Double-trace deformations, holography and the c-conjecture
aa r X i v : . [ h e p - t h ] J u l MIT-CTP/4163
Double-trace deformations, holography andthe c -conjecture Andrea Allais
Center for Theoretical Physics,Massachusetts Institute of Technology,Cambridge, MA 02139, USA
Abstract
A double-trace deformation is the simplest perturbation of a con-formal field theory that has a gravity dual. In this paper we review theexisting results for the case in which the deformation is composed froma scalar operator, and extend them to the case of a spinor operator. Inparticular we check the validity of the c -conjecture along the RG flowinduced by the deformation, using both Cardy’s c -function and therecent proposal by Myers and Sinha of a c -function from entanglemententropy. An interesting way of perturbing a conformal field theory is to add theintegral of a local operator to the conformal action. A particularly simplechoice of perturbation, in the framework of the holographic correspondence,is to add the square of an operator O which is dual to a fundamental fieldin the gravity theory.This kind of perturbation is called double trace deformation and it hasbeen extensively studied on both sides of the duality, at least in the case inwhich the operator is a scalar operator. It has been shown [1][2][3][4] that thedual operation of perturbing the conformal field theory is to appropriatelychange the boundary conditions imposed on bulk fields. The two pointfunction of the operator O in the perturbed field theory has been shown[2][3] to agree at leading order in N on both sides of the duality, andthe renormalization group flow induced by the perturbation has been givena geometric description on the gravity side [5]. In fact, if the conformal The large N limit and the associated factorization of correlation functions are crucialin simplifying the effects of the perturbation. O is less then d/
2, the operator O is relevant ,and it triggers a renormalization group flow away from the conformal point,which can be shown to end on another conformal point, where the operatorhas dimension d − ∆ > d/
2. It has also been possible to compute thedifference in central charge between these two conformal field theories, andthe same result has been obtained from both sides of the duality. The changeturned out to be negative, in support of Cardy’s c -conjecture [6][5][7].The aim of this paper is to extend these results to the case of a spinorfield. In doing so we will find convenient to review some of the resultsobtained for the scalar field, and to discuss and clarify the way in whichthe perturbation is dualized as a change in the boundary conditions on thefields.The paper is organized as follows. In the next section we will discuss theissue of boundary conditions, and show how the two point function of theoperator O can be obtained from both sides of the duality. We will reviewthe known results on the scalar field and extend them to the spinor field.In the following section we will discuss the c -conjecture. We will reviewCardy’s proposal for the c -function as well as the proposal in [8] for con-structing a c -function from the entanglement entropy that is meaningful inany (even and odd) dimensions. We will show how both kinds of c -functionscan be computed from holography, and we will compute the difference intheir value between the two conformal points connected by the RG flow in-duced by the double trace deformation. We will carry out the holographiccomputation for both the scalar field and the spinor field, and, in even di-mension we will compute the same quantity also by conformal field theorymethods. Note added:
While our work was being completed, the paper [9] ap-peared, and we found some overlap between the results contained in thereand our treatment of the boundary conditions as discussed in Section 2.
Usually, boundary conditions on the fields in the gravity theory are imposedby hand. The pure conformal field theory corresponds to either Dirichletboundary conditions or mixed boundary condition with a specific value ofthe coefficient. Displacing this coefficient by a certain amount is dual to Because of the large N limit, the dimensions of operators are additive, so that thedimension of O is 2∆. > d/ Let us now quickly review the properties of the two point function of thescalar operator O as computed with CFT methods. Then we will showhow the same result arises from the gravity theory, by imposing appropriateboundary conditions through the variational principle. Let h i f denote the expectation value in the perturbed CFT hQi f = D Q e − f R d d x O ( x ) E CFT , and let ∆ be the conformal dimension of the operator O . By introducing aHubbard-Stratonovich auxiliary field and exploiting the large N limit, onecan obtain the two point function [7] D O ( k ) O † ( q ) E f = (2 π ) d δ ( k − q ) 1 f + k d − . If ∆ < d/ O added to the conformal action starts a renormalization groupflow away from the conformal point, which ends on another conformal point,where the operator has dimension d − ∆ > d/
2. In fact, at high energy, wehave D O ( k ) O † ( q ) E f ∼ (2 π ) d δ ( k − q ) k − d , The operator O is normalized so that hO ( x ) O ( y ) i CFT = A (∆) | x − y | where A (∆) =(4 π ) d/ − Γ( d/ − ∆) / Γ(∆)
3s one would expect from an operator of dimension ∆, whereas, at lowenergy, D O ( k ) O † ( q ) E f ∼ (2 π ) d δ ( k − q ) (cid:20) f − k d − f + . . . (cid:21) , that is, in position space, hO ( x ) O ( y ) i f ∼ f A ( d − ∆) 1 | x − y | d − ∆) , as is appropriate for an operator of dimension d − ∆.This derivation of the correlation function still works when ∆ > d/
2, butthis is an artifact of the large N approximation. In fact, in this case, theperturbation is irrelevant, and it means that the CFT is at the end point ofsome RG flow. However, the RG flow is irreversible, there are many highenergy theories that can flow to the same IR effective field theory (CFT +irrelevant operators). To be able to follow backwards the flow additionalinformation is needed that is not obtainable from the IR CFT, because theirrelevant operator will couple to other unspecified operators, turning on abeta function for them. The large N approximation discards the coupling of O to other operators, and gives the illusion of being able to follow backwardsthe RG flow. Now let us review how the gravity theory can yield the same physics. Thecorrelation function can be obtained from the generating functional W [ J, f ] = log D e R d d x [ − f O ( x )+ J ( x ) O ( x ) ] E CFT . Both the perturbation f O and the source term J O can be consideredperturbations of the conformal fixed point, so let us at first focus to the casein which they are zero, and the theory is purely conformal. Then the dualgeometry will be pure AdS space, for which we choose Poincar´e coordinates g = d r + P di =1 d x i r . Here we set the AdS radius L = 1. φ , dual to the scalaroperator O , and an appropriate action for the scalar field is S [ φ ] = Z ∞ ε d r Z d d x √ g (cid:20) g µν ∂ µ φ∂ ν φ + 12 m φ (cid:21) ++ (cid:18) d − ν (cid:19) Z d d x ε − d φ ( ε, x ) , where ν = r m + d . As usual, since the metric is singular at r = 0, it is necessary to cut offspacetime at r = ε . Eventually we will take the limit ε →
0, with appro-priate renormalization of some quantities. The boundary term, often calledholographic renormalization term, is necessary to have conformal invariancein the dual theory . In fact, when it is not present, the on-shell action doesnot have a good ε → .Now we can introduce the perturbations f O and J O . This breaksconformal invariance, and consequently the geometry changes. However,since the dimension of O will turn out to be∆ = ∆ − ≡ d − ν ≤ d , both perturbations are relevant, they become negligible in the u.v. limit,and we can still assume asymptotic AdS geometry. The perturbations are in-troduced as additional boundary terms in the action, so that the generatingfunctional is W − [ J, f ] = log Z [ D φ ] e − S − [ φ,J,f ] , where S − [ φ, J, f ] = S [ φ ] + Z d d x (cid:20)(cid:0) λε − ∆ − (cid:1) f φ ( ε, x ) − λε − ∆ − J ( x ) φ ( ε, x ) (cid:21) . Since we are interested in the two-point function, we may limit ourselves to the studyof a quadratic action. This is sufficient when ν ∈ [0 , More on this in the section about the c -conjecture. − . Later we will discuss how toobtain a conformal dimension greater than d/ ε − ∆ − are renormalization factors, needed to have a good ε → λ is a finite renormalization factor needed toexactly reproduce the field theory result. It is interesting to note that thesame factor renormalizes φ in both terms. Apart from these renormalizationissues, the way in which the perturbations of the conformal field theory areintroduced is very transparent, and it is easily generalized to more compli-cated perturbations.We use the saddle point approximation in evaluating the path integral,which amounts to disregarding subleading corrections in the large N limit.We have W − [ J, f ] = − min φ S − [ φ, J, f ] . We want to extremize S − . Let us first look at the variation of S . Wehave S [ φ + δφ ] − S [ φ ] = Z d r d d x √ g (cid:2) − g µν D µ D ν φ + m φ (cid:3) δφ −− Z d d x ε − d π ( ε, x ) δφ ( ε, x ) , where π ( r, x ) = ε d δS δ [ ∂ r φ ( r, x )] = (cid:18) r∂ r − d ν (cid:19) φ ( r, x ) . The variation of the other terms is easy to compute and, putting allterms together we have S − [ φ + δφ, J, f ] − S − [ φ, J, f ] = Z d r d d x √ g [e . o . m . ] δφ ++ Z d d xε − d h − π ( ε, x ) + λ ε ν f φ ( ε, x ) − λε d + ν J ( x ) i δφ ( ε, x ) . Then the field configuration φ J that extremizes the action is the one thatsatisfies ( (cid:0) − g µν D µ D ν + m (cid:1) φ J ( r, x ) = 0 − π J ( ε, x ) + λ ε ν f φ J ( ε, x ) = λε d + ν J ( x ) , so that W − [ J, f ] = − S − [ φ J , J, f ] . S − [ φ J , J, f ], bysubstituting the solution of the equations of motion. However, since we aremainly interested in the correlation function hO ( x ) O ( y ) i f = δ W − δJ ( x ) δJ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 , we will vary the sources first, and substitute later. Let us look at the varia-tion of W − when J changes: W − [ J + δJ, f ] − W − [ J, f ] = − S − [ φ J + δJ , J + δJ, f ] + S − [ φ J , J, f ]= Z d d xλε − d + ν φ J ( ε, x ) δJ ( x ) , where we have used the fact that the action is at a stationary point withrespect to variations of φ . Then we have hO ( x ) O ( y ) i f = λε − d + ν δφ J ( ε, x ) δJ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 . To proceed further we need the small r behavior of the solution φ J , solet us look at the equations of motion near the boundary (cid:2) − r ∂ r + ( d − r∂ r − r ∂ + m (cid:3) φ ( r, x ) = 0 . We can expand in plane waves in the transverse directions and find thatthe solution has the following small r behavior φ J ( r, k ) ∼ A J ( k ) h ( kr ) d − ν − a ( kr ) d + ν i for r → , where the constant a is fixed by the requirement that the solution be regularin the interior of space. For example, in the case of pure AdS geometry, wehave a = − − ν Γ( − ν )Γ( ν ) . Substituting the expansion in the definition of π we have π J ( r, k ) ∼ A J ( k ) h − aν ( kr ) d + ν i for r → . It is important that a > a is positive for ν ∈ [0 , φ J ( r, k ) = λJ ( k ) k − d + ν ( kr ) d − ν − a ( kr ) d + ν λ f [1 − a ( kε ) ν ] + 2 aνk ν , and we have D O ( k ) O † ( q ) E f = (2 π ) d λε − d + ν δφ J ( ε, k ) δJ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 = (2 π ) d δ ( k − q ) λ (cid:2) − a ( kε ) ν (cid:3) λ f [1 − a ( kε ) ν ] + 2 aνk ν . Setting λ = √ aν and taking the ε → − D O ( k ) O † ( q ) E f = (2 π ) d δ ( k − q ) 1 f + k d − − . How can the previous approach extend to operators of conformal dimen-sion greater than d/
2? First of all we remark that this is meaningful onlywhen f = 0, otherwise we would be following backwards in the RG an irrel-evant perturbation . Then we notice that the most naive approach wouldbe to just send ν → − ν . However it is easy to see that in this case the termsthat were suppressed in the ε → ν > W + [ J ] = − log Z [ D φ ] e − S + [ φ,J ] ≃ min φ S + [ φ, J ] , where S + [ φ, J, f ] = S [ φ ] + Z d d x h ε − d π ( ε, x ) φ ( ε, x ) + λε − ∆ + J ( x ) π ( ε, x ) i . Varying this action we have S + [ φ + δφ, J ] − S + [ φ, J ] = Z d r d d x √ g [e . o . m . ] δφ ++ Z d d xε − d h φ ( ε, x ) + λε d − ν J ( x ) i δπ ( ε, x ) . As with the field theory computation, it is formally possible to make the perturbation f work even when ∆ > d/
2, by adding a π term to the action, but we stress that this isan artifact of the large N limit. ( (cid:2) − g µν D µ D ν + m (cid:3) φ J ( r, x ) = 0 φ J ( ε, x ) = − λε d − ν J ( x ) . Using the same approach as before we have hO ( x ) O ( y ) i f = λε − d − ν δπ J ( ε, x ) δJ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) J =0 , and D O ( k ) O † ( q ) E f = (2 π ) d δ ( k − q ) λ aν k ν − a ( kε ) ν . Taking the ε → λ = (2 aν ) − / we recover the fieldtheory result with ∆ = d + ν and f = 0. We will now show how the results of the previous section extend to the caseof a spinor field.
Consider a conformal field theory containing a spinor field χ of dimension∆. The partition function of the perturbed theory is Z [ η, ¯ η ; f ] = Z D e − f ¯ χχ +¯ ηχ +¯ χη E CFT , where h i CFT denotes the expectation value in the unperturbed field theory,and where we used the short hand notation ¯ χη = Z d d x p h ( x ) ¯ χ ( x ) η ( x ) . We want to compute this partition function exploiting the large N limit,by the same trick used in [7] for the scalar field. We introduce a couple ofauxiliary, Grassmann-valued fields σ , ¯ σ , and we have Z [ η, ¯ η ; f ] = ˜ Z Z [ D σ ] [ D ¯ σ ] D e ¯ σσ − f ¯ χχ +¯ ηχ +¯ χη E CFT . h is the fixed background metric on which the field theory lives. f or η . Now we make the shift σ → σ + p f χ , ¯ σ → ¯ σ + p f ¯ χ , and we have Z [ η, ¯ η ; f ] = ˜ Z Z [ D σ ] [ D ¯ σ ] D e ¯ σσ +(¯ η + √ f ¯ σ ) χ +¯ χ ( η + √ fσ ) E CFT . Now we use the fact that, for large N , (cid:10) e ¯ ηχ +¯ χη (cid:11) ≃ e ¯ ηGη , where G denotes the convolution with h χ ¯ χ i CFT ( Gσ )( x ) = Z d d y h χ ( x ) ¯ χ ( y ) i CFT σ ( y ) . Then we have Z [ η, ¯ η ; f ] ≃ ˜ Z Z [ D σ ] [ D ¯ σ ] e ¯ σσ +(¯ η + √ f ¯ σ ) G ( η + √ fσ ) . Now we make the shift σ → σ − √ f G f G η , ¯ σ → ¯ σ − √ f G f G ¯ η , and we get Z [ η, ¯ η ; f ] = ˜ Z Z [ D σ ] [ D ¯ σ ] e ¯ σ (1+ fG ) σ +¯ η G fG η = Z det(1 + f G ) e ¯ η G fG η . (1)From this, taking derivatives with respect to the sources, we have h χ ( x ) ¯ χ ( y ) i f = G f G δ ( x − y ) . On flat space we have h χ ( x ) ¯ χ ( y ) i CFT = 1 B (∆) γ · ( x − y ) | x − y | , B (∆) = (4 π ) d/ − ∆ Γ( d/ − ∆ + 1 / / . Then the operator G is diagonal in the basis of momentum eigenstates G ( k, q ) = (2 π ) d δ ( k − q ) iγ · ˆ k k − d , where ˆ k = k/ | k | , and we have h χ ( k ) ¯ χ ( q ) i f = (2 π ) d δ ( k − q ) 1 f − iγ · ˆ k k d − . If ∆ < d/ χ has dimension ∆ to a conformal point in which the operator χ has dimension d − ∆. In the gravity theory, the operator χ is dual to a spinor field ψ . The actionfor the spinor field that is dual to a pure conformal field theory is S [ ψ, ¯ ψ ] = Z ∞ ε d r Z d d x √ g ¯ ψ (cid:20) (cid:16) −→ D/ − ←− D/ (cid:17) − m (cid:21) ψ ++ Z d d xε − d
12 ¯ ψψ (cid:12)(cid:12) r = ε . Also in this case, the holographic renormalization term is necessary tohave a good ε → − ≡ d/ − m < d/ , later we will show how to obtain a conformal dimension larger than d/ W [ η, ¯ η, f ] = log Z [ η, ¯ η, f ] = log D e R d d x [ − f ¯ χχ +¯ ηχ +¯ χη ] E CFT , This action is valid for m ∈ [0 , /
11e need to add some boundary terms to the action. We have W − [ η, ¯ η, f ] = log Z [ D ψ ] (cid:2) D ¯ ψ (cid:3) e − S − [ ψ, ¯ ψ,η, ¯ η,f ] ≃ − min ψ S − [ ψ, ¯ ψ, η, ¯ η, f ] , with S − [ ψ, η, f ] = S [ ψ ] + Z d d x (cid:2) ( λε − ∆ − ) f ¯ ψψ − λε − ∆ − (¯ ηψ + ¯ ψη ) (cid:3) . We want to extremize this action, so let us first look at the variation ofthe action S S [ ψ + δψ ] − S [ ψ ] = Z d r d d x h δ ¯ ψ ( −→ D/ − m ) ψ + ¯ ψ ( −←− D/ − m ) δψ i ++ Z d d xε − d (cid:2) δ ¯ ψ ( ε, x ) π ( ε, x ) + ¯ π ( ε, x ) δψ ( ε, x ) (cid:3) , where π ( r, x ) = − ε d δS δ [ ∂ r ¯ ψ ( r, x )] = P + ψ, ¯ π ( r, x ) = − ε d δS δ [ ∂ r ψ ( r, x )] = ¯ ψP − , and P ± = 1 ± γ r . Varying the action S − we have S − [ ψ + δψ, η ] − S − [ ψ, η ] = e . o . m . + Z d d xε − d δ ¯ ψ ( ε, x ) h π ( ε, x ) + λ ε m f ψ ( ε, x ) − λε d + m η ( x ) i + Z d d xε − d h ¯ π ( ε, x ) + λ ε m f ¯ ψ ( ε, x ) − λε d + m ¯ η ( x ) i δψ ( ε, x ) . The field configuration ψ η , ¯ ψ η that extremizes the action is the one thatsatisfies (cid:16) −→ D/ − m (cid:17) ψ η ( r, x ) = 0 π η ( ε, x ) + λ ε m f ψ η ( ε, x ) = λε d + m η ( x ) , and a similar set of equations for ¯ ψ . From now on we will not write explicitly the dependence on the conjugate fields. S − with respect to η and using the fact thatthe action is stationary for variations of ψ and ¯ ψ we get h χ ( x ) ¯ χ ( y ) i f = δ Wδη ( y ) δ ¯ η ( x ) = λε − d + m δψ η ( ε, x ) δη ( y ) . To proceed further we need at least the asymptotic behavior for small r of the solution to the equation of motion (cid:16) −→ D/ − m (cid:17) ψ = 0 , or, in coordinates (cid:20) γ r (cid:18) r∂ r − d (cid:19) + rγ · ∂ − m (cid:21) ψ ( r, x ) = 0 . To reduce clutter we will use the exact analytic solution in pure AdSspace, instead of just the asymptotic behavior, but it is understood that theresults do not rely on the assumption of having pure AdS background.It is easy to check by substitution that the most general solution to theequations of motion is ψ ( r, k ) = ( kr ) d +12 h K m − ( kr ) − K m + ( kr ) iγ · ˆ k i A ( k ) , where A ( k ) is a constant spinor that satisfies γ r A ( k ) = A ( k ) . Imposing the boundary conditions we have ψ η ( r, k ) = λε d + m (cid:16) rε (cid:17) d +12 ×× K m − ( kr ) − K m + ( kr ) iγ · ˆ kK m − ( kε ) + λ ε m f h K m − ( kε ) − K m + ( kε ) iγ · ˆ k i η ( k ) , and hence h χ ( k ) ¯ χ ( q ) i f =(2 π ) d δ ( k − q ) λ ε m ×× K m − ( kε ) − K m + ( kε ) iγ · ˆ kK m − ( kε ) + λ ε m f h K m − ( kε ) − K m + ( kε ) iγ · ˆ k i .
13o take the ε → K ν ( z ) ∼ a ν z ν + a − ν z − ν , a ν = 2 − ν − Γ( − ν ) . In the range m ∈ [0 , /
2] the leading contributions come from the terms( kε ) − m − and ( kε ) m − , so that, after some manipulations we have h χ ( k ) ¯ χ ( q ) i f = (2 π ) d δ ( k − q ) λ " a m − a − m − k m iγ · ˆ k + λ f − , and, upon setting λ = s a m − a − m − , the field theory result is recovered with ∆ = ∆ − .To obtain a conformal dimension greater than d/ W + [ η, ¯ η, f ] = − log Z [ D ψ ] (cid:2) D ¯ ψ (cid:3) e − S + [ ψ, ¯ ψ,η, ¯ η,f ] ≃ min ψ S + [ ψ, ¯ ψ, η, ¯ η, f ] , where S + [ ψ, η ] = S [ ψ ] − Z d d xε − d (cid:2) ¯ π ( ε, x ) ψ ( ε, x ) + ¯ ψ ( ε, x ) π ( ε, x ) (cid:3) + Z d d xλε − ∆ + [¯ η ( x ) π ( ε, x ) + ¯ π ( ε, x ) η ( x )] . Varying the action S + we have S + [ ψ + δψ, η ] − S + [ ψ, η ] = e . o . m . + Z d d xε − d δ ¯ π ( ε, x ) h − ψ ( ε, x ) + ε d/ − m η ( x ) i + Z d d xε − d h − ¯ ψ ( ε, x ) + ε d/ − m ¯ η ( x ) i δπ ( ε, x ) . The field configuration ψ η , ¯ ψ η that extremizes the action is the one thatsatisfies (cid:16) −→ D/ − m (cid:17) ψ η ( r, x ) = 0 ψ η ( ε, x ) = λε d − m η ( x ) . h χ ( x ) ¯ χ ( y ) i f = − λε − d − m δπ η ( ε, x ) δη ( y ) , and the solution that satisfies the boundary conditions has π η ( r, k ) = λε d − m (cid:16) rε (cid:17) d +12 K m − ( kr ) K m − ( kε ) − K m + ( kε ) iγ · ˆ k η ( k ) , so that h χ ( k ) ¯ χ ( q ) i f = − (2 π ) d δ ( k − q ) λ ε − m K m − ( kε ) K m − ( kε ) − K m + ( kε ) iγ · ˆ k . Taking the ε → h χ ( k ) ¯ χ ( q ) i f = (2 π ) d δ ( k − q ) λ " a − m − a m − k − m iγ · ˆ k − , and, upon setting λ = s a − m − a m − , we recover the field theory result for ∆ = d/ m , f = 0. c -conjecture We will now review some results on the holographic c -conjecture. Accordingto Zamolodchikov’s famous c -theorem [13], in a two dimensional field theorythere exists a function of all the possible couplings that is monotonically de-creasing along the trajectories of the renormalization group flow. Moreover,at the fixed points of the RG, where the theory is conformal, the value ofthis function coincides with the central charge of the CFT.In another famous paper [6] Cardy conjectured that the same may betrue for higher dimensional field theories, and proposed the following formfor the function c ≡ ( − d a d Z S d d d x √ h h ij (cid:10) T ij (cid:11) , The coefficient a d is a normalization factor. It is fixed by the requirement that, for asingle free massless scalar field, c = 1, in any dimension. In particular, with our definitionof the stress energy tensor, a free massless scalar field in two dimensions has (cid:10) T ii (cid:11) = − R π and hence a = 3. S d , with round metric h and where T is the stress energy tensor ofthe field theory.In two dimensions, Cardy’s definition reduces to Zamolodchikov’s c -function, but his definition is interesting also in higher dimensions. Infact, in even-dimensional conformal field theories, the conformal symme-try is anomalous, and hence the c -function can take a non-trivial (non-zero)value at a fixed point of the RG . Moreover Cardy gives evidence that the c -function may be monotonically decreasing along the RG trajectories .For odd dimensional conformal field theories, instead, there is no conformalanomaly, and the c -function is just zero at every fixed point.More recently, another possible definition of a c -function that works inany dimension and reduces to Zamolodchikov’s definition in two dimensionshas been proposed [8]. According to this proposal, one has to put the fieldtheory on the spacetime R × S d − , i.e. choose the boundary metric h = − d t + R dΩ d − and compute the entanglement entropy in the ground state of half the sphere.A certain subleading universal contribution to this entanglement entropy,which we will discuss later, is conjectured to be decreasing along the trajec-tories of the RG-flow .The renormalization group flow induced by double trace deformationsis a case in which the c -conjecture and the various proposals of c -functionscan be tested. In the case of perturbation by a scalar operator, it has beenpossible to compute the change in the value of Cardy’s c -function betweenthe two end points of the flow, by both holographic [5] and CFT [7] methods,showing that it is negative.In the following we will show how to compute the c -function from theholographic principle, review the results on the scalar field and extend themto the spinor field. We will show that the c -function is decreasing, in anydimension, for both the scalar and the spinor field. In addition, we will showthat the same results can obtained, in even dimension, by CFT methods. Often people still give to this value of c the name of central charge, even when dimen-sionality is higher than two. Recently a counterexample has been found [14] in 4 dimensions, in which this c -function increases. This probably means that the c -theorem in higher dimensions holdsonly under more stringent conditions than in two dimensions. In this paper d always refers to the total number of dimensions of the boundary fieldtheory. At least under some suitable conditions, to exclude the counterexample [14]. .1 Cardy’s c -function from holography Let us briefly review how the holographic computation of Cardy’s c -functioncan be accomplished [15][3]. The partition function Z of the field theorycan be computed from the dual gravity theory, following the holographicprinciple.In a semiclassical approach in which the metric is treated classically andthe matter content is given a full quantum treatment, we can write thegravity action as S gr [ g ] = 116 πG Z ε d r Z d d x √ g ( −R + 2Λ + 16 πGV [ g ]) , where r is the holographic direction, the boundary being at r = ε , and where V is the effective potential Z ε d r Z d d x √ g V [ g ] = − log Z [ D φ ] e − S [ φ,g ] . where φ represents the matter content of the theory.Then − log Z is given by the on-shell value of the gravity action, withthe boundary condition that, at r = ε , the transverse part of the metriccoincides with the background metric h of the dual field theory ε g ij ( ε, x ) = h ij ( x ) . In general, the matter content can act as a source for the gravitationalfield in an arbitrarily complicated way, but, for certain choices of the mat-ter action, the AdS geometry turns out to be an extremum of V [ g ]. Thisis the case, in particular, for the actions discussed before, when both theperturbation and the source term are set to zero. In these particular cases,if 2Λ + 16 πGV <
0, the geometry that extremizes the entire gravity actionis the AdS geometry, and hence the dual field theory is a conformal fieldtheory in its ground state.Adding a double trace deformation to the dual field theory breaks confor-mal invariance and triggers a renormalization group flow. Correspondingly, Since we are in euclidean space, curvature must come with the negative sign, becausethe Wick rotation of the Minkowski action has to be consistent with what is conventionallydone with the matter action [16]. The effective potential V is a divergent quantity, but its divergences are proportionalto geometric quantities, and can be absorbed in the renormalization of the correspondentterms in the gravity action This usually brings about higher derivatives terms, but herewe will stick to Einstein’s gravity.
17n the gravity theory, adding a boundary term to the matter action yieldsan effective potential that is not extremized by AdS geometry. As a conse-quence, also the geometry that extremizes the entire gravity action deviatesfrom pure AdS. If one could compute the new geometry, as a function ofthe renormalized parameter f , he could have a full geometric picture of theRG flow. In particular, he could obtain c ( f ), and explicitly check that itis monotonously decreasing. However, this program is probably too ambi-tious. What is possible to do is to compare the end points of the RG flow,i.e. compute the change in quantities when the matter action changes fromthe ∆ − action to the ∆ + action.In particular, we are interested in computing the change in the value of c between the two conformal points, so we want the boundary metric to bethe spherical metric h = R dΩ d . Then we have
R ∂∂R ( − log Z ) = Z d d x R ∂h ij ( x ) ∂R δδh ij ( x ) ( − log Z )= Z d d x h ij ( x ) 12 p h ( x ) (cid:10) T ij ( x ) (cid:11) , that is c = ( − d a d R ∂∂R ( − log Z ) . The metric that solves Einstein’s equations with the given boundaryconditions is the metric of AdS space in hyperbolic coordinates , g = L r d r + R r (cid:18) − L r R (cid:19) dΩ d , where L = − d ( d −
1) 12Λ + 16 πGV .
Clearly the scalar curvature R of the AdS metric is constant, and wehave −R + 2Λ + 16 πGV = 2 dL . In this section of the paper, we find convenient to write explicitly the AdS radius. − log Z = S gr [ g ] = 116 πG dL Z RL ε d r Z d d x √ g = 116 πG dL Z RL ε d r Z dΩ d LR d r d +1 (cid:18) − L r R (cid:19) d = 2 − d d Σ d L d − πG Z Lε R d rr − d − (cid:0) − r (cid:1) d , where Σ d is the surface of the d -sphere.The integrand contains negative powers of r , and the integral is clearlynot finite in the limit of ε →
0. The action needs to be regularized by addingsome ε -dependent boundary counterterms . The dependence on R is onlyin the integration limit, in the form εL/R . After regularization and the ε → R/L ,that could come from integrating the term proportional to 1 /r . Such a termis present only for even d . Explicitly, in even d , the R dependent term is − log Z → ( − d/ − d d Σ d (cid:18) d d (cid:19) L d − πG log R , and hence c = 2 − d d Σ d (cid:18) d d (cid:19) a d L d − πG , where Σ d is the surface of the d -sphere. In particular, in two dimension, thisreproduces the well known result [17] c = 3 L G .
When the effective potential changes by an amount∆ V = V + − V − , the relative change in central charge is∆ cc ≃ ( d −
1) ∆ LL ≃ − d −
12 16 πG ∆ V
2Λ + 16 πGV = 12 d πGL ∆ V = 1 N L d +1 ∆ V d , It can be shown that this can be done in a fully covariant way [15]. πGL d − = 1 N . c -function from holography According to the proposal of [8], we choose the boundary metric to be h = − d t + R dΩ d − , and the metric that satisfies Einstein equations with negative cosmologicalconstant and this boundary condition is g = L r d r + 1 r ( − (cid:18) L r R (cid:19) d t + R (cid:18) − L r R (cid:19) dΩ d − ) . We want to compute the entanglement entropy of half the sphere. This isproportional to the area of the minimal surface in the bulk whose boundaryis the equator of the sphere [18]. The minimal surface is clearly the diskthat cuts the AdS space in two. The metric induced on this surface is k = L r d r + R r (cid:18) − L r R (cid:19) dΩ d − , and its area is A = Z RL ε d r Z d d − x √ k . We are then instructed to discard all the contributions that are powerdivergent when ε →
0. When d is odd the result is finite, when d is eventhere is still a divergence proportional to log ε . The proposed c function isthe finite term in the first case, and the coefficient that multiplies log ε in thesecond. It is apparent that the computation is identical, modulo an overallpositive constant, to the one for Cardy’s c -function, but this proposal alsogives a meaning to the constant term that appears when d is odd. Since thearea A is proportional to L d − , exactly as Cardy’s c function, we find thatthe c -function constructed from the entanglement entropy also changes as∆ cc = 1 N L d +1 ∆ V d , for any d . This expression for N is true in N = 4 SYM theory. The coefficient may change inother theories with gravity duals. In this paper, d will always refer to the total number of dimensions of the boundaryfield theory. .3 Change in central charge from holography - scalar field Let us now review how the change in the effective potential can be computedfor the case of a scalar field [5]. According to our definition we have Z d r d d x √ g V = − log Z [ D φ ] e − R d r d d x √ g h ( ∂φ ) + m L φ i . Since we are computing a density, we are free to use any coordinates weprefer to parametrize the space, even if they don’t cover the whole space.Using Poincar´e coordinates g = L r d r + d X i =1 d x i ! is the most convenient choice.We are interested in computing the difference∆ V = V + − V − . Computing V directly is difficult, it is much easier to compute the derivativeof V with respect to m . At the Breitenlohner-Freedman bound m = − d / φ ∼ r ∆ ± of the fields become identical, andit can be argued [5] that also the respective potentials become equal. Thenwe have ∆ V ( m ) = Z m − d / d ( µ )∆ V ′ ( µ ) , where ∆ V ′ ( µ ) ≡ d V + d( m ) (cid:12)(cid:12)(cid:12)(cid:12) m = µ − d V − d( m ) (cid:12)(cid:12)(cid:12)(cid:12) m = µ . Taking the derivative with respect to m of both sides of the equationdefining V we have Z d r d d x √ g d V ± d( m ) = 12 L Z d r d d x √ g (cid:10) φ ( r, x ) (cid:11) , Here we use a quadratic action, i.e. we compute the effective potential at one loop.Moreover, for brevity we drop the boundary terms in the action. However, they stillcontrol the boundary conditions on the fields, and this will become important later in thederivation. d V ± d( m ) = 12 L G ± ( r, x ; r, x ) , where G is the propagator of the theory. Its computation is reported in theappendix, it has the form G ± ( r, x ; s, y ) = r d s d L d − Z d d k (2 π ) d e ik · ( x − y ) Z d ω ω J ± ν ( ωr ) J ± ν ( ωs ) ω + k . Then we haved V + d( m ) = 12 L d +1 Z d d k (2 π ) d
11 + k Z d ω ω d − J ν ( ω ) , where we have rescaled the integration variables appropriately. Both inte-grals can be done exactly in dimensional regularization, and we haved V + d( m ) = 12 L d +1 (4 π ) − d +12 Γ (cid:18) − d (cid:19) Γ (cid:0) d + ν (cid:1) Γ (cid:0) − d + ν (cid:1) . Now we subtract the expression with ν → − ν and set d = integer − ε .The divergent contributions cancel and we obtain a finite expression for∆ V ′ ( m ). After a few manipulations we get∆ V ′ ( m ) = − d − d L d +1 ν d − Y p =1 ( p − ν )( p + ν )for even d and∆ V ′ ( m ) = − d − d L d +1 tan( πν ) d − Y p =1 (cid:18) p − − ν (cid:19) (cid:18) p −
12 + ν (cid:19) for odd d .Now we should integrate in m from the Breitenlohner-Freedman bound ν = 0. However, here we are mainly interested in showing that ∆ V < ν = 0, ∆ V ′ <
0, and the first zero of∆ V ′ is at ν = 1. Therefore, at least for 0 < ν ≤
1, that is d < ∆ + ≤ d + 22 , d − ≤ ∆ − < d , we have ∆ V < Because of the symmetry of AdS space, the effective potential does not depend on r , x . .4 Change in central charge from holography - spinor field Now we will carry out a similar computation for the spinor field. In thiscase the relation between V − and V + is quite explicit. When the sources areturned off, we have S + [ ¯ ψ, ψ, m ] = S − [ ¯ ψ, ψ, m ] − Z d d xε − d (cid:2) ¯ π ( ε, x ) ψ ( ε, x ) + ¯ ψ ( ε, x ) π ( ε, x ) (cid:3) , and it is easy to check that S + [ ¯ ψ, ψ, m ] = S − [ ψ † , ¯ ψ † , − m ] . Since the fields are integrated over, we have V + ( m ) = V − ( − m ) , and we are interested in the difference between the effective potentials. Wehave: ∆ V ( m ) = V + ( m ) − V − ( m ) = Z m d µ ∆ V ′ ( µ ) , where ∆ V ′ ( µ ) = dd m [ V + ( m ) − V + ( m )] m = µ = d V + d m (cid:12)(cid:12)(cid:12)(cid:12) m = µ + d V + d m (cid:12)(cid:12)(cid:12)(cid:12) m = − µ . According to our definition, we have Z d r d d x √ g V + [ g ] = − log Z (cid:2) D ¯ ψ D ψ (cid:3) e − R d r d d x √ g ¯ ψ ( D/ − mL ) ¯ ψ . Taking the derivative with respect to m of both sides we haved V + d m = − L (cid:10) ¯ ψ ( r, x ) ψ ( r, x ) (cid:11) = 1 L Tr (cid:2)(cid:10) ψ ( r, x ) ¯ ψ ( r, x ) (cid:11)(cid:3) = 1 L Tr [ G ( r, x ; r, x )] , Also here, for shortness, we dropped the boundary terms in the action. They will playan important role later when we will discuss the boundary conditions, but no role in thepresent derivation. G is the propagator of the field theory. The computation of thepropagator is reported in the appendix, it has the form G ( r, x ; s, y ) = − r d +12 s d +12 L d Z d d k (2 π ) d e ik · ( x − y ) Z ∞ d ω ω ×× h J m − ( ωr ) P + + J m + ( ωr ) P − i ω + iγ · kk + ω h P + J m + ( ωs ) + P − J m − ( ωs ) i . Therefore, rescaling the integration variables appropriately we haved V + d m = − dim γL d +1 Z d d k (2 π ) d
11 + k Z ∞ d ω ω d J m − ( ω ) J m + ( ω ) , where dim γ is the dimension of the representation of the Clifford algebra.Both integrals can be done exactly, to getd V + d m = − dim γL d +1 (4 π ) − d +12 Γ (cid:18) − d (cid:19) Γ (cid:0) d + m (cid:1) Γ (cid:0) − d + m (cid:1) . Now we add the expression with m → − m , set d = integer − ε , thedivergent contributions cancel and we obtain a finite expression for ∆ V ′ ( m ).After a few manipulations we get∆ V ′ ( m ) = − dim γ ( d − d L d +1 d Y p =1 (cid:18) p − − m (cid:19) (cid:18) p −
12 + m (cid:19) for even d and∆ V ′ ( m ) = − dim γ ( d − d L d +1 cot( πm ) m d − Y p =1 ( p − m ) ( p + m )for odd d .From this expression it is already manifest that ∆ V is negative. In fact,at m = 0, ∆ V ′ <
0, and the first zero of ∆ V ′ is at m = 1 /
2. Therefore, atleast for 0 < m ≤ /
2, that is d < ∆ + ≤ d + 12 , d − ≤ ∆ − < d , we have ∆ V < cc = 1 N L d +1 ∆ V d for the first few even dimensions. 24 N ∆ c/c − π (cid:18) m − m (cid:19) − π (cid:18) m − m + 1645 m (cid:19) − π (cid:18) m − m + 112225 m − m (cid:19) Table 1: Relative change in central charge as computed from the gravitytheory.
Let us now see how the same results can be obtained directly from theconformal field theory (we follow closely [7]). According to our large N computation of the partition function we have Z [ f ] = Z det(1 + f G ) , where G ( θ, θ ′ ) = (cid:10) χ ( θ ) ¯ χ ( θ ′ ) (cid:11) CFT . Now, according to Cardy’s proposal for the c -function: c ( f ) − c (0) = ( − d R ∂∂R ( − log Z [ f ])= ( − d (cid:20) − R ∂∂R log det(1 + f G ) (cid:21) , where the R -dependence comes from the operator G .In order to compute the determinant we will diagonalize G and obtaineigenvalues and degeneracies. The simplest conformal field theory with aspinor field is the free Dirac field, and we assume that G has the samesymmetry properties of the Green’s function of the Dirac operator. Hence,since the space is compact, the spectrum is discrete. Since the operator isantihermitian, the eigenvalues are pure imaginary numbers, and we assumethat they come in conjugate pairs ± ig n . With this information, and defining25 ( d ) ( n ) to be the degeneracy of each eigenspace, we have c ( f ) − c (0) = ( − d " − R ∂∂R X n M ( d ) ( n ) log (cid:2) f g n ) (cid:3) . Computing the c -function for generic f presents difficulties of the samedegree as the corresponding holographic computation. We will do what ispossible and consider the limit of this expression for f → ∞ . This amountsto compute the difference in the central charge between the end points ofthe RG flow: ( − d ∆ c = − lim f →∞ R ∂∂R X n M ( d ) ( n ) log( f g n ) . The term proportional to log f in this expression does not depend on R or ∆, and therefore does not survive the derivative with respect to R .Then we have ( − d ∆ c = − R ∂∂R X n M ( d ) ( n ) log g n . To proceed further we need an explicit expression for the spectrum of G and the degeneracies. Their computation is reported in the appendix, theyhave the form g n ∝ R d − Γ (cid:0) n + ∆ + (cid:1) Γ (cid:0) n + d − ∆ + (cid:1) ,M ( d ) ( n ) = dim γ ( n + d − n !( d − , where dim γ is the dimension of the representation of the Clifford algebra.Since our starting point, the unperturbed CFT, must have ∆ < d/
2, letus set ∆ = d − m . We have to compute( − d ∆ c = − R ∂∂R ∞ X n =0 M ( d ) ( n ) log " R m Γ (cid:0) n + d +12 − m (cid:1) Γ (cid:0) n + d +12 + m (cid:1) . The sum is divergent, and needs to be regulated. In general, there willbe two contributions to the logarithmic derivative R∂ R [7]. One will comefrom the term log R explicitly present:( − d ∆ c = − m ∞ X n =0 M ( d ) ( n ) , This can be checked explicitly later when a regulated sum is introduced. − d ∆ c = − R ∂∂R ∞ X n =0 M ( d ) ( n ) log Γ (cid:0) n + d +12 − m (cid:1) Γ (cid:0) n + d +12 + m (cid:1) that is logarithmically divergent. In fact, any regulator will refer to somefixed energy scale Λ, so that such divergent contribution to the sum wouldbe proportional to log R Λ, and give a finite contribution to the logarithmicderivative. We will use zeta function regularization and a couple of wordsmust be spent on how to extract either contributions from this regulator.By means of an appropriate shift of n and using some relations betweenspecial functions, we will cast the sums in the form∆ c = ∞ X k =0 d − X r =0 a r k r ∆ c = R ∂∂R ∞ X k =0 ∞ X r =0 b r k d − r − . The first one will be evaluated with the zeta function,∆ c = d − X r =0 a r ζ ( − r ) . In the second one, the derivative with respect to R extracts the coefficientof log Λ R . When using the zeta function regularization, this coefficient isthe coefficient that multiplies 1 /k in the sum, i.e. b d . In the following wewill keep the derivative R∂ R with this meaning of an operator extractingthe appropriate coefficient from the series.Let us now show how to carry out what outlined above. We havelog Γ (cid:0) n + d +12 − m (cid:1) Γ (cid:0) n + d +12 + m (cid:1) = − ∞ X p =0 m p +1 (2 p + 1)! ψ (2 p ) (cid:18) n + d + 12 (cid:19) , where ψ ( z ) = Γ ′ ( z ) / Γ( z ) is the digamma function. We will use the followingasymptotic expansion ψ ( z ) = log( z ) + ∞ X s =0 ζ ( − s ) z − s − , It is crucial that the shift be the same for both sums. ψ (cid:18) z + 12 (cid:19) = 2 ψ (2 z ) − ψ ( z ) − . For d even we can set n = k − d/
2, and we have( − ) d ∆ c = − m ∞ X k =0 M ( d ) (cid:18) k − d (cid:19) ( − ) d ∆ c = 4 R ∂∂R ∞ X k =0 M ( d ) (cid:18) k − d (cid:19) ∞ X p =0 m p +1 (2 p + 1)! ψ (2 p ) (cid:18) k + 12 (cid:19) , where the sum can start from k = 0 and not from k = d/ M ( d ) ( k − d/
2) is zero for all the additional terms. The summand in c is already apolynomial in k . We can use the asymptotic expansion of ψ to express also∆ c as a power series in k . Then we have to compute ∆ c with the zetafunction regularization and to extract the coefficient that multiplies 1 /k inthe series for ∆ c . This is best done with the aid of the computer. Table2 reports the results for the first few dimensions, in agreement with theresults from the gravity theory reported in table 1 up to an overall positiveconstant. d ∆ c − (cid:18) m − m (cid:19) − (cid:18) m − m + 1645 m (cid:19) − (cid:18) m − m + 11215 m − m (cid:19) Table 2: Change in central charge as computed from the conformal fieldtheory.In odd dimensions we set n = k − ( d + 1) /
2, and we have( − ) d ∆ c = − m ∞ X k =0 M ( d ) (cid:18) k − d + 12 (cid:19) , ( − ) d ∆ c = 4 R ∂∂R ∞ X k =0 M ( d ) (cid:18) k − d + 12 (cid:19) ∞ X p =0 m p +1 (2 p + 1)! ψ (2 p ) ( k ) .
28n this case, as expected, we find each contribution to vanish. Thecontribution of the linear term in m can easily be shown to vanish. In fact,if we expand M ( d ) (cid:18) k − d + 12 (cid:19) = d − X r =0 a r k r , then ( − ) d ∆ c = − m d − X r =0 a r ζ ( − r ) , and ( − ) d ∆ c =4 R ∂∂R m ∞ X k =0 d − X r =0 a r k r ψ ( k )= 4 R ∂∂R m ∞ X k =0 d − X r =0 a r k r " log ( k ) + ∞ X s =0 ζ ( − s ) k − s − = 4 m d − X r =0 a r ζ ( − r ) . The coefficients of the higher powers of m also turn out to be zero, dueto nontrivial cancellations in the expansion in k . It may be possible to givea general argument that shows this is true in any dimension, but we thinkthis would add little understanding, and we were content of checking thatthe coefficients vanish for the first few dimensions. Let us summarize the content of this paper • We have discussed how to impose the boundary conditions on the fieldsof the gravity theory. We have shown that the appropriate boundaryconditions arise in a very natural way from the variational principle,if the appropriate boundary terms are added to the action. In thisframework, the gravity action that is dual to a pure conformal fieldtheory is not just the naive action, but needs to be complemented byan holographic renormalization boundary term. This term, besidesbeing important for having a finite on-shell action, is also importantin the determination of the boundary conditions. Perturbations of theconformal field theory like double trace deformations or source terms29an then be introduced by adding additional boundary terms, whoseform exactly matches the form of the CFT perturbation. • We have reviewed the c -conjecture, both Cardy’s proposal and therecent proposal in [8] of a c -function from entanglement entropy thatis meaningful in any dimensions. In even dimensions, when both c -functions can constructed, we have shown that they coincide, at leastfrom their holographic construction. • We have verified that, for this proposal of c function, the c -conjectureholds along the RG flow induced by a double trace deformation, at leastin the sense that the difference in central charge between the end pointand the start point of the flow is negative. This result was alreadyknown for the case of a double trace deformation by a scalar operator.We have shown that it also holds for a spinor operator, and we havecomputed the difference in central charge from both holographic andCFT methods, finding that the two results agree. Acknowledgments
Work supported in part by funds provided by the U.S. Department of En-ergy (D.O.E.) under cooperative research agreement DE-FG0205ER41360.We would like to thank John McGreevy for his help through many usefuldiscussions and Igor Klebanov for important pieces of advice.
The propagator is defined by √ g (cid:18) − g µν D µ D ν + m L (cid:19) G ( r, x ; s, y ) = δ ( r − s ) δ d ( x − y ) , or, more explicitly, L d − r d +1 (cid:2) − r ∂ r + ( d − r∂ r − r ∇ + m (cid:3) G ( r, x ; s, y ) = δ ( r − s ) δ d ( x − y ) . Rescaling the propagator as G ( r, x ; s, y ) = r d s d L d − D ( r, x ; s, y ) , D ( r, x ; s, y ) = Z d d k (2 π ) d e ik · ( x − y ) D ( r, s ; k ) , the equation becomes (cid:18) − ∂ r − r ∂ r + ν r + k (cid:19) D ( r, s ; k ) = 1 s δ ( r − s ) . Now we want to expand in eigenfunctions of the differential operator,and we have (cid:18) − ∂ r − r ∂ r + ν r (cid:19) J ± ν ( ωr ) = ω J ± ν ( ωr ) . Now the question of the boundary conditions on fields becomes impor-tant, in order to choose the sign on ν . If we use the ∆ + action, then thefield must go to zero approaching the boundary. Since J − ν ( r ) diverges as r →
0, this implies that we have to expand over J ν . If we use the ∆ − action,then we want π = (cid:20) r∂ r − d ν (cid:21) φ to go to zero. Taking into account the fact that we rescaled the fields, itis not difficult to see that this forces us to expand over J − ν . This meansthat we can just compute for the ∆ + case, the ∆ − case will be obtained bysimply sending ν → − ν .Then let us then expand over J ν : D ( r, s ; k ) = Z d ω ω J ν ( ωr ) D ( w, s ; k ) , and, using the completeness relation Z d ω ω J ν ( ωr ) J ν ( ωs ) = 1 s δ ( r − s ) , we have ( ω + k ) D ( ω, s ; k ) = J ν ( ωs ) , that is D ( r, s ; k ) = Z d ω ω J ν ( ωr ) J ν ( ωs ) ω + k , The extra ω factor is the weight with respect to which the Bessel functions are or-thonormal G ( r, x ; s, y ) = r d s d L d − Z d d k (2 π ) d e ik · ( x − y ) Z d ω ω J ν ( ωr ) J ν ( ωs ) ω + k . The propagator is defined by √ g (cid:16) D/ − mL (cid:17) G ( r, x ; s, y ) = δ ( r − s ) δ ( x − y ) , or, more explicitly L d r d (cid:20) γ r (cid:18) ∂ r − d r (cid:19) + γ · ∂ − mr (cid:21) G ( r, xs, y ) = δ ( r − s ) δ ( x − y ) . Let us now rescale the propagator and expand in plane waves G ( r, x ; s, y ) = r d +12 s d +12 L d D ( r, x ; s, y ) ,D ( r, x ; s, y ) = Z d d k (2 π ) d e ik · ( x − y ) D ( r, s ; k ) . Then we have (cid:20) γ r (cid:18) ∂ r + 12 r (cid:19) + iγ · k − mr (cid:21) D ( r, s ; k ) = 1 s δ ( r − s ) . Now let us consider all the projections of this equation over the eigenspacesof γ r . We have ± (cid:18) ∂ r + 1 / ∓ mr (cid:19) D ±± + iγ · kD ∓± = P ± s δ ( r − s ) ± (cid:18) ∂ r + 1 / ∓ mr (cid:19) D ±∓ + iγ · kD ∓∓ = 0 , where D ( r, s ; k ) = P α D αβ ( r, s ; k ) P β , α, β ∈ { + , −} , and P ± = 1 ± γ r . D − + and D + − and substitutingin the inhomogeneous we get (cid:20) ∂ r + 1 r ∂ r − ( m + 1 / r − k (cid:21) D − + ( r, s ; k ) = P − iγ · k P + s δ ( r − s ) , (cid:20) ∂ r + 1 r ∂ r − ( m − / r − k (cid:21) D + − ( r, s ; k ) = P + iγ · k P − s δ ( r − s ) . The differential operator on the left hand side is hermitian, so we wouldlike to expand in its eigenfunctions. In fact we have (cid:20) ∂ r + 1 r ∂ r − ν r (cid:21) J ± ν ( ωr ) = − ω J ± ν ( ωr ) , and the Bessel functions J ν ( ωr ) are orthonormal and complete Z ∞ d ω ω J ν ( ωr ) J ν ( ωs ) = 1 s δ ( r − s ) . At this point boundary conditions become important, in order to un-derstand what sign of ν to choose for each equation. Looking again at thevariation of the ∆ + action we have: δS + = e . o . m . + Z d d x ¯ ψP + δψ + δ ¯ ψP − ψ . So we have to impose the following boundary conditions P − ψ = 0 , ¯ ψP + = 0 , and this leads to the requirementlim r → D − + ( r, s ; k ) = lim s → D − + ( r, s ; k ) = 0 . Therefore we have to use the expansion D − + ( r, s ; k ) = Z ∞ d ω ω D − + ( ω, s ; k ) J m + ( ωr ) , whereas the opposite choice of using J − m − would give a divergent r → D + − , andboth expansions can be used. We choose to expand over J m − , because it33eads to slightly more symmetric expressions, but the same results could beobtained by expanding over J − m + D + − ( r, s ; k ) = Z ∞ d ω ω D + − ( ω, s ; k ) J m − ( ωr ) . Substituting the expansion in the equations we get D − + ( r, s ; k ) = − Z ∞ d ω ω P − iγ · k P + ω + k J m + ( ωr ) J m + ( ωs ) ,D + − ( r, s ; k ) = − Z ∞ d ω ω P + iγ · k P − ω + k J m − ( ωr ) J m − ( ωs ) . Now we can substitute in the homogeneous equations and get an expres-sion for D ++ and D −− as well. The final form of the propagator is G ( r, x ; s, y ) = − r d +12 s d +12 L d Z d d k (2 π ) d e ik · ( x − y ) Z ∞ d ω ω ×× h J m − ( ωr ) P + + J m + ( ωr ) P − i ω + iγ · kk + ω h P + J m + ( ωs ) + P − J m − ( ωs ) i . Since the theory is supposed to be Weyl invariant, i.e. invariant upon thesimultaneous transformation h ( x ) → h ′ ( x ) = e ω ( x ) h ( x ) χ ( x ) → χ ′ ( x ) = e − ∆ ω ( x ) χ ( x )¯ χ ( x ) → ¯ χ ′ ( x ) = e − ∆ ω ( x ) ¯ χ ( x ) , we can obtain an expression for the two point function G ( θ, θ ′ ) = (cid:10) χ ( θ ) ¯ χ ( θ ′ ) (cid:11) CFT on the sphere by operating a Weyl transformation on the flat space result.Consider the flat metric in polar coordinates h E d = d r + r dΩ d − , r = R tan θ d , wecan show that the flat metric is conformally equivalent to the metric on thesphere h E d = 1 (cid:0) θ (cid:1) R (cid:0) d θ d + sin θ d dΩ d − (cid:1) = 1 (cid:0) θ (cid:1) h S d . Up to an irrelevant overall normalization factor, the correlation functionin flat space is h χ ( x ) ¯ χ (0) i E d = γ · x | x | . Going to polar coordinates and then to the projective coordinates wehave h χ ( θ d , Ω d − ) ¯ χ (0 , i E d = γ d cos θ d − + γ d − sin θ d − cos θ d − + . . . (cid:16) R tan θ d (cid:17) , and, performing the Weyl rescaling, we can obtain the correlator on thesphere h χ ( θ d , Ω d − ) ¯ χ (0 , i S d = γ d cos θ d − + γ d − sin θ d − cos θ d − + . . . (cid:16) R sin θ d (cid:17) . In particular we will need the result on the principal meridian Ω d − = 0 G ( θ ) = γ d (cid:0) R sin θ (cid:1) . We assume that G is diagonal if expanded over the eigenfunctions of theDirac operator D/ ( d ) on the sphere D/ ( d ) = γ d (cid:18) ∂∂θ d + d −
12 cot θ d (cid:19) + 1sin θ d D/ ( d − ; D/ (0) = 0 . The eigenfunctions are known [19], let us quickly review their properties.They are labeled by two set of quantum numbers n = { n , . . . , n d } , ≤ n ≤ n ≤ . . . ≤ n d ,σ = { σ , . . . , σ d } , σ i ∈ {− , } , For the moment we will work on a sphere of unit radius D/ ( d ) ψ ( d ) σn (Ω d ) = iσ d (cid:18) n d + d (cid:19) ψ ( d ) σn (Ω d ) . They can be constructed by recursion: ψ ( d ) σn ( θ d , Ω d − ) = c ( d ) n d n d − h φ ( d ) σ d − n d n d − ( θ d )(1 + iγ d )++ σ d σ d − φ ( d ) − σ d − n d n d − ( θ d )(1 − iγ d ) i ψ ( d − σn (Ω d − ) ,ψ (1) σn ( θ ) = 1 √ πs e iσ ( n + ) θ , s = dim γ , where φ ( d )+ nl ( θ ) = (cid:18) cos θ (cid:19) l (cid:18) sin θ (cid:19) l +1 P ( d + l, d + l − n − l (cos θ ) ,φ ( d ) − nl ( θ ) = (cid:18) cos θ (cid:19) l +1 (cid:18) sin θ (cid:19) l P ( d + l − , d + l ) n − l (cos θ ) , and where P ( a,b ) n are the Jacobi polynomials. With this construction not allvalues of the quantum numbers σ label different eigenfunctions. In fact, for i < d/ σ i = ± Z dΩ d ψ ( d ) † σn (Ω d ) ψ ( d ) σn (Ω d ) = 1 , and we have c ( d ) nl = p Γ( n − l + 1)Γ( d + n + l )2 d Γ (cid:0) d + n (cid:1) . Now we can write explicitly the expansion of G over the eigenfunctions: G (Ω d , Ω ′ d ) = X n,σ ( − i ) σ d g n d R − d ψ ( d ) σn (Ω d ) ψ ( d ) † σn (Ω ′ d ) , where the factor R − d is necessary because the eigenfunctions ψ ( d ) σn are nor-malized on the unit sphere.In order to get a simple expression, we let θ ′ d = 0, Ω ′ d − = Ω d − = 0, i.e.we take the second point to be the north pole of the sphere and the first36oint to lie on the principal meridian. In this case, with some tedious work,it can be shown that X σ σ d ψ ( d ) nσ ( θ, ψ ( d ) † nσ (0 ,
0) = δ n d − δ n d − . . . δ n iγ d Σ d − ×× (cid:16) c ( d ) n d (cid:17) sin θ P ( d , d − n d (cos θ ) P ( d − , d ) n d (1) , where Σ d is the volume of the d -sphere of unit radius, and hence G ( θ ) ≡ G ( θ,
0; 0 , ∞ X n =0 g n R − d γ d Σ d − (cid:16) c ( d ) n (cid:17) sin θ P ( d , d − n (cos θ ) P ( d − , d ) n (1) . Now we can extract the eigenvalues g n by exploiting the orthogonalityrelation of the Jacobi polynomials Z − d x (1 − x ) a (1 + x ) b P ( a,b ) n ( x ) P ( a,b ) m ( x ) = δ mn ×× a + b +1 n + a + b + 1 Γ( n + a + 1)Γ( n + b + 1)Γ( n + 1)Γ( n + a + b + 1) , so that we have g n = Σ d − R d P ( d − , d ) n (1) Z − d x (1 − x ) d − r − x G ( x ) γ d P ( d , d − n ( x ) , where x = cos θ and by G ( x ) /γ d we mean the coefficient that multiplies γ d in G .Explicitly g n ∝ R d − P ( d − , d ) n (1) Z − d x (1 − x ) d − (1 − x ) − ∆ P ( d , d − n ( x ) , where we have dropped an overall prefactor that does not depend on n or R . Such factor only amounts to a change in the normalization of theoperator and can be ignored. Or, from an alternative point of view, whencomputing the central charge, this factor would produce an additive constantthat would not survive the derivative with respect to R . The integral canbe done exactly, and the n - and R -dependent part of the result is g n ∝ R d − Γ (cid:0) n + ∆ + (cid:1) Γ (cid:0) n + d − ∆ + (cid:1) . M ( d ) ( n ) = dim γ ( n + d − n !( d − . References [1] Edward Witten. Multi-trace operators, boundary conditions, andAdS/CFT correspondence. 2001.[2] Igor R. Klebanov and Edward Witten. AdS/CFT correspondence andsymmetry breaking.
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