 9 Appenix B. Some Useful Formulae 9. Series Expansions for the Moifie Bessel Functions K ν For ν not integer ( K ν (z) = ( ) n ( z ν z Γ(ν) Γ( ν) ( ) n ) z ν z ) n 0 n! Γ(n + ν). n 0 n! Γ(n + + ν) (90) For ν integer an non-zero K ν (z) = ( z ν ν ( n Γ(ν n) z n ( ) ) n=0 n! ) ( ) [ z ν ( ] ( n z z ( ) ν λ(n + ) + λ(ν + n + ) ) ln n 0 ) n! Γ(n + + ν), (9) where an γ is the Euler constant. For ν = 0 λ() = γ λ(n) = γ + K 0 (z) = n 0 n m= [ ( ] z ln λ(n + ) ) m ( z ) n (n ), (9) n! Γ(n + ). (93) 8

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