Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization
aa r X i v : . [ h e p - t h ] A p r SISSA 13/2017/FISI
Anomaly-corrected supersymmetry algebraand supersymmetric holographic renormalization
Ok Song An
Abstract
We present a systematic approach to supersymmetric holographic renormalization for ageneric 5D N = 2 gauged supergravity theory with matter multiplets, including its fermionicsector, with all gauge fields consistently set to zero. We determine the complete set of super-symmetric local boundary counterterms, including the finite counterterms that parameterize thechoice of supersymmetric renormalization scheme. This allows us to derive holographically thesuperconformal Ward identities of a 4D superconformal field theory on a generic background,including the Weyl and super-Weyl anomalies. Moreover, we show that these anomalies satisfythe Wess-Zumino consistency condition. The super-Weyl anomaly implies that the fermionicoperators of the dual field theory, such as the supercurrent, do not transform as tensors underrigid supersymmetry on backgrounds that admit a conformal Killing spinor, and their anticom-mutator with the conserved supercharge contains anomalous terms. This property is explicitlychecked for a toy model. Finally, using the anomalous transformation of the supercurrent,we obtain the anomaly-corrected supersymmetry algebra on curved backgrounds admitting aconformal Killing spinor. ontents N = 2 gauged SUGRA action in 5D 53 Radial Hamiltonian dynamics 6 B.1 ADM decomposition of vielbein and the strong Fefferman-Graham gauge . . . . . . 38B.2 Decomposition of the covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . 40B.3 Equations of motion and leading asymptotics of fermionic fields . . . . . . . . . . . . 41B.4 Generalized PBH transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
C Decomposition of the action and the fermion boundary terms 44
C.1 Decomposition of the kinetic action of the hyperino field . . . . . . . . . . . . . . . . 44C.2 Gravitino part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45C.3 Decomposition of the other terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
D Variation of the canonical momenta under the generalized PBH transformation 45E Derivation of the SUSY algebra without using Poisson bracket 46References 47 Introduction
Supersymmetric (SUSY) field theories in curved backgrounds [1–3] (see also [4] for a recent review)have received much attention in recent years, since they provide a playground where physicallyinteresting, non-perturbative, results can often be obtained through localization techniques [5, 6].Formulating consistent SUSY field theories in curved space usually consists of two steps [1];the first one is to find the classical supergravity theory (SUGRA) by coupling a flat-space SUSYfield theory to the gravity multiplet, and the second one is to take a rigid limit of SUGRA suchthat the gravity multiplet becomes non-dynamical, but maintains a non-trivial background value.Consistency requires that there exists at least one SUSY transformation of the SUGRA under whichthis background gravity multiplet should be invariant, namely δ η e a (0) i = 0 , δ η Ψ (0)+ i = 0 , · · · , (1.1)where e a (0) i refers to the vielbein and Ψ (0)+ i is the gravitino field and η refers to the spinor param-eter of the preserved SUSY. We refer to appendix A and B for notations and conventions. Therequirement that variation of the bosonic fields vanish is trivially satisfied on bosonic backgrounds.One then derives the SUSY transformation of the local operators and the SUSY algebra incurved space from the corresponding ones of SUGRA. However, they are classical in the sense thatthe SUSY transformation laws and algebra derived in this way do not reflect any quantum effects.To clarify this point, let us schematically discuss these quantum effects for a theory with an N = 1 4D superconformal field theory (SCFT) as a UV fixed point. For this aim, we derive theWard identities which contain UV data of quantum field theories. These Ward identities can beobtained in a local renormalization group language [7] without relying on a classical Lagrangiandescription, see e.g. section 2.3 in [8] for a recent review. In N = 1 SCFT, we have two localfermionic transformations, supersymmetry and super-Weyl, respectively δ ǫ + e a (0) i = −
12 Ψ (0)+ i Γ a ǫ + , δ ǫ + Ψ (0)+ i = D i ǫ + + · · · , · · · (1.2a) δ ǫ − e a (0) i = 0 , δ ǫ − Ψ (0)+ i = − b Γ i ǫ − + · · · , · · · (1.2b)where the ellipses indicate possible contributions from other fields in the gravity multiplet and higer-order terms in fermions. Requiring the generating functional of connected correlation functions, W [ g (0) ij , Ψ (0)+ i , · · · ], to be invariant under these local transformations up to a possible anomaly,we obtain two local operator equations, namely12 T ia Ψ (0)+ i Γ a − S i ←− D i + · · · = A s , (1.3a) − S i b Γ (0) i + · · · = A sW , (1.3b)where T ia and S i refer to the energy-momentum tensor and supercurrent operator, respectively.Note that the Ward identities hold for generic background such as ones where the fermionic sourcesare turned on. Combining these two Ward identities with the parameters η + and η − which satisfyconformal Killing spinor (CKS) condition δ η Ψ (0)+ i ≡ δ η + Ψ (0)+ i + δ η − Ψ (0)+ i = D i η + − b Γ i η − = 0 , (1.4)to the lowest order in fermions, we obtain the SUSY- η Ward identity − T ia Ψ (0)+ i Γ a η + + D i ( S i η + ) + · · · = − ( A s η + + A sW η − ) ≡ A η , (1.5)2here the fermionic sources are still turned on, because the CKS equation (1.4) to the lowestorder in fermions does not require the background to be bosonic. One can see from the operatorequation (1.5) that the SUSY- η anomaly A η should be dependent on fermionic background sourcessuch as the gravitini field Ψ + i . Therefore, one may not notice the existence of A η on the bosonicbackground.The Ward identities such as (1.5) turn out to be rather useful. For instance, they determinethe variation of quantum operators under the corresponding symmetry transformations, see e.g.(2.3.7) in [9]. It then follows from (1.5) that on the (bosonic) supersymmetric vacua the supercurrentoperator S i transforms under the SUSY- η transformation as δ η S i (cid:12)(cid:12)(cid:12) susy − vacua = (cid:16) − T ia Γ a η + − δδ Ψ (0)+ i A η + · · · (cid:17) susy − vacua . (1.6)We emphasize that the anomalous term δδ Ψ (0)+ i A η does not appear in the ‘classical’ SUSY variationof the supercurrent operator S i , and it is non-zero in generic curved backgrounds admitting aconformal Killing spinor. Moreover, by integrating (1.6) over a Cauchy surface, one can obtainthe commutator of two supercharges (see e.g. (2.6.14) and (2.6.15) in [9]) and find that it is alsocorrected by the anomalous term.The upshot is that once the Ward identities (1.3) are found, one can see immediately all thesequantum corrections. The main obstacle in obtaining (1.3) is to find out the anomalies A s and A sW . Fortunately, we have a nice tool for computing the anomalies, i.e. AdS/CFT correspondence[10–12]. The holographic computation of the quantum anomalies, such as the computation of theWeyl anomaly in [13], results in specific values for the anomaly coefficients. For instance, for theWeyl anomaly one gets a = c from a holographic calculation, which is valid for N = 4 superYang-Mills theory in the large N limit. This is due to the fact that we use two-derivative action onthe gravity side. To obtain the whole class of anomalies one should consider the higher-derivativeaction. We emphasize that since the anomalies belonging to the same multiplet are related bySUSY transformations, the super-Weyl anomaly A sW obtained by a holographic computation alsohas specific values for the anomaly coefficients.Henceforth, in order to obtain the Ward identities of 4D N = 1 SCFT by AdS/CFT, weconsider generic N = 2 5D gauged SUGRA including its fermionic sector in asymptotically locallyAdS (AlAdS) spaces, particular examples of which were studied in [14–19]. More specifically, thetheory we consider has a superpotential W and its field content consists of a vielbein, two gravitini,as well as an equal number of spin-1 / W has an isolated local extremum. We also demand that W is a local function around thatpoint.As indicated in [17, 20], the N = 2 5D gauged SUGRA can have a superpotential W in severalcases. A typical case is when there are only vector multiplets and U (1) R (subgroup of SU (2) R R-symmetry group) is gauged [21]. When there are also hypermultiplets, the gauged SUGRA canhave a superpotential under a certain constraint related to the ‘very special geometry’ on the scalarmanifold of the vector multiplets, which we do not discuss here in detail. One should keep in mind that the conservation law which allows to construct the conserved supercharge withnon-covariantly-constant rigid parameter η + is D i ( S i η + ) = 0, not S i ←− D i η + = 0. Even though the solution considered in [19] is not AlAdS due to existence of the massless scalars, the generalform of the action given there is the same with the one here.
3s in field theory, renormalization is indispensable also in the bulk holographic computation.Although it has been studied since the early period of AdS/CFT, many works on holographic renor-malization (HR) [13, 22–30] have focused on the bosonic sector. [31–36] obtained some boundarycounterterms for the fermionic sector, but typically these were limited to either lower dimensionalspacetime (mainly 3 or 4 dimensions) or to homogeneous solutions which do not depend on thetransverse directions. We note that in a context different from this paper, 4D N = 1 SUGRAincluding the fermionic sector was treated in [37] by a somehow ad hoc method.We perform HR along the lines of [23, 30, 38]. By formulating the theory in radial Hamiltonianlanguage, we obtain the radial Hamiltonian, which gives the first class constraints. From theHamiltonian constraint we obtain the Hamilton-Jacobi (HJ) equation, enabling us to determinethe divergent counterterms in a covariant way without relying on a specific solution of the classicalSUGRA. We emphasize that the counterterms, as the solution of HJ equation, should satisfy otherfirst class constraints. General covariance of the counterterms is a necessary and sufficient conditionto satisfy diffeomorphism constraint, which is one of the first class constraints.Once the counterterms are obtained, one can renormalize the canonical momenta of the radialHamiltonian and thus obtain the renormalized canonical momenta . According to AdS/CFT dic-tionary, the renormalized canonical momenta correspond to local operators of the field theory inthe local renormalization group language [7]. The first class constraints turn out to be relationsbetween local sources and operators, from which we obtain the Ward identities (see (5.2)) that infact reflect the symmetries of the dual field theory and do not rely on a Lagrangian descriptionof the quantum field theory. Since the bulk theory is 5D N = 2 SUGRA, the dual field theoryis supposed to have 4D N = 1 superconformal symmetry and we obtain the corresponding Wardidentities. Note that here we cannot see the U (1) R symmetry because we truncate all gauge fields.In a related work [39] the U (1) R gauge field is included in the model.It turns out that the N = 1 superconformal symmetry is broken by anomalies. From the bulkpoint of view, these anomalies are due to the fact that the first class constraints are non-linearfunctions of canonical momenta, implying that corresponding symmetries are broken by the radialcut-off. From the dual field theory point of view, of course, the global anomalies are a quantumeffect. We obtain not only the SUSY-completion of the trace-anomaly but also the holographicsuper-Weyl anomaly, which are rather interesting by themselves, since they can provide anothertool for testing AdS/CFT. As discussed before, we find that due to the anomaly operators donot transform as tensors under super-Weyl transformation and the variation of operators getsan anomalous contribution, see (5.22). Hence, the Q -transformation of the operators also becomesanomalous, since it is obtained by putting together supersymmetry and super-Weyl transformations.Here Q refers to the preserved supercharge. This is rather remarkable, since it implies that the‘classical’ SUSY variation cannot become a total derivative in the path integral of SUSY fieldtheories in curved space, unless the anomaly effects disappear. In this regard, it is shown in [39]that the ‘new’ non-covariant finite counterterms suggested in [43, 44] should be discarded since theywere introduced in order to match with field theory without taking into account the anomaly-effect.From the anomalous transformation of the supercurrent operator, we find that the supersymmetryalgebra in curved space is corrected by anomalous terms, see (5.56).We finally note that the boundary conditions consistent with SUSY should be specified before Notice that the existence of super-Weyl anomaly is natural, due to the existence of Weyl anomaly that is relatedto the super-Weyl anomaly by SUSY transformation. As we will see in the main text, our result of the super-Weyl anomaly is different from [40] where they obtainedit on the field theory by using Feynman diagrams. In [41], they tried to obtain the holographic super-Weyl anomaly.In any case, we show that our result satisfies Wess-Zumino (WZ) consistency conditions. One can check that theresult of [40] does not satisfy the consistency condition. See [42] for a review of WZ consistency condition. a priori . In this work we always keep Dirichletboundary conditions for the metric and gravitino field. As we will see, consistency with SUSYrequires that either Dirichlet or Neumann boundary conditions should be imposed for scalars andtheir SUSY-partner spin 1 / N = 2 5Dgauged SUGRA action and SUSY variation of the fields. In section 3, we first present the radialHamiltonian and other first class constraints for Dirichlet boundary conditions. We then explaina systematic way of holographic renormalization and obtain the flow equations. In section 4 wedetermine the divergent counterterms and possible finite counterterms. In particular, the completeset of counterterms are shown for a toy model. By means of these counterterms, in section 5 weobtain the holographic Ward identities and anomalies and show that the anomalies satisfy the Wess-Zumino consistency condition. We then define constraint functions of the phase space by using Wardidentities and see that symmetry transformation of the sources and operators are simply describedin terms of the Poisson bracket and constraint functions. Finally, we provide important subsequentresults which hold on supersymmetric backgrounds and present anomaly-corrected supersymmetryalgebra. In section 6 we observe from consistency with SUSY that scalars and their SUSY-partnerfields should have the same boundary condition. In appendix A, we describe our notations andpresent some useful identities, and in appendix B we develop the preliminary steps to obtain radialHamiltonian, which contains definition of ADM decomposition, strong Fefferman-Graham (FG)gauge and generalized Penrose-Brown-Henneaux (gPBH) transformations. In appendix C, we showADM decomposition of the radial Lagrangian part by part and in appendix D we prove that gPBHtransformation of the operators can be obtained from the holographic Ward identities. In appendixE we derive the anomaly-corrected SUSY algebra in an alternative way. N = 2 gauged SUGRA action in 5D The action of gauged on-shell SUGRA possessing a single superpotential with all gauge fieldsconsistently truncated ( D = d + 1 = 5) is given by [19] S = S b + S f , (2.1)where S b = 12 κ Z M d d +1 x √− g (cid:0) R [ g ] − G IJ ( ϕ ) ∂ µ ϕ I ∂ µ ϕ J − V ( ϕ ) (cid:1) , (2.2) S f = − κ Z M d d +1 x √− g (cid:26) (cid:16) Ψ µ Γ µνρ ∇ ν Ψ ρ − Ψ µ ←−∇ ν Γ µνρ Ψ ρ − W Ψ µ Γ µν Ψ ν (cid:17) + (cid:16) i G IJ ζ I Γ µ (cid:0) /∂ϕ J − G JK ∂ K W (cid:1) Ψ µ − i G IJ Ψ µ ( /∂ϕ I + G IK ∂ K W )Γ µ ζ J (cid:17) + (cid:16) G IJ ζ I (cid:0) δ JK / ∇ + Γ JKL [ G ] /∂ϕ L (cid:1) ζ K − G IJ h ζ I / ←−∇ ζ J + ζ K ( /∂ϕ L )Γ JKL ζ I i(cid:17) + 2 M IJ ( ϕ ) ζ I ζ J + quartic terms (cid:27) , (2.3)and the scalar potential and the mass matrix M IJ are expressed in terms of the superpotential as V ( ϕ ) = G IJ ∂ I W ( ϕ ) ∂ J W ( ϕ ) − dd − W ( ϕ ) , (2.4)5 IJ ( ϕ ) = ∂ I ∂ J W − Γ KIJ [ G ] ∂ K W − G IJ W . (2.5)Here κ is related to the gravitational constant by κ = 8 πG ( d +1) . Note that in AlAdS spaces (withradius 1) which we are interested in the scalar potential and the superpotential are given by V ( ϕ ) = − d ( d −
1) + O (cid:0) ϕ (cid:1) , W ( ϕ ) = − ( d −
1) + O (cid:0) ϕ (cid:1) . (2.6)The action (2.1) is invariant under the supersymmetry transformation δ ǫ ϕ I = i ǫζ I + h.c. = i (cid:16) ǫζ I − ζ I ǫ (cid:17) , (2.7a) δ ǫ E αµ = 12 ¯ ǫ Γ α Ψ µ + h.c. = 12 (cid:0) ǫ Γ α Ψ µ − Ψ µ Γ α ǫ (cid:1) , (2.7b)where h . c . refers to hermitian conjugation, and δ ǫ ζ I = − i (cid:0) /∂ϕ I − G IJ ∂ J W (cid:1) ǫ, (2.8a) δ ǫ Ψ µ = (cid:18) ∇ µ + 12( d − W Γ µ (cid:19) ǫ. (2.8b)for any value of d .Two comments are in order about the action (2.1). Firstly, all the fermions here including thesupersymmetry transformation parameter ǫ are Dirac fermions. In fact, in N = 2 5 dimensionalSUGRA, the gravitino field is expressed in terms of a symplectic Majorana spinor [46], which canalso be described in terms of Dirac fermion [16]. Other fermions in the theory can also be expressedin the same way. Secondly, we would like to be as general as possible and thus, we keep d genericin most of the following computations. According to the holographic dictionary [12] the on-shell action of the supergravity theory is thegenerating functional of the dual field theory. Therefore, the first step of the holographic compu-tation is usually to consider the on-shell action on the bulk side. As is well-known, this on-shellaction always suffers from the long-distance divergence which corresponds to the UV divergenceof the dual field theory, and thus we need to renormalize the on-shell action of the supergravitytheory, which is called as holographic renormalization [13].The Hamiltonian formulation is one of the powerful approach in holographic renormaliza-tion [23, 30]. The Hamiltonian constraint, one of the first class constraints obtained from theradial Hamiltonian, gives the Hamilton-Jacobi (HJ) equation by which we can obtain all the in-finite counterterms for generic sources and curved background. The holographic renormalizationis basically done once we find all the divergent counterterms and subtract them from the on-shellaction for generic background and sources. Depending on the problem under consideration onecan add some extra finite counterterms which actually correspond to the choice of scheme in theboundary field theory.In this section we obtain the radial Hamiltonian, from which we extract the first class con-straints. Afterwards, we explain a general algorithm for obtaining the full counterterms from theHJ equation. We then present the flow equations which are needed to form a complete set ofequations of motion. In [19] the transformation rule of the gravitino field is given by δ ǫ Ψ µ = ( ∇ µ + W Γ µ ) ǫ , which is obtained bysetting D = 5 explicitly in (2.8). .1 Radial Hamiltonian The Gibbons-Hawking term [47] 1 κ Z ∂ M d d x √− γ K, (3.1)where K is the extrinsic curvature on the boundary ∂ M , was introduced to have a well-definedvariational problem for the Einstein-Hilbert action S EH = 12 κ Z M d d +1 x √− g R. (3.2)As indicated in [31–33, 35], by the same reason some additional boundary terms are needed whenthe theory involves the fermionic fields. It turns out that regarding the action (2.1) we have to addthe boundary terms (for details, see appendix C.1 and C.2) ± κ Z ∂ M d d x √− γ Ψ i b Γ ij Ψ j , (3.3a) ± κ Z ∂ M d d x √− γ G IJ ζ I ζ J , (3.3b)where the signs in front of the terms bilinear in fermionic fields fixes which radiality (see (B.10)) ofthe fermion should be used as a generalized coordinate. Note that, however, the sign depends onmass of the fields and choice of the boundary condition [33]. Since mass of the gravitino Ψ µ in ourcase is ( d − / >
0, sign of (3.3a) should be positive (see also appendix B.3 and B.4). Sign of themass of ζ I changes according to the model, and thus we can not choose sign of (3.3b) a priori .For the time being, however, let us pick the + sign. As we will discuss the opposite case insection 6, picking minus sign corresponds to imposing Neumann boundary condition on spin-1/2field ζ I . We emphasize that this choice of the sign will not affect our claim later about determinationof the scalar fields’ leading asymptotics. The whole action including the terms (3.1) and (3.3) isthen given by S full = S + 12 κ Z ∂ M d d x √− γ (cid:16) K + Ψ i b Γ ij Ψ j + G IJ ζ I ζ J (cid:17) . (3.4)The full action S full can be written as S full = R dr L , where the radial Lagrangian L is L = 12 κ Z Σ r d d x N √− γ ( R [ γ ] − G IJ ∂ i ϕ I ∂ i ϕ J − V ( ϕ ) + ( γ ij γ kl − γ ik γ jl ) K ij K kl − G IJ N ( ˙ ϕ I − N i ∂ i ϕ I )( ˙ ϕ J − N j ∂ j ϕ J ) + 2 N (cid:16) ˙Ψ + i b Γ ij Ψ − j + Ψ − i b Γ ij ˙Ψ + j (cid:17) + 1 N ˙ e ia e jb (cid:16) Ψ i Γ ab Ψ j + Ψ j Γ ba Ψ i (cid:17) + (cid:18) K + 1 N D k N k (cid:19) Ψ i b Γ ij Ψ j + 14 N e ak ˙ e kb Ψ i Γ { b Γ ij , Γ ab } Ψ j + 12 N K kl h(cid:0) Ψ r − N i Ψ i (cid:1) [ b Γ kj , b Γ l ]Ψ j − Ψ j [ b Γ kj , b Γ l ] (cid:0) Ψ r − N i Ψ i (cid:1)i + 14 N Ψ i (cid:16) ∂ k N [ b Γ ij , b Γ k ] − ( D k N l )Γ { b Γ ij , b Γ kl } (cid:17) Ψ j − N i N (cid:16) Ψ j Γ b Γ jk D i Ψ k − Ψ j ←− D i Γ b Γ jk Ψ k (cid:17) − Ψ i b Γ ijk D j Ψ k + Ψ i ←− D j b Γ ijk Ψ k − N Ψ k ←− D j Γ b Γ jk (cid:0) Ψ r − N i Ψ i (cid:1) − N (cid:0) Ψ r − N i Ψ i (cid:1) Γ b Γ jk D j Ψ k + 1 N Ψ k Γ b Γ jk (cid:0) D j Ψ r − N i D j Ψ i (cid:1) + 1 N (cid:16) Ψ r ←− D j − N i Ψ i ←− D j (cid:17) Γ b Γ jk Ψ k N W h(cid:0) Ψ r − N i Ψ i (cid:1) Γ b Γ j Ψ j + Ψ j b Γ j Γ (cid:0) Ψ r − N i Ψ i (cid:1)i + W Ψ i b Γ ij Ψ j + 2 N G IJ (cid:16) ζ I + ˙ ζ J − + ˙ ζ I − ζ J + (cid:17) + (cid:18) K + 1 N D k N k (cid:19) G IJ ζ I ζ J − N G IJ e ai ˙ e ib ζ I Γ ab Γ ζ J + 1 N (cid:0) ˙ ϕ K − N i ∂ i ϕ K + N i ∂ i ϕ K (cid:1) ∂ K G IJ ζ I ζ J − G IJ (cid:16) ζ I b Γ i D i ζ J − ζ I ←− D i b Γ i ζ J (cid:17) − N G IJ (cid:20) − D i N j (cid:16) ζ I b Γ ij Γ ζ J (cid:17) − N i ζ I Γ D i ζ J + N i ( ζ I ←− D i )Γ ζ J (cid:21) − iN G IJ " N (cid:0) ˙ ϕ J − N j ∂ j ϕ J (cid:1) h ζ I (cid:16) Ψ r − N i Ψ i + N b Γ i ΓΨ i (cid:17) − (cid:16) Ψ r − N i Ψ i + N Ψ i Γ b Γ i (cid:17) ζ I i + ∂ i ϕ J h ζ I Γ b Γ i (cid:0) Ψ r − N j Ψ j (cid:1) − (cid:0) Ψ r − N j Ψ j (cid:1) b Γ i Γ ζ I i + N ∂ i ϕ J (cid:16) ζ I b Γ j b Γ i Ψ j − Ψ j b Γ i b Γ j ζ I (cid:17) + iN ∂ I W h ζ I Γ (cid:0) Ψ r − N i Ψ i (cid:1) + (cid:0) Ψ r − N i Ψ i (cid:1) Γ ζ I + N (cid:16) Ψ i b Γ i ζ I + ζ I b Γ i Ψ i (cid:17)i − N ∂ K G IJ h(cid:0) ˙ ϕ J − N i ∂ I ϕ J (cid:1) (cid:16) ζ I Γ ζ K − ζ K Γ ζ I (cid:17) + N ∂ i ϕ J (cid:16) ζ I b Γ i ζ K − ζ K b Γ i ζ I (cid:17)i − M IJ ζ I ζ J ) . (3.5)Given the radial Lagrangian L we can derive the canonical momenta π ia = δLδ ˙ e ai = (cid:0) δ ij e ak + δ ik e aj (cid:1) √− γ κ " (cid:16) γ jk γ lm − γ jl γ km (cid:17) K lm + 12 γ jk (cid:16) G IJ ζ I ζ J + Ψ p b Γ pq Ψ q (cid:17) − N (cid:16) Ψ p [ b Γ jp , b Γ k ] (cid:16) Ψ r − N l Ψ l (cid:17) − (cid:16) Ψ r − N l Ψ l (cid:17) [ b Γ jp , b Γ k ]Ψ p (cid:17) − √− γ κ " e bi (cid:18)
14 Ψ j Γ { b Γ jk , Γ ab } Ψ k − G IJ ζ I Γ ab Γ ζ J (cid:19) + e aj (cid:16) Ψ j b Γ ik Ψ k + Ψ k b Γ ki Ψ j (cid:17) , (3.6a) π ϕI = δLδ ˙ ϕ I = √− γ N κ h − G IJ (cid:0) ˙ ϕ J − N i ∂ i ϕ J (cid:1) + N ∂ I G JK ζ J ζ K − N ∂ K G IJ (cid:16) ζ J Γ ζ K − ζ K Γ ζ J (cid:17) − i G IJ (cid:16) ζ J (cid:16) Ψ r − N i Ψ i + N b Γ i ΓΨ i (cid:17) − (cid:16) Ψ r − N i Ψ i + N Ψ i Γ b Γ i (cid:17) ζ J (cid:17) i , (3.6b) π ζI = L ←− δδ ˙ ζ I − = √− γκ G IJ ζ J + , (3.6c) π ζI = −→ δδ ˙ ζ I − L = √− γκ G IJ ζ J + , (3.6d) π i Ψ = L ←− δδ ˙Ψ + i = √− γκ Ψ − j b Γ ji , (3.6e) π i Ψ = −→ δδ ˙Ψ + i L = √− γκ b Γ ij Ψ − j . (3.6f)One should keep in mind that π i Ψ and π i Ψ have minus radiality, and π ζI and π ζI have plus radiality.8rom K ij = K ji , we obtain the constraint0 = J ab ≡ κ √− γ ( e ia π bi − e ib π ai ) −
14 Ψ j Γ { b Γ jk , Γ ab } Ψ k + 12 G IJ ζ I Γ ab Γ ζ J − e ia e jb (Ψ i b Γ jk Ψ k + Ψ k b Γ kj Ψ i − Ψ j b Γ ik Ψ k − Ψ k b Γ ki Ψ j ) , (3.7)which, as we will see, corresponds to the local Lorentz generator of the frame bundle on the sliceΣ r [35].Taking inverse of the canonical momenta and implementing Legendre transformation we obtainthe radial Hamiltonian H = Z d d x (cid:16) ˙ e ai π ia + ˙ ϕ I π ϕI + π ζI ˙ ζ I − + ˙ ζ I − π ζI + π i Ψ ˙Ψ + i + ˙Ψ + i π i Ψ (cid:17) − L = Z d d x (cid:2) N H + N i H i + (cid:0) Ψ r − N i Ψ i (cid:1) F + F (cid:0) Ψ r − N i Ψ i (cid:1)(cid:3) , (3.8)where H = κ √− γ " (cid:18) d − e ai e bj − e aj e bi (cid:19) π ia π jb − G IJ π ϕI π ϕJ + G IJ (cid:16) π ζI / D π ζJ − π ζI ←− / D π ζJ (cid:17) − d − (cid:0) e aj π ia + e ai π ja (cid:1) h ( d − + i π Ψ j + π Ψ j Ψ + i ) + π p Ψ (cid:16)b Γ pi − ( d − γ pi (cid:17) b Γ jk Ψ + k + Ψ + k b Γ kj (cid:16)b Γ ip − ( d − γ ip (cid:17) π p Ψ i + 1 d − e ai π ia (cid:16) − ζ I − π ζI − π ζI ζ I − + Ψ + j π j Ψ + π j Ψ Ψ + j (cid:17) + 2 G IJ Γ LJK [ G ] π ϕI (cid:16) ζ K − π ζL + π ζL ζ K − (cid:17) + iπ ϕI h d − (cid:16) ζ I − b Γ i π i Ψ + π i Ψ b Γ i ζ I − (cid:17) − G IJ (cid:16) π ζI b Γ i Ψ + i + Ψ + i b Γ i π ζJ (cid:17) i − π k Ψ (cid:20)(cid:18) d − b Γ k b Γ j − γ kj (cid:19) / D − ←− / D (cid:18) d − b Γ k b Γ j − γ kj (cid:19)(cid:21) π j Ψ + id − (cid:16) π ζI /∂ϕ I b Γ i π i Ψ − π i Ψ b Γ i /∂ϕ I π ζI (cid:17) − i∂ i ϕ I (cid:16) π ζI π i Ψ − π i Ψ π ζI (cid:17) + G IM G KN ∂ i ϕ J ( ∂ K G IJ − ∂ I G KJ ) π ζM b Γ i π ζN − W (cid:0) Ψ + i π i Ψ + π i Ψ Ψ + i (cid:1) + M IJ (cid:16) G IK π ζK ζ J − + G JK ζ I − π ζK (cid:17) − i ∂ I W (cid:20) G IJ (cid:16) Ψ + i b Γ i π ζJ + π ζJ b Γ i Ψ + i (cid:17) + 1 d − (cid:16) π i Ψ b Γ i ζ I − + ζ I − b Γ i π i Ψ (cid:17)(cid:21) + √− γ κ " − R [ γ ] + G IJ ∂ i ϕ I ∂ i ϕ J + V ( ϕ ) + G IJ ζ I − (cid:16) / D − ←− / D (cid:17) ζ J − + Ψ + i b Γ ijk (cid:16) D j − ←− D j (cid:17) Ψ + k + D k (cid:16) Ψ + i (cid:16) γ jk b Γ i − γ ik b Γ j (cid:17) Ψ + j (cid:17) + i G IJ ∂ i ϕ J (cid:16) ζ I − b Γ j b Γ i Ψ + j − Ψ + j b Γ i b Γ j ζ I − (cid:17) + ∂ K G IJ ∂ i ϕ J (cid:16) ζ I − b Γ i ζ K − − ζ K − b Γ i ζ I − (cid:17) , (3.9) H i = − e ai D j π ja + ( ∂ i ϕ I ) π ϕI + ( ζ I − ←− D i ) π ζI + π ζI ( D i ζ I − ) + π j Ψ (cid:0) D i Ψ + j (cid:1) + (cid:16) Ψ + j ←− D i (cid:17) π j Ψ For instance, the inverse of the canonical momentum π i Ψ is Ψ − i = κ √− γ d − [ b Γ ij − ( d − γ ij ] π j Ψ . D j ( π j Ψ Ψ i + + Ψ i + π j Ψ ) , (3.10) F = 2 κ √− γ ( d − b Γ i π i Ψ e aj π ja −
18 Γ a γ ik π k Ψ π ia − e al b Γ i π l Ψ π ia + i G IJ π ϕI π ζJ ) + 14 Γ a Ψ + i π ia + 14 b Γ i Ψ + j e aj π ia + i π ϕI ζ I − − D i π i Ψ − d − W b Γ i π i Ψ − i ∂ i ϕ I b Γ i π ζI − i G IJ ∂ I W π ζJ + √− γ κ (cid:16) b Γ ij D i Ψ + j + W b Γ i Ψ + i + i G IJ ∂ i ϕ J b Γ i ζ I − + i ( ∂ I W ) ζ I − (cid:17) . (3.11)We note that in the above computations we used the local Lorentz constraint (3.7).By radiality we split F into two parts F + ≡ Γ + F = κ √− γ " π ja e ak (cid:18) d − γ jk b Γ i − γ ij b Γ k − γ ik b Γ j (cid:19) π i Ψ + i G IJ π ϕI π ζJ − d − W b Γ i π i Ψ − i G IJ ∂ I W π ζJ + √− γ κ (cid:16) b Γ ij D i Ψ + j + i G IJ ∂ i ϕ J b Γ i ζ I − (cid:17) , (3.12)and F − ≡ Γ − F = 14 (cid:16)b Γ i Ψ + j + b Γ j Ψ + i (cid:17) e aj π ia + i π ϕI ζ I − − D i π i Ψ − i ∂ i ϕ I b Γ i π ζI + √− γ κ (cid:16) W b Γ i Ψ + i + i∂ I W ζ I − (cid:17) . (3.13)The canonical momenta for N , N i and Ψ r vanish identically, and it then follows from theHamilton’s equation that H = H i = F − = F + = 0 . (3.14)These first class constraints reflect respectively radial reparameterization and diffeomorphism andsupersymmetry and super-Weyl invariance along the radial cut-off Σ r , which can be seen by com-paring with (B.37).Inserting (3.6) in (3.7), we obtain0 = e ia π bi − e ib π ai + 12 ζ I − Γ ab π ζI − π ζI Γ ab ζ I − − π i Ψ Γ ab Ψ + i + 12 Ψ + i Γ ab π i Ψ , (3.15)which reflects the local frame rotation symmetry of the theory according to (B.37). We emphasizethat on the bosonic level this local Lorentz constraint is reduced into e ia π bi = e ib π ai , (3.16)which implies that we can define symmetric canonical momenta for the metric such as δδ ˙ γ ij L B ≡ π ij = 12 e aj δδ ˙ e ai L B . (3.17)Here L B denotes bosonic part of the radial Lagrangian (3.5).We emphasize that linearity of the constraints H i = F − = 0 and local Lorentz constraint reflectthe fact that their corresponding symmetries are not broken by the cut-off. Meanwhile, H = 0 and F + = 0 constraints look much different from their corresponding symmetries, due to the quadraticterms, implying that in fact the cut-off breaks these symmetries, though they are non-linearlyrealized in the bulk. 10 .2 Hamilton-Jacobi equations and the holographic renormalization The HJ equations are obtained by plugging π ia = δδe ai S , π ϕI = δδϕ I S , π ζI = S ←− δδζ I − , π ζI = −→ δδζ I − S , π i Ψ = S ←− δδ Ψ + i , π i Ψ = −→ δδ Ψ + i S (3.18)into the first class constraints (3.14). Here S [ e, ϕ, ζ − , Ψ + ] is the Hamilton’s principal functional.The Hamilton’s principal functional S is particularly important since it can be identified as theon-shell action evaluated on the radial slice Σ r . For the sake of renormalization of the on-shellaction we have only to solve these HJ equations for S up to the finite terms without relying on thespecific solution of the equations of motion. Because this asymptotic solution of the HJ equationsare obtained for the generic sources we can identify them as the counterterms for cancellation ofthe divergence of the on-shell action as well as all the correlation functions.As pointed out in [48], the constraint H i = 0 and the local Lorentz constraint (3.15) whichreflects the bulk diffeomorphism invariance along the transverse direction is automatically satisfiedas long as we look for a local and covariant solution. Hence the equations which we have to solveare the constraints H = F − = F + = 0.Let us briefly review the algorithm of solving the HJ equation in AlAdS geometry. In general,the Hamiltonian constraint is solved asymptotically by using the formal expansion of S with respectto the dilatation operator δ D [30] (see section 5.2 of [8] for a recent review) δ D = Z d d x X Φ (∆ Φ − d ) δδ Φ , (3.19)where Φ refers to every field in the theory and ∆ Φ denotes to the scaling dimension of the operatordual to Φ. The solution takes form of S = Z Σ r d d x √− γ L = Z Σ r d d x √− γ (cid:16) L [0] + L [1] + · · · + e L [ d ] log e − r + L [ d ] + · · · (cid:17) , (3.20)where δ D L [ n ] = − n L [ n ] , ≤ n < d, δ D e L [ d ] = − d e L [ d ] . (3.21)Since the dilatation operator δ D is asymptotically identical to the radial derivative δ r = Z Σ r d d x X Φ ˙Φ δδ Φ (3.22)in AlAdS, one can see that L [ n ] for n < d and e L [ d ] are asymptotically divergent, which we canidentify as counterterms, namely S ct = − Z Σ r d d x √− γ (cid:16) L [0] + L [1] + · · · + e L [ d ] log e − r (cid:17) . (3.23)By construction, this is collection of all possible divergent terms.This general argument of finding S ct is not suitable in our case, since the operator δ D requiresknowledge of scaling dimensions in the theory a priori. To avoid this disadvantage, here we employa universal operator δ e = Z d d x e ai δδe ai + 12 Ψ + i δ Ψ + i + 12 ←− δδ Ψ + i Ψ + i ! , (3.24)11ather than δ D [38, 48], since we know that the scaling dimension of the operators dual to e ai andΨ + i in AlAdS are d + 1 and d + 1 / δ e basically countsthe number of vielbein and gravitino. The formal expansion of the Hamilton’s principal function S [ e, ϕ, ζ − , Ψ + ] with respect to δ e is thus S = S (0) + S (1) + S (2) + · · · , S ( k ) ≡ Z d d x L ( k ) , (3.25)where δ e S ( k ) = ( d − k ) S ( k ) . This implies that π i ( k ) a e ai + 12 π i ( k )Ψ Ψ + i + 12 Ψ + i π i ( k )Ψ = ( d − k ) L ( k ) + ∂ i v i ( k ) , (3.26)for certain v i ( k ) . However, the Lagrangian L ( k ) is defined up to a total derivative, and thus we canput [38] π i ( k ) a e ai + 12 π i ( k )Ψ Ψ + i + 12 Ψ + i π i ( k )Ψ := ( d − k ) L ( k ) . (3.27)As we see later, this identification of L ( k ) greatly simplifies the HJ equation and makes it almostalgebraic.By using (3.27) we can solve the HJ equation recursively, but this procedure stops at S ( d ) due topoles. The reason why higher-order terms, which are finite in r → ∞ limit, cannot be determinedin this recursive procedure is that they are related to the arbitrary integration constants whichform a complete integral together with the integration constants from the flow equations.Since the scaling dimension of the other operators is less than d under the general assumptionthat there is neither exactly marginal nor irrelevant deformation, all the divergent terms appearup to S ( d ) so that we can identify the counterterms as S ct = − d X k =0 S ( k/ , (3.28)if we consider the theory only up to quadratic terms in fermions. Note that the logarithmicallydivergent terms are distributed in almost all of S ( k ) s. Since our radial slice is 4 dimensional, theseterms appear with the pole 1 / ( d − d − → −
12 log e − r (3.29)and summing up all of them, we obtain the logarithmically divergent terms e L [ d ] . We emphasizethat the two algorithms we described in fact give the same result for S ct .Once the local counterterms S ct are obtained, we renormalize the on-shell action by b S ren = lim r → + ∞ ( S full + S ct ) = lim r → + ∞ Z Σ r d d x L [ d ] . (3.30)The canonical momenta are automatically renormalized by S ct , namely b π Φ ≡ π Φ + δδ Φ S ct , for every field Φ , (3.31)and variation of the renormalized on-shell action under any symmetry transformation is given bythe chain rule δ b S ren = lim r → + ∞ Z d d x (cid:16)b π ia δe ai + b π ϕI δϕ I + δζ I − b π ζI + b π ζI δζ I − + δ Ψ + i b π i Ψ + b π i Ψ δ Ψ + i (cid:17) . (3.32)12 .3 Flow equations and leading asymptotics After Solving the HJ equations we insert (3.18) into (3.6)s to get the flow equations for the sources.Whereas for the bosonic sector it works, for the fermionic sector this procedure does not since (3.6c),(3.6d), (3.6e), (3.6f) just play a role of field-redefinition and are just reminiscence of the second-class constraints in the Lagrangian (3.5), Notice that the second-class constraints are completelyeliminated from the radial Hamiltonian H . Therefore from now on we regard that the theory isoriginally defined in the Hamiltonian formalism and the radial Hamiltonian H is more fundamentalobject than the radial Lagrangian L , even though both the Hamiltonian and Lagrangian formalismbasically give the same equations of motion.The flow equations are then obtained by substituting (3.18) into the Hamilton’s equations ofmotion ˙ e ai = δHδπ ia , ˙ π ia = − δHδe ai , (3.33a)˙ ϕ I = δHδπ ϕI , ˙ π ϕI = − δHδϕ I , (3.33b)˙ ζ I − = δδπ ζI H, ˙ π ζI = − H δδζ − , ˙ ζ I − = H δδπ ζI , ˙ π ζI = − δδζ I − H, (3.33c)˙Ψ + i = δδπ i Ψ H, ˙ π i Ψ = − H δδ Ψ + i , ˙Ψ + i = H δδπ i Ψ , ˙ π i Ψ = − δδ Ψ + i H, (3.33d)namely ˙ e ai = κ √− γ ( (cid:18) d − e ai e bj − e aj e bi (cid:19) π jb − d − e aj h ( d − + i π Ψ j + π Ψ j Ψ + i ) − π p Ψ [ b Γ pi − ( d − γ pi ] b Γ j k Ψ + k + Ψ + k b Γ kj [ b Γ ip − ( d − γ ip ] π p Ψ + ( i ↔ j ) i + 1 d − e ai (cid:16) − ζ I − π ζI − π ζI ζ I − + Ψ + j π j Ψ + π j Ψ Ψ + j (cid:17) ) , (3.34)˙ ϕ I = κ √− γ G IJ " − π ϕJ + Γ KJL [ G ] (cid:16) π ζK ζ L − + ζ L − π ζK (cid:17) − i (cid:16) π ζJ b Γ i Ψ + i + Ψ + i b Γ i π ζJ (cid:17) + κ √− γ i d − (cid:16) ζ I − b Γ i π i Ψ + π i Ψ b Γ i ζ I − (cid:17) , (3.35)˙Ψ + i = κ √− γ " − (cid:16) δ ki e aj + γ jk e ai (cid:17) π ka Ψ + j + 1 d − e aj π ja Ψ + i + iπ ϕI b Γ i ζ I − − d − (cid:16) e aj π la + e al π ja (cid:17) (cid:16)b Γ il − ( d − γ il (cid:17) b Γ j k Ψ + k − id − b Γ i /∂ϕ I π ζJ − d − (cid:16)b Γ ijk − ( d − γ ij b Γ k (cid:17) D k π j Ψ + 2 i∂ i ϕ I π ζI i − W Ψ + i − i d − ∂ I W b Γ i ζ I − . (3.36)13nd ˙ ζ I − = κ √− γ " G IJ / D π ζJ + ∂ i G IJ b Γ i π ζJ − d − e ai π ia ζ I − + 2 G LJ Γ IJK [ G ] π ϕL ζ K − − i G IJ π Jϕ b Γ i Ψ + i − i∂ i ϕ I π i Ψ + G IM G KN ∂ i ϕ J ( ∂ K G JM − ∂ M G KJ ) b Γ i π ζN + M JK G IK ζ J − − i ∂ I W b Γ i Ψ + i . (3.37)Here for simplicity we choose the gauge (B.12), which makes the radial Hamiltonian H reducedinto H = R d d x H . We emphasize that these flow (3.34), (3.35), (3.36) and (3.37) together withthe HJ equations form a complete set of equations of motion of the theory. To solve the HJ equation efficiently we divide the Hamilton’s principal function into several partsaccording to the structure of terms. Namely, we first split S into two sectors: S B , purely bosonicpart and S F , quadratic in fermions. The terms in S F are further split into 3 parts: S ζζ quadraticterms in ζ I − s, S ΨΨ quadratic terms in Ψ + i and S ζ Ψ bilinear in ζ I − and Ψ + i . In total, S = S B + S ζζ + S ΨΨ + S ζ Ψ . (4.1)Due to radiality and Lorentz structure of the fermionic sources, the asymptotic expansion of S B , S ζ Ψ , S ζζ and S ΨΨ should be S B = S B (0) + S B (2) + S B (4) + · · · , (4.2a) S ζ Ψ = S ζ Ψ(3 / + S ζ Ψ(7 / + · · · , (4.2b) S ζζ = S ζζ (1) + S ζζ (3) + S ζζ (5) + · · · , (4.2c) S ΨΨ = S ΨΨ(2) + S ΨΨ(4) + · · · . (4.2d)How to solve the HJ equation for the bosonic sector has been discussed in many literatureregarding some special models [30, 38, 49], though it is difficult to solve the HJ equation for thegeneral model. The key feature is that after finding solution of the HJ equation to leading order,we only need to solve (almost algebraic) first-order differential equation from the next order, thanksto the relation (3.27). Nevertheless, these first-order differential equations are not easy to solve atthe first attempt.Here we have another set of first-order differential equations, namely F − = F + = 0. These arerelatively simpler than the Hamiltonian constraint H = 0, so one can try to solve these constraintsfirst. Not surprisingly, it works well, in particular for the fermionic sector, and the solution istotally consistent with the other constraints, as we will see soon. One can use the flow equations (3.36) and (3.37) to determine the asymptotic behavior of Ψ + i and ζ I − in appendixB.3 instead of using the Euler-Lagrange equations of motion (B.18) and (B.19). One might try to solve the HJ equation for the general scalar-gravity model by using the argument in [38]. .1 Bosonic sector Let us first consider the bosonic sector. The corresponding Hamiltonian constraint H = 0 is κ √− γ (cid:20) (cid:18) d − γ ij γ kl − γ ik γ jl (cid:19) δ S B δγ ij δ S B δγ kl − G IJ δ S B δϕ I δ S B δϕ J (cid:21) + √− γ κ (cid:0) − R [ γ ] + G IJ ∂ i ϕ I ∂ i ϕ J + V ( ϕ ) (cid:1) = 0 . (4.3)One can readily see that the HJ equation for S (0) is κ √− γ (cid:20) (cid:18) d − γ ij γ kl − γ ik γ jl (cid:19) δ S (0) δγ ij δ S (0) δγ kl − G IJ δ S (0) δϕ I δ S (0) δϕ J (cid:21) + √− γ κ V ( ϕ ) = 0 . (4.4)The leading term of S , S (0) should not contain any derivatives and must be purely bosonic sothat its ansatz becomes S (0) = − κ Z d d x √− γ U ( ϕ ) . (4.5)Substituting this ansatz into the constraint F − = 0, we obtain14 (cid:16)b Γ i Ψ + j + b Γ j Ψ + i (cid:17) e aj δ S (0) δe ai + √− γ κ W b Γ i Ψ + i = 0 , (4.6)and find the unique solution for U ( ϕ ) given by U = W ( ϕ ), or S (0) = − κ Z d d x √− γ W . (4.7)As promised, we obtain (4.7) regardless of the sign of (3.3b). It follows that leading asymptoticsof the scalar field ϕ I is also determined whatever the sign of (3.3b) was chosen, as we see in (4.8c).From (4.7) we can now determine the leading asymptotics of the fields by using the above flowequations, namely e ai ( r, x ) ∼ e r e a (0) i ( x ) , (4.8a)Ψ + i ( r, x ) ∼ e r/ Ψ (0)+ i ( x ) , (4.8b)˙ ϕ I ∼ G IJ ∂ J W , or ϕ I ∼ e − µ I r ϕ I (0) , (4.8c)˙ ζ I − ∼ − ζ I − + ( G IK ∂ J ∂ K W ) ζ J − , or ζ I − ∼ e − ( µ I + ) r ζ I − (0) , (4.8d)where µ I stands for radial weight of ϕ I when the scalars are properly diagonalized.Now let us go to the next level of the bosonic sector. The HJ equation for S B (2) is then − d − W γ ij δδγ ij S B (2) + G IJ ∂ I W δδϕ J S B (2) + √− γ κ (cid:0) − R [ γ ] + G IJ ∂ i ϕ I ∂ i ϕ J (cid:1) = 0 . (4.9)The most general ansatz for S B (2) is as follows: S B (2) = 1 κ Z d d x √− γ (cid:0) Ξ( ϕ ) R + A IJ ( ϕ ) ∂ i ϕ I ∂ i ϕ J (cid:1) . (4.10)15hen, γ ij δδγ ij S B (2) = √− γκ d − (cid:0) Ξ R + A IJ ∂ i ϕ I ∂ i ϕ J (cid:1) − √− γκ ( d − ✷ Ξ , (4.11) δδϕ J S B (2) = √− γκ (cid:0) R∂ J Ξ + ∂ J A IK ∂ i ϕ I ∂ i ϕ K − D i (cid:0) A JK ∂ i ϕ K (cid:1)(cid:1) , (4.12)where we used the relation γ ij δR ij = D i D j δγ ij − γ ij ✷ ( δγ ij ) . (4.13)One can notice from (4.11) that L B (2) = √− γκ (cid:18) Ξ R + A IJ ∂ i ϕ I ∂ i ϕ J − d − d − ✷ Ξ (cid:19) . (4.14)Therefore, (4.9) becomes0 = R (cid:18) − d − d − W Ξ [1] + G IJ ∂ I W ∂ J Ξ − (cid:19) + ∂ i ϕ I ∂ i ϕ J (cid:16) − d − d − W A IJ + 2 W ∂ I ∂ J Ξ+ G KL ∂ L W ∂ K A IJ − G KL ∂ K W ∂ I A LJ + 12 G IJ (cid:17) + 2 ✷ ϕ I (cid:0) W ∂ I Ξ − G JK ∂ J W A IK (cid:1) , (4.15)and we obtain the equations for Ξ and A IJ − d − d − V I ∂ I Ξ − W , (4.16a)0 = − d − d − A IJ + V K ∂ K A IJ + ∂ I V K A JK + ∂ J V K A IK + 12 W G IJ , (4.16b)0 = ∂ I Ξ − V J A IJ , (4.16c)where V I ≡ W G IJ ∂ J W . (4.17)Note that A IJ should satisfy the condition ∂ I ( V K A JK ) = ∂ J ( V K A IK ) . (4.18)We emphasize that we do not discuss existence of the solution for A IJ and Ξ here. Nevertheless,the equations (4.16) are useful for determination of S ζζ (1) , S ΨΨ(2) and S ζ Ψ(3 / . S B (2 n ) ( n ≥
2) is obtained by the following recursive equation0 = − d − W γ ij π B (2 n ) ij + W V I π B (2 n ) I + κ √− γ n − X m =1 (cid:20) (cid:18) d − γ ij γ kl − γ ik γ jl (cid:19) π ijB (2 m ) π klB (2 n − m ) − G IJ π B (2 m ) I π B (2 n − m ) J (cid:21) . (4.19)In particular, when d = 4 the inhomogeneous terms on the RHS become2 κ √− γ (cid:18) d − γ ij γ kl − γ ik γ jl (cid:19) π ij (2) π kl (2) = √− γκ Ξ (cid:18) d d − R − R kl R kl (cid:19) , (4.20)where Ξ = 12( d −
2) + O ( ϕ ) (4.21)is the solution of (4.16a), while other inhomogeneous terms are asymptotically suppressed.16 .2 Fermionic sector After substituting the leading order solution (4.7) into the Hamiltonian constraint (3.9), we get thefollowing first-order differential equation for e S ≡ S − S (0) W (cid:18) − d − e ai e π ia + V I e π ϕI (cid:19) − d − W (cid:0) Ψ + i π i Ψ + π i Ψ Ψ + i (cid:1) + W (cid:18) d − δ JI + ∂ I V J (cid:19) ×× (cid:16) ζ I − π ζJ + π ζJ ζ I − (cid:17) + κ √− γ ( (cid:18) d − e ai e bj − e aj e bi (cid:19) e π ia e π jb − G IJ e π ϕI e π ϕJ + G IJ (cid:16) π ζI / D π ζJ − π ζI ←− / D π ζJ (cid:17) − e π ij h (Ψ + i π Ψ j + π Ψ j Ψ + i ) + 1 d − π p Ψ (cid:16)b Γ pi − ( d − γ pi (cid:17) b Γ j k Ψ + k + 1 d − + k b Γ kj (cid:16)b Γ ip − ( d − γ ip (cid:17) π p Ψ − i ∂ I WW γ jk (cid:16) ζ I − b Γ i π k Ψ + π k Ψ b Γ i ζ I − (cid:17) i + 2 d − γ ij e π ij (cid:18) − ζ I − π ζI − π ζI ζ I − + Ψ + k π k Ψ + π k Ψ Ψ + k − i ∂ I WW ζ I − b Γ k π k Ψ − i ∂ I WW π k Ψ b Γ k ζ I − (cid:19) + (cid:20) G IJ G LM ( ∂ J G MK − ∂ M G JK ) − W ∂ K (cid:18) G IL W (cid:19)(cid:21) e π ϕI (cid:16) ζ K − π ζL + π ζL ζ K − (cid:17) + i e π ϕI h d − (cid:16) ζ I − b Γ i π i Ψ + π i Ψ b Γ i ζ I − (cid:17) − G IJ (cid:16) π ζJ b Γ i Ψ + i + Ψ + i b Γ i π ζJ (cid:17) i − π k Ψ (cid:20)(cid:18) d − b Γ k b Γ j − γ kj (cid:19) / D − ←− / D (cid:18) d − b Γ k b Γ j − γ kj (cid:19)(cid:21) π j Ψ + id − (cid:16) π ζI /∂ϕ I b Γ i π i Ψ − π i Ψ b Γ i /∂ϕ I π ζI (cid:17) − i∂ i ϕ I (cid:16) π ζI π i Ψ − π i Ψ π ζI (cid:17) + G IM G KN ∂ i ϕ J ( ∂ K G IJ − ∂ I G KJ ) π ζM b Γ i π ζN ) + √− γ κ " − R [ γ ] + G IJ ∂ i ϕ I ∂ i ϕ J + G IJ ζ I − (cid:16) / D − ←− / D (cid:17) ζ J − + Ψ + i b Γ ijk (cid:16) D j − ←− D j (cid:17) Ψ + k + D k (cid:16) Ψ + i (cid:16) γ jk b Γ i − γ ik b Γ j (cid:17) Ψ + j (cid:17) + i G IJ ∂ i ϕ J (cid:16) ζ I − b Γ j b Γ i Ψ + j − Ψ + j b Γ i b Γ j ζ I − (cid:17) + ∂ K G IJ ∂ i ϕ J (cid:16) ζ I − b Γ i ζ K − − ζ K − b Γ i ζ I − (cid:17) − iV I (cid:16) ζ I − b Γ ij D i Ψ + j + Ψ + j ←− D i b Γ ij ζ I − (cid:17) + V I G JK ∂ i ϕ J (cid:16) ζ I − b Γ i ζ K − − ζ K − ˆΓ i ζ I − (cid:17) , (4.22)where e π ia ≡ δ e S δe ai , e π ϕI ≡ δ e S δϕ I . (4.23)From this one could write the recursive equation for every S ( k ) . It, however, looks too complicated,and thus we first write down equations for S ζζ (1) , S ζ Ψ(3 / and S ΨΨ(2) , namely0 = − W L ζζ (1) + W ( V I ∂ I + 12( d − ζ I − δδζ I − + ←− δδζ I − ζ I − ! + ∂ I V J ζ I − δδζ J − + ←− δδζ J − ζ I − ! ) S ζζ (1) + √− γ κ h G IJ (cid:16) ζ I − / D ζ J − − ζ I − ←− / D ζ J − (cid:17) + ( V I G JK + ∂ K G IJ ) ∂ i ϕ J (cid:16) ζ I − b Γ i ζ K − − ζ K − ˆΓ i ζ I − (cid:17) i . (4.24a)0 = − d − d − W L ζ Ψ(3 / + W " V I ∂ I + 12( d − ζ I − δδζ I − + ←− δδζ I − ζ I − ! + ∂ L V K ζ L − δδζ K − + ←− δδζ K − ζ L − ! S ζ Ψ(3 / √− γ κ i h − V I (cid:16) ζ I − b Γ ij D i Ψ + j + Ψ + j ←− D i b Γ ij ζ I − (cid:17) + G IJ (cid:16) ζ I − b Γ j /∂ϕ J Ψ + j − Ψ + j /∂ϕ J b Γ j ζ I − (cid:17) i , (4.24b)0 = − d − d − W L ΨΨ(2) + G IJ ∂ I W δδϕ J S ΨΨ(2) + √− γ κ h Ψ + i b Γ ijk ( D j − ←− D j )Ψ + k + D k (cid:16) Ψ + i ( γ jk b Γ i − γ ik b Γ j )Ψ + j (cid:17) i , (4.24c)where we used (3.27).While (4.24a) and (4.24b) are not so easy to treat at first sight, the solution of (4.24c) is obvious,namely L ΨΨ(2) = − √− γκ Ξ h Ψ + i b Γ ijk ( D j − ←− D j )Ψ + k + D k (cid:16) Ψ + i ( γ jk b Γ i − γ ik b Γ j )Ψ + j (cid:17)i , (4.25)once we take into account (4.16a). Instead of solving (4.24a) and (4.24b) directly, we now try F + constraint (3.13), which greatly reduces the amount of efforts. They are respectively at the ’level’1 and 3/2 i G IJ ∂ I W δδζ J − S ζζ (1) + 1 d − W b Γ i δδ Ψ + i S ζ Ψ(3 / = √− γ κ i G IJ ∂ i ϕ J ˆΓ i ζ I − , (4.26a)1 d − W b Γ i δδ Ψ + i S ΨΨ(2) + i G IJ ∂ I W δδζ J − S ζ Ψ(3 / = √− γκ b Γ ij D i Ψ + j . (4.26b)The solution (4.25) allows us to solve (4.26b) immediately and we obtain δδζ I − S ζ Ψ(3 / = i √− γκ (cid:16) − ∂ I Ξ ζ I − b Γ ij D i Ψ + j + A IJ ζ I − b Γ i /∂ϕ J Ψ + i (cid:17) . (4.27)One can readily see that S ζ Ψ(3 / = iκ Z d d x √− γ h ∂ I Ξ (cid:16) Ψ + i ←− D j b Γ ij ζ I − − ζ I − b Γ ij D i Ψ + j (cid:17) ++ A IJ (cid:16) ζ I − b Γ i /∂ϕ J Ψ + i − Ψ + i /∂ϕ I b Γ i ζ J − (cid:17) i . (4.28)In the same way, we find from (4.26a) that S ζζ (1) = 1 κ Z d d x √− γ (cid:16) A IJ ζ I − ( / D − ←− / D ) ζ J − + ( ∂ J A Ik − ∂ I A JK ) ζ I − /∂ϕ K ζ J − (cid:17) . (4.29)Moreover, we can confirm that the solutions (4.29) and (4.28) satisfy the Hamiltonian constraints(4.24a) and (4.24b) respectively. That is not the whole story, and one has to convince himself that F − = 0 constraint also holds for these solutions. From (3.13), we obtain0 = D i δδ Ψ + i S ΨΨ(2 k ) + i ∂ i ϕ I b Γ i δδζ I − S ζ Ψ(2 k − / − b Γ i Ψ + j δδγ ij S B (2 k ) , (4.30a)0 = D i δδ Ψ + i S ζ Ψ(2 k − / + i ∂ i ϕ I b Γ i δδζ I − S ζζ (2 k − − i ζ I − δδϕ I S B (2 k ) , (4.30b)where k is an arbitrary positive integer. It is not so difficult to check the solutions we obtainedsatisfy the constraints (4.30b) and (4.30a) for k = 1, implying that the combination S B (2) + S ΨΨ(2) + S ζζ (1) + S ζ Ψ(3 / (4.31)18s ( ǫ + ) supersymmetric.We have seen how to obtain the Hamilton’s principal function in the fermionic sector from itsbosonic supersymmetric partner, but at the lower ’level’. It was relatively easy because we couldgive the most general ansatz for S B (2) which has a small number of terms. To go further we shouldfirst find out S B (4) , S B (6) , · · · and obtain their SUSY partners by using the above trick. The ansatzfor S B (2 n ) ( n ≥ d = 4. This is because in the generic case that there are no scalar fieldsdual to marginal operators, S ζζ (3) and S ζ Ψ(7 / are asymptotically suppressed in 4 dimensions. As aresult, what is remained in the case of d = 4 is only to find out S ΨΨ(4) that are the logarithmicallydivergent terms, which are directly related to the holographic Weyl anomaly [13].We should emphasize that from the general analysis here the divergent counterterms (exceptfor S (0) ) always satisfy the constraint F − = 0 and so is the renormalized on-shell action b S ren .We finish this subsection by presenting the recursive relation obtainted from (3.12), namely0 = − d − W b Γ i π i Ψ( n − / − i G IJ ∂ I W π ζJ ( n − + κ √− γ ⌊ n ⌋ − X m =1 h i G IJ π ϕI (2 m ) π ζJ ( n − m − + π jk (2 m ) (cid:18) d − γ jk b Γ i − γ ij b Γ k (cid:19) π i Ψ( n − m − / i , (4.32)where (integer or half-integer) n ≥
4. This will be useful for determination of the super-Weylanomaly in section 5.1.2.
As mentioned in subsection 3.2, every S ( k ) in the asymptotic expansion (3.25) with respect to theoperator δ e contains the poles related to the logarithmically divergent terms. Let us denote suchterms by e S ( k ) . Whereas e S B (4) and e S ΨΨ(4) are purely gravitational (meaning that they are related onlyto the metric and the gravitino field) and universal, e S ζζ (1) , e S ζ Ψ(3 / , e S B (2) and e S ΨΨ(2) are model-dependent.We first discuss the former and then study the latter for a simple model. e S B (4) is easily obtained from (4.19) and (4.20), namely e S B (4) ≡ Z d d x √− γ e L B (4) log e − r = 14 κ ( d − Z d d x √− γ (cid:18) d d − R − R ij R ij (cid:19) log e − r , (4.33)which is already well-known. Meanwhile, e S ΨΨ(4) is determined by the inhomogeneous terms of theHamiltonian constraint (4.22) at the ’level’ 4, namely e S ΨΨ(4) ≡ Z d d x √− γ e L ΨΨ(4) log e − r = Z d d x κ √− γ ( (cid:18) d − γ ij γ kl − γ ik γ jl (cid:19) e π ij (2) e ak π l Ψ a (2) − e π ij (2) (Ψ + k b Γ j b Γ k π (2)Ψ i + π (2)Ψ i b Γ k b Γ j Ψ + k ) When the boundary metric is flat, (4.34) matches with the result in [18]. d − γ ij e π ij (2) (cid:16) Ψ k + b Γ kl π l (2)Ψ + π (2) l Ψ b Γ lk Ψ k + (cid:17) + 12( d −
1) ( π (2) k Ψ b Γ k / D b Γ j π (2) j Ψ − π (2) k Ψ b Γ k ←− / D b Γ j π (2) j Ψ )+ 12 ( π (2) i Ψ / D π (2)Ψ i − π (2) i Ψ ←− / D π (2)Ψ i ) ) log e − r = 18( d − κ Z d d x √− γ ( ( d − R (Ψ + i b Γ ijk D j Ψ + k − Ψ + i ←− D j b Γ ijk Ψ + k )+ dd − RD j h Ψ + i ( γ ij b Γ k − γ jk b Γ i )Ψ + k i − ( d − R (Ψ + i b Γ i b Γ jk D j Ψ + k − Ψ + i ←− D j b Γ ij b Γ k Ψ + k )+ ( d − d − R h Ψ k + b Γ j D k Ψ + j − Ψ i + / D Ψ + i − Ψ + i ←− D k b Γ i Ψ + k + Ψ k + ←− / D Ψ + k i + 2 R kl h Ψ + i [( γ ip b Γ k − γ ik b Γ p ) D l − ←− D l ( γ ip b Γ k − γ pk b Γ i )]Ψ + p − Ψ + i b Γ i b Γ jl D j Ψ k + + Ψ k + ←− D j b Γ lj b Γ i Ψ + i − D j [Ψ l + b Γ kji Ψ + i − Ψ + i b Γ ijk Ψ l + − Ψ + i ( γ jk γ pl b Γ i − γ jk γ il b Γ p + γ jp γ il b Γ k − γ pl γ ij b Γ k )Ψ + p ] i − d − d − + i ←− D j b Γ ij / D b Γ kl D k Ψ + l − Ψ + i ←− D j b Γ ij ←− / D b Γ kl D k Ψ + l ) − + p ←− D q b Γ pqi / D b Γ ijk D j Ψ + k − Ψ + p ←− D q b Γ pqi ←− / D b Γ ijk D j Ψ + k ) ) log e − r . (4.34)Although nontrivial, one can show that e S B (4) + e S ΨΨ(4) satisfies the constraints H = F − = F + = 0 (i.e.conformal, supersymmetry and super-Weyl invariance), namely0 = e ai δδe ai + 12 Ψ + i δδ Ψ + i + ←− δδ Ψ + i Ψ + i ! e S ΨΨ(4) , (4.35a)0 = b Γ i Ψ + j δδγ ij e S B (4) + D i δδ Ψ + i e S ΨΨ(4) , (4.35b)0 = b Γ i δδ Ψ + i S ΨΨ(4) . (4.35c) Up to now we obtained generic part of the divergent counterterms. S ct can involve additionalfinite terms which satisfy the first class constraints (3.14), though. The possible bosonic finitecounterterms are Euler density and Weyl invariant in 4D, namely, E (4) = 164 (cid:16) R ijkl R ijkl − R ij R ij + R (cid:17) , I (4) = − (cid:18) R ijkl R ijkl − R ij R ij + 13 R (cid:19) (4.36)Integral of the Euler density E (4) by itself satisfies all the first class constraints, since it is topologicalquantity, any local variation of which vanishes. Therefore, we find that the possible supersymmetricfinite counterterms are linear combination of X I = 64 I (4) + ( d − R (Ψ + i b Γ ijk D j Ψ + k − Ψ + i ←− D j b Γ ijk Ψ + k ) + dd − RD j h Ψ + i ( γ ij b Γ k − γ jk b Γ i )Ψ + k i − ( d − R (Ψ + i b Γ i b Γ jk D j Ψ + k − Ψ + i ←− D j b Γ ij b Γ k Ψ + k ) Otherwise, these finite terms would generate trivial cocycle terms, which do not have any physical implication.
20 ( d − d − R h Ψ k + b Γ j D k Ψ + j − Ψ i + / D Ψ + i − Ψ + i ←− D k b Γ i Ψ + k + Ψ k + ←− / D Ψ + k i + 2 R kl h Ψ + i [( γ ip b Γ k − γ ik b Γ p ) D l − ←− D l ( γ ip b Γ k − γ pk b Γ i )]Ψ + p − Ψ + i b Γ i b Γ jl D j Ψ k + + Ψ k + ←− D j b Γ lj b Γ i Ψ + i − D j [Ψ l + b Γ kji Ψ + i − Ψ + i b Γ ijk Ψ l + − Ψ + i ( γ jk γ pl b Γ i − γ jk γ il b Γ p + γ jp γ il b Γ k − γ pl γ ij b Γ k )Ψ + p ] i − d − d − + i ←− D j b Γ ij / D b Γ kl D k Ψ + l − Ψ + i ←− D j b Γ ij ←− / D b Γ kl D k Ψ + l ) − + p ←− D q b Γ pqi / D b Γ ijk D j Ψ + k − Ψ + p ←− D q b Γ pqi ←− / D b Γ ijk D j Ψ + k ) , (4.37)and X E = E (4) , X P = P = 164 ǫ ijkl R ijpq R klpq , (4.38)where P is the Pontryagin density. Notice that integral of P is the topological quantity and thus itcan be a finite counterterm as in the case of the Euler density, as long as there is no other symmetrywhich prevents its appearance.In summary, collecting all of these finite counterterms and the previous divergent ones we obtain S ct = − (cid:0) S (0) + S (1) + S (2) (cid:1) − (cid:16)e S B (4) + e S ΨΨ(4) (cid:17) + · · · = 1 κ Z d d x √− γ n W − Ξ R − A IJ ∂ i ϕ I ∂ i ϕ J − A IJ ζ I − ( / D − ←− / D ) ζ J − − ( ∂ J A IK − ∂ I A JK ) ζ I − /∂ϕ K ζ J − − i∂ I Ξ(Ψ + i ←− D j b Γ ij ζ I − − ζ − b Γ ij D i Ψ + j ) − iA IJ ( ζ I − b Γ i /∂ϕ J Ψ + i − Ψ + i /∂ϕ I b Γ i ζ J − ) + ΞΨ + i b Γ ijk ( D j − ←− D j )Ψ + k + Ψ + i ( ∂ i Ξ b Γ j − ∂ j Ξ b Γ i )Ψ + j o − e S B (4) − e S ΨΨ(4) + 1 κ Z d d x √− γ ( α I X I + α E X E + α P X P ) + · · · , (4.39)where e S B (4) and e S ΨΨ(4) are given in (4.33) and (4.34) and α I , α E and α P are arbitrary constants. Herethe ellipsis stand for the model-dependent terms, which we discuss in section 4.5 for a simple toymodel. For completeness, we present an application of our general procedure to a simple toy model.In the toy model there is only one scalar field, which corresponds to the operator with thescaling dimension ∆ = d − d = 4. It then implies that W = − ( d − − ϕ + k ϕ + k ϕ + O ( ϕ ) , (4.40)where k and k are arbitrary constants, and therefore the solution of (4.16a), (4.16b) and (4.16c)becomes Ξ = 12( d − − d − · d − ϕ + · · · , A IJ = − d − ·
12 + · · · . (4.41)The divergent counterterms that we need other than those in (4.39) are only the logarithmicallydivergent terms. Following the argument in section 3.2 again we can see them from the poles (when d = 4) in Ξ and A IJ and are responsible for additional logarithmically divergent terms.21e thus obtain e S B (2) = 14 κ Z d d x √− γ (cid:18) d − ϕ R + ∂ i ϕ∂ i ϕ (cid:19) log e − r , (4.42a) e S ζζ (1) = 14 κ Z d d x √− γ ( ζ − / D ζ − + h . c . ) log e − r , (4.42b) e S ζ Ψ(3 / = i κ Z d d x √− γ (cid:18) ζ − b Γ i /∂ϕ Ψ + i − d − ϕζ − b Γ ij D i Ψ + j + h . c . (cid:19) log e − r , (4.42c) e S ΨΨ(2) = − κ Z d d x √− γ d − (cid:18) ϕ Ψ + i b Γ ijk D j Ψ + k + ϕ Ψ + i ∂ i ϕ b Γ j Ψ + j + h . c . (cid:19) log e − r . (4.42d)One can easily check that e S B (2) + e S ζζ (1) + e S ζ Ψ(3 / + e S ΨΨ(2) again satisfies the constraints H = F − = F + = 0.Besides X I and X E , the possible finite counterterms (conformal and ǫ + supersymmetric) are X = 12( d − ϕ R + ∂ i ϕ∂ i ϕ + ζ − / D ζ − + iζ − b Γ i /∂ϕ Ψ + i − id − ϕζ − b Γ ij D i Ψ + j − d − ϕ Ψ + i b Γ ijk D j Ψ + k − d − ϕ Ψ + i ∂ i ϕ b Γ j Ψ + j + h . c ., (4.43)and the finite term k ϕ in W should be in the counterterms without any ambiguity, due to the F − constraint.In total, the divergent counterterms for the toy model are S div ct = − (cid:0) S (0) + S (1) + S (3 / + S (2) (cid:1) − Z d d x √− γ e L [4] log e − r , (4.44)where the logarithmically divergent counterterms are Z d d x √− γ e L [4] log e − r = e S ζζ (1) + e S ζ Ψ(3 / + e S B (2) + e S ΨΨ(2) + e S B (4) + e S ΨΨ(4) . (4.45)Adding possible finite ones, the whole counterterms are S ct = 1 κ Z d d x √− γ h − ( d − − ϕ + k ϕ + k ϕ − d − R + 12( d −
2) Ψ + i b Γ ijk ( D j − ←− D j )Ψ + k i − Z d d x √− γ e L [4] log e − r + 1 κ Z d d x √− γ ( α I X I + α E X E + α P X P + α X ) , (4.46)where α E , α I , α P and α are arbitrary constants and determine the renormalization scheme. Now that all the counterterms are determined, we can relate by the holographic dictionary [12] therenormalized canonical momenta to the renormalized local operators of the boundary field theory,namely T ia = − lim r →∞ e ( d +1) r √− γ (cid:18) π ia + δS ct δe ai (cid:19) := − | e (0) | Π ia , (5.1a)22 ϕI = lim r →∞ e ( d + µ I ) r √− γ (cid:18) π ϕI + δS ct δϕ I (cid:19) := 1 | e (0) | Π ϕI , (5.1b) O ζI = lim r →∞ e ( d + µ I + ) r √− γ π ζI + δS ct δζ I ! := 1 | e (0) | Π ζI , (5.1c) S i = lim r →∞ e ( d + ) r √− γ (cid:18) π i Ψ + δS ct δ Ψ + i (cid:19) := 1 | e (0) | Π i Ψ , (5.1d)where T ia is the energy-momentum tensor, S i is the supercurrent and e (0) = det( e a (0) i ). We notethat since these local renormalized operators are obtained in the presence of arbitrary sources wecan obtain higher-point functions simply by taking functional derivative of them with respect tothe sources. One can find from the computation of section 4 and 4.5 that S ct satisfies the first class constraints H i = F − = 0 and local Lorentz constraint (3.7), and so does the renormalized on-shell action b S ren .This is also related to the fact that these constraints are linear functional derivative equations.Since H and F + are not linear constraints, one should expect that the counterterms do not satisfythe constraints H = 0 and F + = 0 in general and thus generate the non-trivial cocycle terms, whichappear in the constraints for the renormalized on-shell action. Also, the poles appearing in solvingthe constraints contribute to the corresponding anomaly. In total, after removing all divergentcounterterms, the first class constraints (3.12), (3.13), (3.9), (3.10) and (3.15) are reduced into0 = −
12 Γ a Ψ (0)+ i T ia + i ζ I (0) − O ϕI − i /∂ϕ I (0) O ζI − D i S i , (5.2a) A sW = − i G IJ ∂ I WO ζJ + b Γ i S i , (5.2b) A W = e a (0) i T ia − G IJ ∂ I WO ϕJ − (cid:0) Ψ (0)+ i S i + h . c . (cid:1) ++ (cid:18) δ JI − ∂ I ∂ J W (cid:19) (cid:16) ζ I (0) − O ζJ + h . c . (cid:17) , (5.2c)0 = e ai (0) D j T ja + ∂ i ϕ I (0) O ϕI + (cid:16) ζ I (0) − ←− D i O ζI + h . c . (cid:17) + (cid:16) Ψ (0)+ j ←− D i S j + h . c . (cid:17) − D j (cid:16) Ψ i (0)+ S j + h . c . (cid:17) , (5.2d)0 = e (0) ai T ib − e (0) bi T ia + 12 (cid:16) ζ I (0) − Γ ab O ζI + Ψ (0)+ i Γ ab S i + h . c . (cid:17) , (5.2e)where A sW and A W are super-Weyl and Weyl anomaly densities respectively. In (5.2) we keep onlyup to the quadratic order and zero order in ϕ I in the Taylor expansion of W and G IJ respectively.We call (5.2)s as Ward identities which relate the local sources and their dual operators of thefield theory. These Ward identities that play a key role in the following discussions reflect theremained local symmetries of the bulk SUGRA after fixing the strong FG gauge (B.13), on whichwe did HR for the bulk theory in section 4. The remaining local symmetry transformations ofSUGRA are called generalized Penrose-Brown-Henneaux (gPBH) transformations, whose actionon the sources are carefully treated in appendix B.4. The resulting expressions are (B.37). Before Definition of the energy-momentum tensor is modified when the vielbein is used instead of the metric, see e.g.(2.198) in [50]. Spinor index of the supercurrent S i is implicit. d = 4. Although there are many ways to find the Weyl anomaly, a direct way is to read it from the HJequation. One can see that in (4.22) substituted by L [4] , the first linear terms are indeed the RHSof the trace Ward identity (5.2c) and the rest of the terms give us part of the trace anomaly. Theterms with pole 1 / ( d −
4) which appeared in the HJ equations for S (1) , · · · , S (4) are also inheritedinto (4.22) for S [4] . These non-homogeneous terms are already identified to the logarithmicallydivergent terms and thus we only need to multiply them by 2 to obtain the trace anomaly [8]. Forgraviton and gravitino parts, the trace anomaly density is then A ( G )W [ e, Ψ + ] = 14( d − κ ( d d − R − R ij R ij + ( d − R (Ψ + i b Γ ijk D j Ψ + k − Ψ + i ←− D j b Γ ijk Ψ + k )+ dd − RD j h Ψ + i ( γ ij b Γ k − γ jk b Γ i )Ψ + k i − ( d − R (Ψ + i b Γ i b Γ jk D j Ψ + k − Ψ + i ←− D j b Γ ij b Γ k Ψ + k )+ ( d − d − R h Ψ k + b Γ j D k Ψ + j − Ψ i + / D Ψ + i − Ψ + i ←− D k b Γ i Ψ + k + Ψ k + ←− / D Ψ + k i + 2 R kl h Ψ + i [( γ ip b Γ k − γ ik b Γ p ) D l − ←− D l ( γ ip b Γ k − γ pk b Γ i )]Ψ + p − Ψ + i b Γ i b Γ jl D j Ψ k + + Ψ k + ←− D j b Γ lj b Γ i Ψ + i − D j [Ψ l + b Γ kji Ψ + i − Ψ + i b Γ ijk Ψ l + − Ψ + i ( γ jk γ pl b Γ i − γ jk γ il b Γ p + γ jp γ il b Γ k − γ pl γ ij b Γ k )Ψ + p ] i − d − d − + i ←− D j b Γ ij / D b Γ kl D k Ψ + l − Ψ + i ←− D j b Γ ij ←− / D b Γ kl D k Ψ + l ) − + p ←− D q b Γ pqi / D b Γ ijk D j Ψ + k − Ψ + p ←− D q b Γ pqi ←− / D b Γ ijk D j Ψ + k ) ) . (5.3)The holographic computation of the supersymmetric Weyl anomaly in 4D is quite remarkable;even though its bosonic part has already been known for a long time, it seems really tough to obtainits SUSY partner terms by means of giving an ansatz and finding out the coefficients, whereas theholography enables us to compute them directly.We comment that although the bosonic sector of A G W is the sum of the a anomaly density E (4) and c anomaly one I (4) , the fermionic sector is in fact SUSY partner of c anomaly density up to atotal derivative. This is because integral of E (4) is supersymmetric by itself, as mentioned before.For the toy model of section 4.5, we have additional contribution to the Weyl anomaly density,which is A (model) W [Φ] = 12 κ d − ϕ R + ∂ i ϕ∂ i ϕ + ζ − / D ζ − + iζ − b Γ i /∂ϕ Ψ + i − id − ϕζ − b Γ ij D i Ψ + j The SUSY completion of the Weyl anomaly in the 4 dimensional supersymmetric theory was obtained in [51, 52]by using the superspace formalism. To get the fermionic sector explicitly, however, one has yet to expand it furtheraround the bosonic coordinates. d − ϕ Ψ + i b Γ ijk D j Ψ + k − d − ϕ Ψ + i ∂ i ϕ b Γ j Ψ + j + h . c . ! . (5.4)The total Weyl anomaly density is thus given by A W [Φ] = A ( G ) W [Φ] + A (model) W [Φ]= A ( G ) W + 12 κ d − ϕ R + ∂ i ϕ∂ i ϕ + ζ − / D ζ − + iζ − b Γ i /∂ϕ Ψ + i − id − ϕζ − b Γ ij D i Ψ + j − d − ϕ Ψ + i b Γ ijk D j Ψ + k − d − ϕ Ψ + i ∂ i ϕ b Γ j Ψ + j + h . c . ! . (5.5) Here we compute the super-Weyl anomaly for the toy model. As pointed out in section 4.2, (4.26b)hold up to the finite order. For the toy model, it means that the RHS of (4.26b) is not canceledout and an additional finite term + √− γκ ϕ d − b Γ ij D i Ψ + j (5.6)comes out from the LHS of (4.26b). As in the case of Weyl anomaly, we thus get from (4.32) − i G IJ ∂ J W π ζ (7 / I − d − W b Γ i π i (4)Ψ = − κ √− γ π jk (2) (cid:18) d − γ jk b Γ i − γ ij b Γ k (cid:19) π i (2)Ψ − √− γ κ i∂ i ϕ b Γ i ζ − + √− γκ ϕ d − b Γ ij D i Ψ + j = √− γκ (cid:20) d − (cid:18) dd − R b Γ kl − R ik b Γ il + 2 R il b Γ ik (cid:19) D k Ψ + l − i ∂ i ϕ b Γ i ζ − + 12( d − ϕ b Γ ij D i Ψ + j (cid:21) , (5.7)or A sW [Φ] = 1 κ " d − (cid:18) dd − R b Γ kl − R ik b Γ il + 2 R il b Γ ik (cid:19) D k Ψ + l −− i ∂ i ϕ b Γ i ζ − + 12( d − ϕ b Γ ij D i Ψ + j . (5.8)Notice that terms in the first bracket A ( G )sW [ e, Ψ + ] = 1 κ d − (cid:18) dd − R b Γ kl − R ik b Γ il + 2 R il b Γ ik (cid:19) D k Ψ + l (5.9)are universal, in the sense that they do not depend on the model. Notice that (5.9) is differentfrom the result obtained by using Feynman diagram [40] (see also [41]). The reason seems to bethat here we computed the sum of a -anomaly and c -anomaly, while the super-trace anomaly in [40]is a different linear combination of them. In any case, the result of [40] does not satisfy the WZconsistency condition, as commented in footnote 4. It seems like that the bosonic sector of the conformal anomaly density A here is different from the one given in [8](see between (5.61) and (5.62) there), because of the ϕ term in e L (4) . One can, however, easily check that it actuallyvanishes, taking into account (2.4). This is because in our model the superpotential W is local by construction. .1.3 Wess-Zumino consistency condition From the relation (4.35a) and corresponding equation for the toy model we find that the Weylanomaly (5.3) and (5.5) satisfy Wess-Zumino (WZ) consistency condition, which can be seen asfollows. Defining the Weyl transformation operator δ σ by δ σ ≡ Z ∂ M d d x X Φ (0) δ σ Φ (0) δδ Φ (0) , (5.10)where Φ (0) refers to the source for every field Φ, the WZ consistency condition becomes that[ δ σ , δ σ ] S ren = 0. This is equivalent to δ σ R d d x A W σ is symmetric in σ and σ , which can beseen from (4.35a) since X Φ (0) δ σ Φ (0) δδ Φ (0) Z d d y A W σ = σ ∂ i ( T ∂ i σ ) , (5.11)for a certain scalar function T . We note that the SUSY and super-Weyl invariance of Weyl anomalyfollowed by (4.35b) and (4.35c) can be thought as the WZ consistency checks.In order to see the super-Weyl anomaly (5.8) satisfies WZ consistency condition, first we needto find the algebra of relevant symmetries. From (B.37), one can readily see that [ δ ǫ + , δ ǫ ′− ] e ai = ( δ σ + δ λ ) e ai , [ δ ǫ + , δ ǫ ′− ] ϕ I = ( δ σ + δ λ ) ϕ I , (5.12)with the parameters σ = ǫ ′− ǫ + , λ = ǫ ′− Γ ab ǫ + . Notice that in our stage it is impossible to seethe above commutator for the fermionic sources, since our consideration is limited to quadraticorder in fermions. However, (5.12) provides us the WZ consistency condition for the super-Weylanomaly, namely (cid:18) δ ǫ + Z d d x | e (0) | ǫ ′− A sW [Φ (0) ] (cid:19) (cid:12)(cid:12)(cid:12) bosonic = (cid:16) [ δ ǫ + , δ ǫ ′− ] S ren (cid:17) (cid:12)(cid:12)(cid:12) bosonic = − Z d d x | e (0) | σ A ( B ) W [Φ (0) ] , (5.13)since δ λ S ren = 0. Here A ( B ) W refers to the bosonic sector of the Weyl anomaly. In the following weshow (5.13) in detail, namely δ ǫ + Z d d x √− γ ǫ ′− A sW == 1 κ Z d d x √− γǫ ′− h d − (cid:18) dd − R b Γ kl − R ik b Γ il + 2 R il b Γ ik (cid:19) D k D l ǫ + − ∂ i ϕ b Γ i b Γ j ∂ j ϕǫ + + 12( d − ϕ b Γ ij D i D j ǫ + i = 1 κ Z d d x √− γ ǫ ′− h d − (cid:18) dd − R b Γ kl − R ik b Γ il + 2 R il b Γ ik (cid:19) R mnkl b Γ mn − ∂ i ϕ∂ i ϕ + 116( d − ϕ b Γ ij b Γ kl R ijkl i ǫ + = 1 κ Z d d x √− γ ǫ ′− h d − (cid:18) − dd − R + 8 R ij R ij (cid:19) − ∂ i ϕ∂ i ϕ − d − ϕ R i ǫ + Here the subscript o is omitted again, which was used to denote the leading asymptotics of the variation param-eters in appendix B.4. − Z d d x √− γ σ A ( B ) W , (5.14)where we have again σ = ǫ ′− ǫ + . In the above computation, we omitted the subscript (0) forsimplicity. In the same spirit, one can find another WZ consistency condition for the super-Weylanomaly from [ δ ǫ − , δ ǫ ′− ] e ai = [ δ ǫ − , δ ǫ ′− ] ϕ I = 0 . (5.15)We therefore have (cid:16) [ δ ǫ − , δ ǫ ′− ] S ren (cid:17) (cid:12)(cid:12)(cid:12) bosonic = 0 , (5.16)which can be shown in the same way. Now that the Ward identities are completely given, we can use (5.2) to derive gPBH transformationof the renormalized canonical momenta, without using FG expansion of the induced fields andthemselves [45, 53, 54]. In order to describe the gPBH transformation of the induced fields andtheir renormalized canonical momenta in an integrated way, we introduce concept of the generalizedPoisson bracket, which is defined by (see e.g. (6.30) in [53]) { A [Φ (0) , Π Φ ] , B [Φ (0) , Π Φ ] } ≡ Z ∂ M d d x X Φ (0) (cid:18) δAδ Φ (0) δBδ Π Φ − δBδ Φ (0) δAδ Π Φ (cid:19) = Z ∂ M d d x δAδe a (0) i δBδ Π ia − δBδe a (0) i δAδ Π ia + δAδϕ I (0) δBδ Π ϕI − δBδϕ I (0) δAδ Π ϕI + A ←− δδ Ψ (0)+ i −→ δδ Π i Ψ B − B ←− δδ Ψ (0)+ i −→ δδ Π i Ψ A + A ←− δδζ I (0) − −→ δδ Π ζI B − B ←− δδζ I (0) − −→ δδ Π ζI A + B ←− δδ Π i Ψ −→ δδ Ψ (0)+ i A − A ←− δδ Π i Ψ −→ δδ Ψ (0)+ i B + B ←− δδ Π ζI −→ δδζ I (0) − A − A ←− δδ Π ζI −→ δδζ I (0) − B ! , (5.17)where A [Φ (0) , Π Φ ] and B [Φ (0) , Π Φ ] are arbitrary functions on the phase space (Φ (0) ,Π Φ ). The Wardidentities (5.2) then allow us to define a constraint function on the phase space C [ ξ, σ, ǫ ± , λ ] ≡ Z ∂ M d d x ( ξ i (cid:16) e ai (0) D j Π ja − ( ∂ i ϕ I (0) )Π ϕI − ( ζ I (0) − ←− D i )Π ζI − Π ζI ( D i ζ I (0) − ) − Π j Ψ (cid:0) D i Ψ (0)+ j (cid:1) − (cid:16) Ψ (0)+ j ←− D i (cid:17) Π j Ψ + D j (Π j Ψ Ψ i (0)+ + Ψ i (0)+ Π j Ψ ) (cid:17) + σ h − e a (0) i Π ia − G IJ ∂ I W Π ϕJ −
12 (Ψ (0)+ i Π i Ψ + h . c . )+ (cid:0) δ JI − ∂ I ∂ J W (cid:1)(cid:0) ζ I (0) − Π ζJ + h . c . (cid:1) − | e (0) |A W [Φ (0) ] i + ǫ + (cid:16) −
12 Γ a Ψ (0)+ i Π ia − i ζ I (0) − Π ϕI + i /∂ϕ I (0) Π ζI + D i Π i Ψ (cid:17) + (cid:16)
12 Π ia Ψ (0)+ i Γ a + i ϕI ζ I (0) − + i ζI /∂ϕ I (0) + Π i Ψ ←− D i (cid:17) ǫ + + ǫ − (cid:16) i G IJ ∂ I W Π ζJ − b Γ i Π i Ψ + | e (0) |A sW [Φ (0) ] (cid:17) + (cid:16) Π i Ψ b Γ i − i G IJ ∂ I W Π ζJ + | e (0) |A sW [Φ (0) ] (cid:17) ǫ − λ ab h e (0)[ ai Π ib ] + 14 (cid:0) ζ I (0) − Γ ab Π ζI + Ψ (0)+ i Γ ab Π i Ψ + h . c . (cid:1)i) , (5.18)which generates the gPBH transformation (B.4) through the Poisson bracket δ σ,ǫ ± ,λ Φ (0) = {C [ σ, ǫ ± , λ ] , Φ (0) } , δ σ,ǫ ± ,λ Π Φ = {C [ σ, ǫ ± , λ ] , Π Φ } , (5.19a) δ (cgct) ξ Φ (0) = {C [ ξ ] , Φ (0) } , δ (cgct) ξ Π Φ = {C [ ξ ] , Π Φ } . (5.19b)Here δ (cgct) ξ refers to the covariant general coordinate transformation (see e.g. section 11.3 of [46]),under which variation of the fields is given by δ (cgct) ξ e a (0) i = D i ξ a , δ (cgct) ξ ϕ I (0) = ξ a ∂ a ϕ I (0) ≡ ξ i ∂ i ϕ I (0) , (5.20a) δ (cgct) ξ Ψ (0)+ i = ξ j D j Ψ (0)+ i + ( D i ξ j )Ψ (0)+ j , δ (cgct) ξ ζ I (0) − = ξ a D a ζ I (0) − ≡ ξ i D i ζ I (0) − , (5.20b)where ξ a ≡ ξ i e a (0) i . Meanwhile, δ ξ given in (B.4) is the general coordinate transformation and it isrelated to δ (cgct) ξ by δ (cgct) ξ = δ ξ − δ λ ab = ω jab ξ j . (5.21)The reason why diffeomorphism and local Lorentz transformation appear in a mixed way is that theconstraint function and Poisson bracket can only give the covariant quantity but δ ξ in (B.37) is notcovariant by itself. Moreover, SUSY transformation demands the sources to be covariant and thuswe are forced to see covariant general coordinate transformation rather than general coordinatetransformation.The useful variations of renormalized canonical momenta extracted from (5.19) are δ ǫ + Π i Ψ = δδ Ψ (0)+ i C [ ǫ + ] = 12 Π ia Γ a ǫ + (5.22a) δ ǫ − Π i Ψ = δδ Ψ (0)+ i C [ ǫ − ] = δδ Ψ (0)+ i Z d d x | e (0) | A sW [Φ (0) ] ǫ − = − | e (0) | κ D k (cid:18)h R (0) b Γ ik (0) − R (0) j k b Γ ij (0) + R (0) j i b Γ kj (0) i ǫ − (cid:19) −− | e (0) | κ b Γ ij D j ( ϕ ǫ − ) , (5.22b) δ ǫ + Π ζI = δδζ I (0) − C [ ǫ + ] = i ϕI ǫ + , (5.22c) δ ǫ − Π ζI = δδζ I (0) − C [ ǫ − ] = δδζ I (0) − Z d d x | e (0) | A sW [Φ (0) ] ǫ − = − | e (0) | κ i ∂ i ϕ (0) b Γ i (0) ǫ − , (5.22d) δ ǫ + Π ϕI = δδϕ I (0) C [ ǫ + ] = − i ∂ i (cid:16) Π ζI b Γ i ǫ + (cid:17) , (5.22e) δ ǫ − Π ϕI = δδϕ I (0) C [ ǫ − ] = − i∂ I ( G JK ∂ K W )Π ζJ ǫ − + δδϕ I (0) Z d d x | e (0) | A sW [Φ (0) ] ǫ − It is obvious that variation of the sources can be obtained through this Poisson bracket. In appendix D we showthat the same thing holds for the canonical momenta. i Π ζ ǫ − + | e (0) | κ ϕ (0) Ψ (0)+ j ←− D i b Γ ji (0) ǫ − , (5.22f)where R (0) , R (0) ij and b Γ i (0) denote the Ricci scalar, Ricci tensor, Gamma matrix and determinantof the metric for the vielbein e a (0) i . Here the underline indicates that the terms over it are computedfor the toy model. Notice that due to the super-Weyl anomaly ǫ − variation of the renormalizedcanonical momenta contain bosonic anomalous terms, which have similar origin to the Schwarzianderivative appearing in conformal transformation of the energy-momentum tensor of 2D CFT. A bulk (bosonic) BPS configuration, which is a bosonic solution of the classical SUGRA action aswell as is invariant under bulk SUSY transformation with a certain parameter, corresponds to asupersymmetric vacuum state of the dual field theory. Since vacuum expectation value (vev) ofmany observables are computed in SUSY field theories, it is necessary to pay a special attention tothe bulk BPS solution. Presence of the bulk BPS configuration implies that there exists a boundarySUSY parameter, under the gPBH transformation with which the fermionic sources are invariant,namely δ η Ψ (0)+ i ≡ δ η + Ψ (0)+ i + δ η − Ψ (0)+ i = D i η + − b Γ (0) i η − = 0 , (5.23a) δ η ζ I (0) − = − i b Γ i (0) ∂ i ϕ I (0) η + + i G IJ ∂ J W η − = 0 , (5.23b)where the first equation is usually called as conformal Killing spinor (CKS) condition. Actually,the rigid supersymmetry of the boundary field theory is found by solving (5.23) [1, 3, 55]. Now we show that η -variation of any renormalized canonical momenta vanishes on the BPSsolution, i.e. δ η Π Φ (cid:12)(cid:12)(cid:12) BPS ≡ δ η + Π Φ (cid:12)(cid:12)(cid:12) BPS + δ η − Π Φ (cid:12)(cid:12)(cid:12) BPS = 0 , for any source Φ (0) , (5.24)where for the fermionic operators we have from (5.22) δ η Π i Ψ = 12 Π ia Γ a η + + δ Ψ (0)+ i Z Σ r d d x | e (0) | A sW [Φ (0) ] η − , (5.25a) δ η Π ζI = i ϕI η + + δδζ I (0) − Z Σ r d d x | e (0) | A sW [Φ (0) ] η − . (5.25b)This is in fact holographic version of that vev of any Q -exact operator vanishes on SUSY vacua.We only need to consider variation of the fermionic canonical momenta, since η -variation of bosoniccanonical momenta trivially vanishes on the bosonic solution. One can in principle see (5.24) byexpanding the bulk BPS equations. But since we have SUSY and super-Weyl Ward identities, theform of which are the same for all SCFTs, we take advantage of the Ward identities (5.2a) for η + and (5.2b) for η − . Here we do not discuss the integrability condition of (5.23). For a discussion about some geometry of (5.23a),which is also known as the twistor equation, see e.g. section 3.1 in [3]. More precisely speaking, most of the rigid N = 1 SUSY field theories on curved background are obtained when U (1) R -symmetry gauge field is turned on. In this case, which is discussed in [39], the covariant derivative D i in(5.23a) becomes D i + igA i , where g is the R -charge. Z ∂ M d d x "(cid:16) −
12 Ψ (0)+ i Γ a Π ia − i ζ I (0) − Π ϕI − i ζI /∂ϕ I (0) − Π i Ψ ←− D i (cid:17) η + + (cid:16) i G IJ ∂ I W Π ζJ − Π i Ψ b Γ (0) i − | e (0) |A sW [Φ (0) ] (cid:17) η − = Z ∂ M d d x (cid:16) −
12 Ψ (0)+ i Γ a Π ia η + − i ϕI ζ I (0) − η + − | e (0) |A sW [Φ (0) ] η − (cid:17) . (5.26)We emphasize that because the Ward identities are valid for any background, (5.26) holds at leastto linear order in fermions for any value of Ψ (0)+ i and ζ I (0) − as long as the bosonic sources admit theCKS. There might be correction at order of O (cid:16) (Ψ (0)+ ) , ( ζ (0) − ) (cid:17) , though. Note that non-trivialdependence of bosonic momenta Π ia and Π ϕI on the fermionic sources occurs from the quadraticorder in fermions, i.e. δδ Ψ (0)+ i Π ia (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ (0)+ i = ζ I (0) − = ··· =0 = 0 (5.27)and so on. Therefore, by taking the functional derivative of (5.26) with respect to the fermionicsources and evaluating on the (bosonic) supersymmetric background, we obtain the (bosonic) iden-tities 12 Π ia Γ a η + + δ Ψ (0)+ i Z Σ r d d x | e (0) | A sW [Φ (0) ] η − = 0 , (5.28a) − i ϕI η + − δδζ I (0) − Z Σ r d d x | e (0) | A sW [Φ (0) ] η − = 0 , (5.28b)where we used (3.17). Therefore, we find that on the BPS backgrounds δ η Π i Ψ = 0 , δ η Π ζI = 0 , (5.29)which confirms our claim.From the field theory point of view, (5.24) is quite natural, since supersymmetric vacua areannihilated by the preserved supercharge Q . It has, however, a deep implication. We should firstemphasize that δ η corresponds to the variation of quantum operators acting on Hilbert space and isdifferent from the ’classical’ SUSY variation which is considered in the context of SUSY localization.For instance Π ij b Γ (0) j η + , which is classically Q -exact here, has non-zero vev due to the anomalouscontribution. This implies that the classical SUSY variation cannot become as a total derivative inthe path integral, so long as the anomalous terms in (5.28) do not vanish, see some comments onthe assumption that SUSY should not be anomalous, in SUSY localization reviews such as [56, 57].In order to convince ourselves, let us check (5.24) for the toy model. First, let us remind thatin the toy model, d = 4 and scaling dimension of ϕ is 3. Then, (5.28)s become0 = − Γ a η + Π ia + 1 κ d − D k h(cid:0) dd − R (0) b Γ ki (0) − R (0) j k b Γ ji (0) + 2 R (0) j i b Γ jk (0) (cid:1) η − i + | e (0) | κ d − ϕ b Γ ij (0) D j η − , (5.30)30 = − i η + Π ϕ + | e (0) | κ i b Γ i (0) η − ∂ i ϕ (0) . (5.31)By combining (5.31) with the conformal Killing spinor equation for the toy model D i η + = b Γ (0) i η − , (5.32a)12 b Γ i (0) ∂ i ϕ (0) η + + ϕ (0) η − = 0 , (5.32b)we get − ϕ (0) Π ϕ + | e (0) | κ ∂ i ϕ (0) ∂ i ϕ (0) = 0 . (5.33)This ’strange-looking’ formula can be verified in the toy model, by using the bulk BPS equation.From the bulk BPS equation for ζ with the bulk SUSY parameter b ǫδ b ǫ ζ = (cid:0) /∂ϕ − W ′ (cid:1) b ǫ = 0 , W ′ ≡ ddϕ W ( ϕ ) , (5.34)one can obtain ˙ ϕ = − q ( W ′ ) + ∂ i ϕ∂ i ϕ, (5.35)where we fix the sign from leading asymptotics of ϕ . It then follows from the definition of π ϕ that π ϕ = − √− γκ ˙ ϕ = √− γκ q ( −W ′ ) + ∂ i ϕ∂ i ϕ. (5.36)On the other hand, the full bosonic counterterms are given by S ct = 1 κ Z d d x √− γ h W − R −
12 log e − r (cid:16) ϕ R + ∂ i ϕ∂ i ϕ + · · · (cid:17)i , (5.37)where the ellipsis denote the terms which does not depend on ϕ . The counterterms for the canonicalmomenta π ϕ ct is then given by π ϕ ct = δδϕ S ct = √− γκ h − ( −W ′ ) −
12 log e − r (cid:16) ϕR − ✷ ϕ (cid:17)i . (5.38)Furthermore, from the conformal Killing spinor condition (5.23), we obtain0 = (cid:16) ✷ (0) ϕ (0) − ϕ (0) R (0) (cid:17) η + , (5.39)which implies that the logarithmically divergent terms in (5.38) actually do not contribute to thecounterterms. Eventually, the renormalized canonical momentum Π ϕ becomesΠ ϕ = 1 κ lim r → + ∞ e − r √− γ ∂ i ϕ∂ i ϕ p ( −W ′ ) + ∂ i ϕ∂ i ϕ + ( −W ′ ) = | e (0) | κ ∂ i ϕ (0) ∂ i ϕ (0) ϕ (0) , (5.40)which confirms the result (5.33) as well as the anomalous SUSY variation of the renormalizedcanonical momenta (5.22). 31 .4 Conserved charges and supersymmetry algebra with anomaly correction We recall that given a Killing vector ξ i which satisfies the Killing condition L ξ g (0) ij = D (0) i ξ j + D (0) j ξ i = 0 , (5.41a) L ξ ϕ I (0) = ξ i ∂ i ϕ I (0) = 0 , (5.41b) L ξ ζ I (0) − = ξ i D (0) i ζ I (0) − + 14 D (0) i ξ j b Γ ij (0) ζ I (0) − = 0 , (5.41c) L ξ Ψ (0)+ j = ξ i D (0) i Ψ (0)+ j + ( D (0) j ξ i )Ψ i (0)+ + 14 D (0) k ξ l b Γ kl (0) Ψ (0)+ j = 0 , (5.41d)we obtain a conservation law by combining (5.2d) with (5.2e), namely D i (cid:2) e aj ξ j Π ia + ξ j (Π i Ψ Ψ + j + Ψ + j Π i Ψ ) (cid:3) = 0 . (5.42)Note that we use the Kosmann’s definition for the spinorial Lie derivative (see e.g. [58] and (A.11)of [2] ) and the Lie derivative is related to gPBH transformations by L ξ = δ (cgct) ξ + δ λ ab = − e ia e jb D [ i ξ j ] . (5.43)We emphasize that (5.42) holds for any background. The conservation law (5.42) allows us to definea conserved charge associated with ξ i , namely [59, 60] Q [ ξ ] ≡ Z ∂ M∩C dσ i (cid:0) e aj Π ia + Π i Ψ Ψ + j + Ψ + j Π i Ψ (cid:1) ξ j , (5.44)which is independent on the choice of Cauchy surface C . Note that the conserved charge Q ξ isrelated to the constraint function by Q [ ξ ] = C [ ξ, λ ab = − e ia e jb D [ i ξ j ] ] . (5.45)We have other conservation laws D i (Π i Ψ η + ) = D i ( η + Π i Ψ ) = 0 , (5.46)which follow from the SUSY and super-Weyl Ward identities (5.2a) and (5.2b) for the CKS pa-rameters η + and η + . Note that the conservation laws (5.46) hold only on the bosonic background.This allows us to define conserved charges Q s [ η + ] ≡ Z ∂ M∩C dσ i Π i Ψ η + , Q s [ η + ] ≡ Z ∂ M∩C dσ i η + Π i Ψ . (5.47)On the bosonic background we can identify these conserved charges with the constraint functions,namely Q s [ η + ] = C [ η + , η − ] , Q s [ η + ] = C [ η + , η − ] . (5.48)It then follows from (5.22) that on the bosonic background we have { Q s [ η + ] , Q s [ η + ] } (cid:12)(cid:12)(cid:12) Bosonic = Z ∂ M∩C dσ i η + {C [ η + , η − ] , Π i Ψ } (cid:12)(cid:12)(cid:12) Bosonic g (0) ij ≡ e a (0) i e (0) aj is the induced metric on the boundary ∂ M . In many literatures including [58], the spinoral Lie derivative is defined by L ξ ζ = ξ i D i ζ − D i ξ j b Γ ij ζ . The sign ofthe last term is minus, since the Gamma matrices there follow Grassman algebra in Euclidean signature, while herewe use the Minkowskian signature. Z ∂ M∩C dσ i "
12 Π ia η + Γ a η + + η + (cid:16) δδ Ψ (0)+ i Z ∂ M d d x | e (0) |A sW η − (cid:17) Bosonic . (5.49)In the case where the conformal Killing vector K i ≡ iη + b Γ i η + (5.50)becomes a Killing vector, we can see that on the bosonic background the above commutator becomes { Q s [ η + ] , Q s [ η + ] } = − i Q [ K ] + Z ∂ M∩C dσ i η + (cid:16) δδ Ψ (0)+ i Z ∂ M d d x | e (0) |A sW η − (cid:17) . (5.51)Not surprisingly, the super-Weyl anomaly corrects the supersymmetry algebra, too.We can obtain other commutators such as {Q [ ξ ] , Q s [ η + ] } . It is possible because Q [ ξ ] for theKilling vector ξ i is conserved for any background so that Z ∂ M∩C dσ i {Q [ ξ ] , Π i Ψ } η + = Z ∂ M∩C dσ k {C [ ξ, λ ab = − e ia e jb D [ i ξ j ] ] , Π k Ψ } η + = Z ∂ M∩C dσ i h − Π i Ψ L ξ η + + D j [( ξ j Π i Ψ − ξ i Π j Ψ ) η + ] + ξ i D j (Π j Ψ η + ) i , where the second term vanishes by using Stokes’ theorem. The third term is also zero on thebosonic background, due to the conservation law. Therefore, we have {Q [ ξ ] , Q s [ η + ] } = − Z ∂ M∩C dσ i Π i Ψ L ξ η + = − Q s [ L ξ η + ] , (5.52)and in the same way {Q [ ξ ] , Q s [ η + ] } = − Z ∂ M∩C dσ i ( η + ←−L ξ )Π i Ψ = − Q s [ η + ←−L ξ ] , (5.53)since L ξ η + and η + ←−L ξ become conformal Killing spinors [58], i.e. D i ( L ξ η + ) = 1 d b Γ i b Γ j D j ( L ξ η + ) , ( η + ←−L ξ ) ←− D i = 1 d ( η + ←−L ξ ) ←− D j b Γ j b Γ i . (5.54)We note that (5.52) and (5.53) can be obtained in the way around, namely by computing { Q s [ η + ] , e aj Π ia + Π i Ψ Ψ + j + Ψ + j Π i Ψ } , { Q s [ η + ] , e aj Π ia + Π i Ψ Ψ + j + Ψ + j Π i Ψ } . (5.55)In summary, the supersymmetry algebra on the curved (bosonic) background is { Q s [ η + ] , Q s [ η + ] } = − i Q [ K ] + Z ∂ M∩C dσ i η + (cid:16) δδ Ψ (0)+ i Z ∂ M d d x | e (0) |A sW η − (cid:17) , {Q [ ξ ] , Q s [ η + ] } = − Q s [ L ξ η + ] , {Q [ ξ ] , Q s [ η + ] } = − Q s [ η + ←−L ξ ] . (5.56) One can easily check K i satisfies the conformal Killing condition, by using (5.23). J i (with the anomaly A J ) D i J i = A J , (5.57)from which we derive the variation of any operator O under the symmetry transformation (see e.g.(2.3.7) in [9]), namely δ O ( x ) + Z ∂ M d d y [ D i J i ( y ) − A J ( y )] O ( x ) = 0 , (5.58)where the second term can be computed by differentiating the relevant Ward identities with thesource dual to operator O ( x ). Now one can readily see that the commutator of charges becomes { Q , Q } = Z ∂ M∩C dσ i ( δ J i ) = − Z ∂ M∩C dσ i (cid:18)Z ∂ M d d y [ D j J j ( y ) − A J ( y )] J i (cid:19) , (5.59)and this prescription gives the same result with (5.56). See e.g. appendix E for derivation of {Q [ ξ ] , Q s [ η + ] } .Now that we know from the last section that the LHS of (5.51) vanishes on BPS backgrounds,we can conclude that the conserved charge associate with K i on BPS backgrounds is totally fixedto be a functional derivative of the fermionic anomaly, namely Q [ K ] (cid:12)(cid:12)(cid:12) BPS = − i Z ∂ M∩C dσ i η + ( δδ Ψ (0)+ i Z ∂ M d d x | e (0) |A sW η − ) . (5.60)Depending on the theory, K i can be combination of other Killing vectors such as ∂ t and angularvelocity. If this is the case, (5.60) can be regarded as a relation of the conserved charges on thesupersymmetric background, but accompanied with anomalous contribution. A similar relation isfound in [39], which explains the discrepancy of the BPS condition (see e.g. (C.16) of [44]) h H i + h J i + γ h Q i = 0 , (5.61)for pure AdS is precisely due to the anomalous contribution coming from the fermionic anomalies. Most of computations so far are for plus sign choice of (3.3b) at the beginning of section 3. Thisplus sign is actually equivalent to imposing Dirichlet boundary condition on the spin 1 / ζ .Independently from this choice, we could determine the leading asymptotics of the scalar field, asemphasized before. This allows us to use the result of appendix B.3 and B.4 to conclude that minussign choice can be supersymmetric only when mass of its scalar SUSY-partner field belongs to thewindow [63–65] − (cid:18) d (cid:19) ≤ m ≤ − (cid:18) d (cid:19) + 1 . (6.1)In this window (3.3b) is already finite, implying that the canonical momenta for ζ − is notrenormalized. Since ζ + by itself becomes the renormalized canonical momentum, change of the34ign from plus to minus is in fact Legendre transformation of the renormalized on-shell action b S ren ,which is equivalent to impose Neumann boundary condition on ζ − [66]. We have seen that b S ren inthe case of plus sign choice is ( ǫ + ) supersymmetric (Dirichlet boundary condition for scalar field wasimplicitly imposed). Therefore, in order to preserve SUSY, one can expect that boundary conditionfor the scalar field should also be changed from Dirichlet to Neumann by Legendre transformation.To see this, one has to prove that the total Legendre transformation action S L = − Z Σ r ( b π ζ ζ − + ζ − b π ζ + ϕ b π ϕ ) , b π ζ = √− γκ ζ + (6.2)is invariant under ǫ + transformation. Note that variation of Π ζI gives directly how gPBH transfor-mations act on ζ + . We again consider only one scalar field, and it is straightforward to extend theresult here to the case for several scalar fields. From (5.22), one can find that the action of ǫ + on S L gives δ ǫ + S L ∼ − Z Σ r (cid:16) i b π ϕ ζ − ǫ + − i ∂ i ϕǫ + b Γ i b π ζ − i b π ϕ ζ − ǫ + − i ϕ∂ i ( ǫ + b Γ i b π ϕ ) + h . c . (cid:17) = 0 . This confirms that the total action S + S L for the Neumann boundary condition is still invariantunder ǫ + transformation.When it comes to ǫ − variation of S L , one can find that all the momenta-related terms arecanceled, as before. The anomalous terms in ǫ − variation of the renormalized canonical momenta,however, are not canceled but contribute to ǫ − anomaly of S + S L , together with A sW . Namely,we obtain for the toy model that δ ǫ − ( S + S L ) ∼ Z Σ r d d x √− γ ǫ − (cid:18) A ( G )sW − κ ϕ b Γ ij D i Ψ + j (cid:19) ≡ Z Σ r d d x √− γ ǫ − A N sW , (6.3)where the super-Weyl anomaly for Neumann boundary condition is A N sW = A ( G )sW − κ ϕ b Γ ij D i Ψ + j . (6.4) In this work we have considered a generic N = 2 5D supergravity with its fermionic sector inthe context of holographic renormalization, through which we have obtained a complete set ofsupersymmetric counterterms. We have also found that scalars and their superpartners shouldsatisfy the same boundary condition in order for the theory to be consistent with SUSY.The Ward identities (5.2) and the anomalies lead to rather remarkable consequences. By meansof them, we showed that SUSY transformation of operators and SUSY algebra of a theory whichhas N = 1 4D SCFT in curved space as a UV fixed point become anomalous at the quantum level,see (5.25) and (5.56). We comment that once the R -symmetry gauge field is turned on, R -chargeand the related terms appear on RHS of the first line (5.56). Note that the anomalous terms arenon-vanishing in general on curved backgrounds, even where all anomalies vanish.(5.24) stating that Q -exact operator has vanishing vev on the SUSY vacua, namely h δ Q Oi = h δ cl Q Oi + (Quantum correction) = 0 , (7.1)implies that the classical Q -variation cannot be a total derivative in the path integral unless thequantum correction vanishes, otherwise we have for instance h δ cl Q S i i = Z [ Dφ ] (cid:16) δ cl Q S i (cid:17) e − S = Z [ Dφ ] δ cl Q (cid:0) S i e − S (cid:1) = 0 . (7.2)35his is very important, since SUSY localization technique are justified only when the classical Q -variation is a total derivative in the path integral. See also footnote 10 in [67] stating a fundamentalassumption used in the SUSY localization principle.We should remark the importance of finding SUSY completion of the Weyl anomaly, becausethe supersymmetric Weyl anomaly automatically satisfies all the requirements for higher-derivativeSUGRA in the superconformal way. For construction of higher-derivative SUGRA, see [68] andsome related references.We emphasize that our whole analysis here crucially relies on existence of the superpotential W . If the theory does not possess any superpotential, one could introduce local and approximate superpotential which is sufficient for reproducing all divergent terms of the scalar potential, as donein [69]. Now one can see that the approximate superpotential should meet more restrictive criterionfor the supersymmetric holographic renormalization. To make this point clear, let us discuss aboutthe approximate superpotential suggested in [69], see (5.15) there. One can find from the BPSequations (3.20) and (3.25) and the algebraic equation (3.26) in [69] that the BPS solution’s flowto leading order is dψdr ∼ − ψ, (7.3a) dϕdr ∼ − ϕ + r ψ ! , (7.3b) dχdr ∼ − χ (cid:18) ψ √ ϕ (cid:19) , (7.3c)where RHS of the last equation is a non-local function of ϕ around ϕ = 0. Hence it is impossible tofind the local and approximate superpotential consistent with the BPS flow equations, which meansthat we need more generic N = 2 gauged SUGRA model to study [69]. Notice that this inconsis-tency of the approximate superpotential with the BPS flow equations imply that the supertentialsuggested in [69] is not approximate for the fermionic sector of SUGRA.As long as there exists a superpotential (or at least approximate one for the whole sector ofSUGRA), many of our results here can be extended straightforwardly to other dimensions. A directapplication of the analysis of this paper to other dimensions is to obtain 2D super-Virasoro algebrawith central extension. Let us explain this here schematically. The super-Weyl anomaly in 2DSCFT can be easily found by using the trick of section 5.1.3, namely that the SUSY variation ofthe super-Weyl anomaly is equal to the Weyl anomaly. Since the Weyl anomaly is e ai T ia = c π R ,we see immediately that the super-Weyl anomaly in 2D is Γ i S i ∼ i c π Γ ij D i Ψ j up to a constantcoefficient, depending on the convention. It follows that the anomalous variation of the super-current operator is δ η S i = − i a η T ia + ic π b Γ ij b Γ k D j D k η, (7.4)where η is the 2D CKS, satisfying the condition b Γ i D i η = 12 b Γ i b Γ j D j η, or b Γ j b Γ i D j η = 0 . (7.5)Note that the anomalous term in (7.4) vanishes only when the 2D Ricci scalar R = 0 and η is aspinor, all second derivatives of which vanish. Since (7.5) gives infinite number of solutions, as 2Dconformal Killing vector equation, one gets infinite number of conserved super-charges G r , whichare added to Virasoro algebra to form the super-Virasoro algebra. Now one can see that the central36xtension in (see e.g. (10.2.11b) in [70]) { G r , G s } = 2 L r + s + c
12 (4 r − δ r, − s (7.6)of the super-Virasoro algebra in 2D flat background is derived from the anomalous term of (7.4).One should keep in mind, however, that since representation of the spinor fields strongly dependson the dimension of spacetime it might not be easy to put the SUGRA action into the form of (2.1)in other (especially odd) dimensions. Acknowledgments
I would like to thank Jin U Kang and Ui Ri Mun for interesting discussions. I am grateful to JinU Kang for a careful reading and important comments on the draft.
AppendicesA Notation, conventions for Gamma matrices and useful identi-ties
Throughout this paper Greek indexes µ, ν and α, β, · · · refer to the coordinate and flat directionsin the bulk respectively, and the Latin indexes i, j, m, n, p, q, · · · and a, b, · · · refer to the coordinateand flat directions on the slice respectively. The flat indices which correspond to radial like andtime like directions are special, so we denote them by ¯ r and ¯ t respectively. The capital Latinletters A, B, · · · indicate the coordinate directions on the scalar and hyperino manifold. ∇ µ , D i and D i refer to the covariant derivative in the bulk and the covariant derivatives of the bosonic andfermionic fields on the slice sequentially.We use the hermitian representation of the Lorentzian Gamma matrices, following the conven-tion in [46]. Γ α and Γ a indicate the Gamma matrices along the flat directions in the bulk and theboundary, while Γ µ and b Γ i refer to the Gamma matrices along the coordinate directions in the bulkand the boundary. The relations between these Gamma matrices are provided in appendix C. Bothin the bulk and on the boundary the hermitian conjugation of the Gamma matrix is given byΓ µ † = Γ ¯ t Γ µ Γ ¯ t , b Γ i † = Γ t b Γ i Γ ¯ t . (A.1)The following formulas, which hold in any D dimensional spacetime (see e.g. section 3 in [46]),are frequently used in this paper.Γ µνρ = 12 { Γ µ , Γ νρ } , (A.2a)Γ µνρσ = 12 [Γ µ , Γ νρσ ] , (A.2b)Γ µνρ Γ στ = Γ µνρστ + 6Γ [ µν [ τ δ ρ ] σ ] + 6Γ [ µ δ ν [ τ δ ρ ] σ ] , (A.2c)Γ µνρσ Γ τλ = Γ µνρστλ + 8Γ [ µνρ [ λ δ σ ] τ ] + 12Γ [ µν δ ρ [ λ δ σ ] τ ] , (A.2d)[Γ µν , Γ ρσ ] = 2( g νρ Γ µσ − g µρ Γ νσ − g νσ Γ µρ + g µσ Γ νρ ) , (A.2e)Γ µνρ Γ ρ = ( D − µν , (A.2f)Γ µνρ Γ ρσ = ( D − µν σ + 2( D − [ µ δ ν ] σ , (A.2g)37 µν ∇ µ ∇ ν ζ = − Rζ, (A.2h)Γ µνρ ∇ ν ∇ ρ ζ = −
14 ( R Γ µ − R ν µ Γ ν ) ζ, (A.2i)where δ refers to the Kronecker delta.There are left and right acting functional derivatives with respect to fermionic variable ψ ,namely −→ δδψ , ←− δδψ , (A.3)and in most cases the rightarrow symbol → are omitted. Here ψ denotes the Dirac adjoint of thespinor ψ , namely ψ ≡ ψ † ( i Γ ¯ t ) . (A.4)The affine connection Γ µνρ is related to spin connection by (see e.g. (7.100) in [46])Γ ρµν = E ρα ( ∂ µ E αν + ω µαβ E βν ) . (A.5)In this work we consider the supergravity theory in the second order formalism. This means thatour theory is torsionless and thus the spin connection is reduced into ω µαβ = E να ∂ µ E νβ + Γ ρµν E ρα E νβ . (A.6)Variation of the torsionless spin connection δω µαβ = E ν [ α D µ δE β ] ν − E ν [ α D ν δE β ] µ + e ρα E νβ E γµ D [ ν δE γρ ] . (A.7)is useful for many of our computations. The covariant derivative of the fermionic fields are givenby ∇ µ Ψ ν = ∂ µ Ψ ν + 14 ω µαβ Γ αβ Ψ ν − Γ ρµν Ψ ρ , (A.8) ∇ µ ζ I = ∂ µ ζ I + 14 ω µαβ Γ αβ ζ I . (A.9) B ADM decomposition and generalized PBH transformation
A preliminary step of the Hamiltonian analysis of the gravitational theory is to decompose thevariables of theory including the metric (or the vielbeins) into a radial-like (or time-like) directionand the other transverse directions (a.k.a. ADM decomposition [71]). Coupling gravity to spinorfields require vielbeins to appear in the action explicitly and thus the ADM decomposition of thevielbeins instead of the metric should be done.The ADM decomposition brings us a natural choice of the gauge for variables of the theory,which is called as the Fefferman-Graham (FG) gauge. In the FG gauge, the Hamiltonian analysisbecomes much simple.
B.1 ADM decomposition of vielbein and the strong Fefferman-Graham gauge
We begin with picking up a suitable radial coordinate r and doing the ADM decomposition ofthe metric to run the Hamiltonian formalism. Since the vielbein explicitly appears in the action38hrough the covariant derivative of the spinor fields we need to decompose the vielbein itself ratherthan the metric.Choosing the radial coordinate r , we describe the bulk space as a foliation of the constant r -slices, which we denote by Σ r . Let E α be vielbeins of the bulk and we decompose them as E α = (cid:0) N n α + N j e αj (cid:1) dr + e αj dx j , (B.1)such that g µν = E αµ E βν η αβ , γ ij = e αi e βj η αβ , n α e αi = 0 , η αβ n α n β = 1 , (B.2)where α, β are bulk tangent space indices and η = diag (1 , − , , . . . ,
1) (where η ¯ t ¯ t = − N and N α are called as lapse and shift respectively. One can check that ds ≡ g µν dx µ dx ν = ( N + N i N i ) dr + 2 N i drdx i + γ ij dx i dx j , (B.3)which usually appears in the textbook. The inverse vielbeins are then given by E rα = 1 N n α , E iα = e iα − N i N n α . (B.4)It follows that Γ r = Γ α E rα = 1 N n α Γ α ≡ N Γ . (B.5)The extrinsic curvature on the radial slice Σ r is defined as K ij ≡ N ( ˙ γ ij − D i N j − D j N i ) (B.6)and K ≡ γ ij K ij . Moreover, Γ i = Γ α E αi = b Γ i − N i N Γ , (B.7)where b Γ i ≡ Γ α e iα . These vielbeins satisfy the relation e iα e βi + n α n β = δ βα . (B.8)One can also see that b Γ i s satisfy the Clifford algebra on the slice and Γ anticommutes with all b Γ i s,i.e. { b Γ i , b Γ j } = 2 γ ij , { b Γ i , Γ } = 0 . (B.9)It follows that the matrix Γ can be used to define the ’radiality’ (see e.g. [37]) on the slice, so that,a generic spinor ψ on the slice can be split into two by radiality , ψ ± ≡ Γ ± ψ, (B.10)where Γ ± ≡ (1 ± Γ).We remind that splitting spinor fields by their radiality is inevitable because different radialityleads to different asymptotic behavior [31, 32] as well as the second-class constraints of the fermionaction should be solved in a Lorentz invariant way [35].In order to simplify the calculations that follow it is convenient to pick a particular vielbeinframe so that n α = (1 , , e i ¯ r = 0 , e ri = 0 , (B.11) When d = D − e ai becomes the vielbein on the slice Σ r . We will call the gauge (B.11) combined with thetraditional Fefferman-Graham (FG) gauge N = 1 , N i = 0 , Ψ r = 0 (B.12)as the strong FG gauge. Namely, the strong FG gauge refers to E ¯ rr = 1 , E ar = 0 , E ri = 0 , Ψ r = 0 . (B.13) B.2 Decomposition of the covariant derivatives
We obtain (see also (88) and (89) in [35]) ω rαβ = n [ α ˙ n β ] + e i [ α ˙ e β ] i + 2 n [ α e β ] i (cid:0) ∂ i N − N j K ji (cid:1) − D i N j e [ αi e β ] j , (B.14) ω iαβ = n α ∂ i n β + e jα ∂ i e β j + Γ kij [ γ ] e kα e βj + 2 K ji e j [ α n β ] , (B.15)where we have used the Christoffel symbolsΓ rrr = N − (cid:16) ˙ N + N i ∂ i N − N i N j K ij (cid:17) , Γ rri = N − (cid:0) ∂ i N − N j K ij (cid:1) , Γ rij = − N − K ij , Γ irr = − N − N i ˙ N − N D i N − N − N i N j ∂ j N + ˙ N i + N j D j N i + 2 N N j K ij + N − N i N k N l K kl , Γ irj = − N − N i ∂ j N + D j N i + N − N i N k K kj + N K ij , Γ kij = Γ kij [ γ ] + N − N k K ij . Denoting the spin connection on the radial cut-off as b ω iab , we get b ω iab = e ja ∂ i e jb + Γ kij [ γ ] e ka e jb = ω iab , (B.16a) ω iαβ Γ αβ = b ω iab Γ ab + 2 K ji e jα n β Γ αβ = b ω iab Γ ab + 2 K ji b Γ j Γ , (B.16b) ω rαβ Γ αβ = e iα ˙ e iβ Γ αβ + 2Γ b Γ i (cid:0) ∂ i N − N j K ji (cid:1) − b Γ ij D i N j , (B.16c) ∇ i Ψ j = D i Ψ j + 12 K li b Γ l ΓΨ j + 1 N K ij (Ψ r − N k Ψ k ) , (B.16d) ∇ i Ψ r = D i Ψ r + 12 K ji b Γ j ΓΨ r − Γ jir Ψ j − Γ rir Ψ r , (B.16e) ∇ r Ψ i = ˙Ψ i + 14 h e ai ˙ e ib Γ ab + 2Γ b Γ j (cid:16) ∂ j N − N l K lj (cid:17) − b Γ jl D j N l i Ψ i − Γ jir Ψ j − Γ rir Ψ r , (B.16f) ∇ i ζ = D i ζ + 12 K ji b Γ j Γ ζ, (B.16g) ∇ r ζ = ˙ ζ + 14 h e ai ˙ e ib Γ ab + 2Γ b Γ j (cid:16) ∂ j N − N l K lj (cid:17) − b Γ jl D j N l i ζ, (B.16h)where D i Ψ j = ∂ i Ψ j + 14 b ω iab Γ ab Ψ j − Γ kij [ γ ]Ψ k , (B.17a) D i Ψ r = ∂ i Ψ r + 14 b ω iab Γ ab Ψ r , (B.17b) D i ζ = ∂ i ζ + 14 ω iab Γ ab ζ (B.17c)are the covariant derivatives of the spinors on the slice Σ r . Note that in the final computations weused the gauge (B.11). 40 .3 Equations of motion and leading asymptotics of fermionic fields In order to discuss with the transformation law of the induced fields, we first study the leadingasymptotic behavior of the fields, which can be understood from equations of motion. For Ψ µ and ζ I they are respectively,Γ µνρ ∇ ν Ψ ρ − W Γ µν Ψ ν − i G IJ (cid:0) /∂ϕ I + G IK ∂ K W (cid:1) Γ µ ζ J = 0 , (B.18)and G IJ (cid:0) δ JK / ∇ + Γ JKL [ G ] /∂ϕ L (cid:1) ζ K + M IJ ( ϕ ) ζ J + i G IJ Γ µ (cid:0) /∂ϕ J − G JK ∂ K W (cid:1) Ψ µ = 0 . (B.19)Extracting the relevant terms, we obtain in the gauge (B.12)0 ∼ − b Γ ij (cid:18) ˙Ψ + j −
12 Ψ + j (cid:19) + b Γ ij (cid:18) ˙Ψ − j + 2 d −
32 Ψ − j (cid:19) + b Γ ijk D j (Ψ + k + Ψ − k ) , (B.20)0 ∼ ˙ ζ + + (cid:18) d M ζ (cid:19) ζ + − ˙ ζ − − (cid:18) d − M ζ (cid:19) ζ − + b Γ i D i ζ + − b Γ i D i ζ − + i ϕ + µϕ ) b Γ i Ψ + i + i b Γ i b Γ j ∂ j ϕ Ψ + i , (B.21)where we assume that there is only one scalar ϕ and one spin-1 / ζ for simplicity, and M ζ which is the mass of ζ and µ are respectively µ = − ∂ ϕ ∂ ϕ W (cid:12)(cid:12)(cid:12) ϕ =0 , M ζ = M ϕϕ (cid:12)(cid:12)(cid:12) ϕ =0 , (B.22)under the assumption that the scalar manifold metric is canonically normalized. µ and M ζ arerelated by M ζ = − µ + d − . (B.23)When d >
2, the leading asymptotics of Ψ + i and Ψ − i areΨ + i ( r, x ) ∼ e r Ψ (0)+ i ( x ) , (B.24)Ψ − i ( r, x ) ∼ − e − r (cid:18) d − d − b Γ (0) i b Γ (0) kl − b Γ (0) ikl (cid:19) D (0) k Ψ (0)+ l ( x ) , (B.25)where we used e ai ( r, x ) ∼ e r e a (0) i ( x ) in AlAdS geometry, and Γ (0) i and D (0) refer to the Gammamatrices and the covariant derivative with respect to e a (0) i .We need to be more careful, regarding ζ . First, we note that the leading asymptotics of ϕ should always be ϕ ( r, x ) ∼ e − µr ϕ (0) ( x ) as can be seen from (4.8c). Therefore, the final two termsin (B.21) can be discarded from the argument. Now there are 3 cases to consider:1. M ζ > / µ < d − ζ − and ζ + are respectively ζ − ( r, x ) ∼ e − ( µ + ) r ζ (0) − ( x ) , (B.26) ζ + ( r, x ) ∼ − µ + (cid:18) e − ( µ + ) r b Γ (0) i D (0) i ζ (0) − ( x ) − i b Γ (0) i b Γ (0) j ∂ j ϕ (0) ( x )Ψ (0)+ i ( x ) (cid:19) . (B.27)41. M ζ < − / µ > d )Here the behavior of ζ − and ζ + is opposite to the first case, namely ζ + ( r, x ) ∼ e − ( d − µ − ) r ζ (0)+ ( x ) , (B.28) ζ − ( r, x ) ∼ d − µ + e − ( d − µ + ) r b Γ (0) i D (0) i ζ (0)+ ( x ) . (B.29)3. 1 / ≥ M ζ ≥ − / d ≥ µ ≥ d − ζ − ( r, x ) ∼ e − ( µ + ) r ζ (0) − ( x ) , (B.30) ζ + ( r, x ) ∼ e − ( d − µ − ) r ζ (0)+ ( x ) . (B.31) B.4 Generalized PBH transformations
Let us find the most general bulk symmetry transformations that preserve the strong FG gauge(B.13), which we call as generalized Penrose-Brown-Henneaux (gPBH) transformations. We canimmediately see that local symmetries of the bulk SUGRA action (2.1) are diffeomorphism, localLorentz and supersymmetry transformation. Their infinitesimal action on the bulk fields takes theform δ ξ,λ,ǫ E αµ = ξ ν ∂ ν E αµ + ( ∂ µ ξ ν ) E αν − λ αβ E βµ + 12 ( ǫ Γ α Ψ µ − Ψ µ Γ α ǫ ) , (B.32a) δ ξ,λ,ǫ Ψ µ = ξ ν ∂ ν Ψ µ + ( ∂ µ ξ ν )Ψ ν − λ αβ Γ αβ Ψ µ + ( ∇ µ + 12( d − W Γ µ ) ǫ, (B.32b) δ ξ,λ,ǫ ϕ I = ξ µ ∂ µ ϕ I + i ǫζ I − ζ I ǫ ) , (B.32c) δ ξ,λ,ǫ ζ I = ξ µ ∂ µ ζ I − λ αβ Γ αβ ζ I − i /∂ϕ I − G IJ ∂ J W ) ǫ, (B.32d)with the parameters ξ µ , λ αβ ( λ αβ = − λ βα ) and ǫ respectively. The condition which keeps thestrong FG gauge invariant is then0 = ˙ ξ r , (B.33a)0 = ˙ ξ i e ai − λ a ¯ r , (B.33b)0 = ∂ i ξ r − λ ¯ ra e ai + 12 ( ǫ − Ψ + i + Ψ + i ǫ − − ǫ + Ψ − i − Ψ − i ǫ + ) , (B.33c)0 = ˙ ǫ + + ˙ ǫ − + ˙ ξ i (Ψ + i + Ψ − i ) + 14 e ai ˙ e ib Γ ab ( ǫ + + ǫ − ) + 12( d − W ( ǫ + − ǫ − ) , (B.33d)and its solution is ξ r = σ ( x ) , (B.34a) ξ i ( r, x ) = ξ io ( x ) − Z r dr ′ γ ij ( r ′ , x ) h ∂ j σ + 12 ( ǫ − Ψ + j + Ψ + j ǫ − − ǫ + Ψ − j − Ψ − j ǫ + ) i , (B.34b) λ ¯ ra = e ai h ∂ i σ + 12 ( ǫ − Ψ + i + Ψ + i ǫ − − ǫ + Ψ − i − Ψ − i ǫ + ) i , (B.34c)42 ab = λ oab ( x ) + · · · , (B.34d) ǫ + ( r, x ) = exp " r Z r dr ′ (cid:16) − W + ( d − d −
1) + γ ij ( r ′ , x ) ∂ j σ − e ai ˙ e ib Γ ab + O (Ψ ) (cid:17) ǫ o + ( x ) , (B.34e) ǫ − ( r, x ) = exp " − r Z r dr ′ (cid:16) W + ( d − d −
1) + γ ij ( r ′ , x ) ∂ j σ − e ai ˙ e ib Γ ab + O (Ψ ) (cid:17) ǫ o − ( x ) , (B.34f)where σ ( x ), ξ io ( x ), λ oab ( x ) and ǫ o ± ( x ) are ’integration constants’ which depend only on transversecoordinates. Taking into account leading behavior of the vielbeins and gravitino one can find theintegral terms are subleading in (B.34). It follows that leading asymptotics of the generalized PBHtransformations are parameterized by the arbitrary independent transverse functions σ ( x ) , ξ io ( x ) , λ oab ( x ) , ǫ o ± ( x ) , (B.35)which in fact correspond to the local conformal, diffeomorphism, Lorentz, SUSY, and super-Weyltransformations of the induced fields on the radial slice Σ r respectively, as we see soon.Extracting the leading terms in (B.32) and taking into account asymptotic behavior of theinduced fields, we obtain how the sources transform, namely (from now on and also in the maintext we do not write the subscript o ) δ ξ,λ,ǫ e ai ∼ ξ j ∂ j e ai + ∂ i ξ j e aj + e ai σ − λ ab e bi + 12 ( ǫ + Γ a Ψ + i + h . c . ) , (B.36a) δ ξ,λ,ǫ Ψ + i ∼
12 Ψ + i σ + ξ j ∂ j Ψ + i + ( ∂ i ξ j )Ψ + j + D i ǫ + − b Γ i ǫ − − λ ab Γ ab Ψ + i , (B.36b) δ ξ,λ,ǫ ϕ I ∼ −G IJ ∂ J W σ + ξ i ∂ i ϕ I + i (cid:0) ǫ + ζ I − + h . c . (cid:1) + i (cid:0) ǫ − ζ I + + h . c . (cid:1) , (B.36c)where we do not write down the variation of Ψ − i since unlike Ψ + i its leading term (B.25) does nottransform as a source so that it cannot be used as a generalized coordinate [33].As for ζ I , we need a careful discussion, since its leading behavior changes according to its mass.In the first case where M ζ ≥ / ζ I + cannot be served as a source, like the case of gravitino Ψ − i . Wealso find that in the second case where M Iζ ≤ − / ϕ ∼ e − µr due to the term i ( ǫ − ζ I + + h . c . ) ∼ e − ( d − µ I ) r , which implies ζ I + can not be turned on asa source, in order to have the theory supersymmetric. In the final case where 1 / > M ζ > − / ζ I + and ζ I − can be used as sources. The transformation law in this case is discussed in section6. In summary, what we obtain is δ ξ,λ,ǫ e ai ∼ ξ j ∂ j e ai + ∂ i ξ j e aj + e ai σ − λ ab e bi + 12 ( ǫ + Γ a Ψ + i + h . c . ) , (B.37a) δ ξ,λ,ǫ Ψ + i ∼
12 Ψ + i σ + ξ j ∂ j Ψ + i + ( ∂ i ξ j )Ψ + j + D i ǫ + − b Γ i ǫ − − λ ab Γ ab Ψ + i , (B.37b) δ ξ,λ,ǫ ϕ I ∼ G IJ ∂ J W σ + ξ i ∂ i ϕ I + i (cid:0) ǫ + ζ I − + h . c . (cid:1) , (B.37c) δ ξ,λ,ǫ ζ I − ∼ − (cid:18) d δ IK − G IJ M JK (cid:19) ζ K − σ + ξ i ∂ i ζ I − + i G IJ ∂ J W ǫ − − i b Γ i ∂ i ϕ I ǫ + − λ ab Γ ab ζ I − , (B.37d)where we inverted mass of ζ I − into the (scalar) σ -manifold language.43 Decomposition of the action and the fermion boundary terms
In this appendix we decompose the terms in the fermionic sector of the action (2.1).
C.1 Decomposition of the kinetic action of the hyperino field
The kinetic term for ζ I in the action (2.1) is decomposed as G IJ (cid:16) ζ I Γ µ ∇ µ ζ J − ( ∇ µ ζ I )Γ µ ζ J (cid:17) = G IJ ζ I (cid:0) Γ r ∇ r ζ J + Γ i ∇ i ζ J (cid:1) − G IJ ζ I ←−∇ r Γ r ζ J ζ J − G IJ ζ I ←−∇ i Γ i ζ J = G IJ ζ I " N Γ ˙ ζ J + 14 N Γ (cid:16) e ai ˙ e ib Γ ab + 2Γ b Γ i (cid:0) ∂ i N − N j K ij (cid:1) − b Γ ij D i N j (cid:17) ζ J + (cid:18)b Γ i − N i N Γ (cid:19) (cid:18) D i ζ J + 12 K ij b Γ j Γ ζ J (cid:19) − G IJ " ˙ ζ I − ζ I h e ai ˙ e ib Γ ab + 2Γ b Γ i (cid:0) ∂ i N − N j K ij (cid:1) − b Γ ij D i N j i N Γ ζ J − G IJ (cid:18) ζ I ←− D i − K ij ζ I b Γ j Γ (cid:19) (cid:18)b Γ i − N i N Γ (cid:19) ζ J = 1 N G IJ (cid:16) ζ I − ˙ ζ J + − ζ I + ˙ ζ J − − ˙ ζ I − ζ J + + ˙ ζ I + ζ J − (cid:17) + 12 N G IJ e ai ˙ e ib ζ I ΓΓ ab ζ J − N D i N j G IJ ζ I Γ b Γ ij ζ J + G IJ (cid:16) ζ I / D ζ J − ζ I ←− / D ζ J (cid:17) − N i N G IJ (cid:16) ζ I Γ D i ζ J − ζ I ←− D i Γ ζ J (cid:17) , (C.1)where the terms in the first bracket can be recast into G IJ (cid:16) ζ I − ˙ ζ J + − ζ I + ˙ ζ J − − ˙ ζ I − ζ J + + ˙ ζ I + ζ J − (cid:17) = G IJ ∂ r (cid:16) ζ I − ζ J + + ζ I + ζ J − (cid:17) − G IJ ζ I + ˙ ζ J − − G IJ ˙ ζ I − ζ J + = 1 √− γ ∂ r (cid:16) G IJ √− γζ I ζ J (cid:17) − (cid:16) N K + D k N k (cid:17) G IJ ζ I ζ J − (cid:0) ˙ ϕ K − N i ∂ i ϕ K + N i ∂ i ϕ K (cid:1) ∂ K G IJ ζ I ζ J − G IJ ζ I + ˙ ζ J − − G IJ ˙ ζ I − ζ J + . (C.2)Finally, the hyperino kinetic terms are decomposed into G IJ (cid:16) ζ I Γ µ ∇ µ ζ J − ( ∇ µ ζ I )Γ µ ζ J (cid:17) = 1 N √− γ ∂ r (cid:16) √− γ G IJ ζ I ζ J (cid:17) − N G IJ (cid:16) ζ I + ˙ ζ J − + ˙ ζ I − ζ J + (cid:17) − (cid:18) K + 1 N D k N k (cid:19) G IJ ζ I ζ J + 12 N G IJ e ai ˙ e ib ζ I Γ ab Γ ζ J − N (cid:0) ˙ ϕ K − N i ∂ i ϕ K + N i ∂ i ϕ K (cid:1) ∂ K G IJ ζ I ζ J + G IJ (cid:16) ζ I b Γ i D i ζ J − ζ I ←− D i b Γ i ζ J (cid:17) + 1 N G IJ (cid:20) − D i N j (cid:16) ζ I b Γ ij Γ ζ J (cid:17) − N i ζ I Γ D i ζ J + N i ( ζ I ←− D i )Γ ζ J (cid:21) . (C.3)44 .2 Gravitino part Repeating the same computation for the kinetic terms for gravitino as before, we obtain (cid:16) Ψ µ Γ µνρ ∇ ν Ψ ρ − Ψ µ ←−∇ ν Γ µνρ Ψ ρ (cid:17) + 1( D −
2) Ψ µ Γ µνρ ( W Γ ν ) Ψ ρ = 1 N √− γ ∂ r (cid:16) √− γ Ψ i b Γ ij Ψ j (cid:17) − N (cid:16) ˙Ψ + i b Γ ij Ψ − j + Ψ − i b Γ ij ˙Ψ + j (cid:17) − (cid:18) K + 1 N D k N k (cid:19) Ψ i b Γ ij Ψ j − N e ak ˙ e kb Ψ i Γ { b Γ ij , Γ ab } Ψ j + 12 N K lk Ψ i (cid:16) N [ b Γ ikj , b Γ l ]Γ + N i [ b Γ kj , b Γ l ] − N j [ b Γ ki , b Γ l ] (cid:17) Ψ j + 12 N K ki (cid:16) Ψ j [ b Γ ij , b Γ k ]Ψ r − Ψ r [ b Γ ij , b Γ k ]Ψ j (cid:17) + 1 N (cid:16) Ψ j ←− D i Γ b Γ ij Ψ r + Ψ r Γ b Γ ij D i Ψ j − Ψ j Γ b Γ ij D i Ψ r − Ψ r ←− D i Γ b Γ ij Ψ j (cid:17) − N W (cid:16) Ψ r Γ b Γ i Ψ i + Ψ i b Γ i ΓΨ r (cid:17) − N Ψ i (cid:16) ∂ k N [ b Γ ij , b Γ k ] − ( D k N l )Γ { b Γ ij , b Γ kl } (cid:17) Ψ j + 1 N Ψ j (cid:16) N b Γ jik − N j Γ b Γ ik − N i Γ b Γ kj − N k Γ b Γ ji (cid:17) D i Ψ k + 1 N Ψ k ←− D i (cid:16) N b Γ jik − N j Γ b Γ ik − N i Γ b Γ kj − N k Γ b Γ ji (cid:17) Ψ j − N W Ψ i (cid:16) N b Γ ij − N i Γ b Γ j + N j Γ b Γ i (cid:17) Ψ j . (C.4) C.3 Decomposition of the other terms
For the other terms, we get i G IJ ζ I Γ µ (cid:0) /∂ϕ J − G JK ∂ K W (cid:1) Ψ µ − i G IJ Ψ µ ( /∂ϕ I + G IK ∂ K W )Γ µ ζ J = iN G IJ ( N (cid:0) ˙ ϕ J − N j ∂ j ϕ J (cid:1) h ζ I (cid:16) Ψ r − N i Ψ i + N b Γ i ΓΨ i (cid:17) − (cid:16) Ψ r − N i Ψ i + N Ψ i Γ b Γ i (cid:17) ζ I i + ∂ i ϕ J h ζ I Γ b Γ i (cid:0) Ψ r − N j Ψ j (cid:1) − (cid:0) Ψ r − N j Ψ j (cid:1) b Γ i Γ ζ I i + N ∂ i ϕ I (cid:16) ζ I b Γ j b Γ i Ψ j − Ψ j b Γ i b Γ j ζ I (cid:17) ) − iN ∂ I W h ζ I Γ (cid:0) Ψ r − N i Ψ i (cid:1) + (cid:0) Ψ r − N i Ψ i (cid:1) Γ ζ I + N (cid:16) Ψ i b Γ i ζ I + ζ I b Γ i Ψ i (cid:17)i , (C.5)and G IJ h ζ I (cid:0) Γ JKL /∂ϕ L (cid:1) ζ K − ζ K (cid:0) Γ JKL /∂ϕ L (cid:1) ζ I i = 1 N ∂ K G IJ h(cid:0) ˙ ϕ J − N i ∂ I ϕ J (cid:1) (cid:16) ζ I Γ ζ K − ζ K Γ ζ I (cid:17) + N ∂ i ϕ J (cid:16) ζ I b Γ i ζ K − ζ K b Γ i ζ I (cid:17)i . (C.6) D Variation of the canonical momenta under the generalized PBHtransformation
By chain rule, δ b S ren = Z d d x X Φ Π Φ δ Φ , (D.1)45nd let us define a symmetry transformation of b S ren by δ ξ = Z d d x X Φ δ ξ Φ( x ) δδ Φ( x ) . (D.2)Let us also assume that this symmetry has anomaly, i.e. δ ξ b S ren = Z d d x X Φ Π Φ δ ξ Φ = Z d d x | e (0) | ξ A ξ . (D.3)Then, definition of the constraint function C [ ξ ] (5.18) can be written as C [ ξ ] = − Z d d x X Φ Π Φ δ ξ Φ − | e (0) | ξ A ξ ! . (D.4)Now we derive how ξ -symmetry acts on Π Φ . It is δ ξ Π Φ ( x ) = δ ξ δδ Φ( x ) b S ren = (cid:20) δ ξ , δδ Φ( x ) (cid:21) b S ren + δδ Φ( x ) δ ξ b S ren = − Z d d y X Φ ′ (cid:18) δδ Φ( y ) δ ξ Φ ′ ( x ) (cid:19) Π Φ ′ ( x ) + δ Φ( x ) Z d d y | e (0) | ξ A ξ = − δδ Φ( x ) Z d d y X Φ ′ (cid:0) Π Φ ′ ( y ) δ ξ Φ ′ ( y ) − | e (0) | ξ A ξ (cid:1) = {C [ ξ ] , Π Φ } , (D.5)(D.5) confirms (5.19). E Derivation of the SUSY algebra without using Poisson bracket
In this appendix we derive {Q [ ξ ] , Q s [ η + ] } . By differentiating the diffeomorphism Ward identity(5.2d) in the integral form with respect to Ψ + k ( y ), we get0 = Z ∂ M d d x ξ i h e ai (0) D j Π ja − ( ∂ i ϕ I (0) )Π ϕI − ( ζ I (0) − ←− D i )Π ζI − Π ζI ( D i ζ I (0) − ) − Π j Ψ (cid:0) D i Ψ (0)+ j (cid:1) − (cid:16) Ψ (0)+ j ←− D i (cid:17) Π j Ψ + D j (Π j Ψ Ψ i (0)+ + Ψ i (0)+ Π j Ψ ) i x Π k Ψ ( y )+ (cid:16) ξ i Π k Ψ (cid:17) ←− D i ( y ) − D j ξ k Π j Ψ ( y ) . (E.1)From the local Lorentz Ward identity (5.2e), we obtain0 = Z ∂ M d d x λ ab h e (0)[ ai Π ib ] + 14 (cid:0) ζ I (0) − Γ ab Π ζI + Ψ (0)+ i Γ ab Π i Ψ + h . c . (cid:1)i x Π k Ψ ( y ) − λ ab Π k Ψ Γ ab ( y ) . (E.2)Summing these two expressions for the parameter λ ab = e ia e jb D [ i ξ j ] , we obtain0 = Z ∂ M d d x D j [ ξ i ( e ai Π ja + Π j Ψ Ψ (0)+ i + Ψ (0)+ i Π j Ψ )] x Π k Ψ ( y )++ (cid:16) ξ i Π k Ψ (cid:17) ←− D i ( y ) − D j ξ k Π j Ψ ( y ) − D i ξ j Π k Ψ b Γ ij ( y ) . (E.3)46t follows from (5.59) that {Q [ ξ ] , Q s [ η + ] } = − Z ∂ M∩C dσ k ( y ) Z ∂ M d d xD j [ ξ i ( e ai Π ja + Π j Ψ Ψ (0)+ i + Ψ (0)+ i Π j Ψ )] x (Π k Ψ η + ) y = Z ∂ M∩C dσ k (cid:20)(cid:16) ξ i Π k Ψ (cid:17) ←− D i − D j ξ k Π j Ψ − D i ξ j Π k Ψ b Γ ij (cid:21) η + = Z ∂ M∩C dσ k h D i ( ξ i Π k Ψ η + − ξ k Π i Ψ η + ) + ξ k D j (Π j Ψ η + ) − Π k Ψ L ξ η + i = − Q s [ L ξ η + ] , (E.4)where the first term in the third line is zero by using the Stokes’ theorem and the second termvanishes due to the conservation law. One can confirm that the other commutators in (5.56) canbe obtained in the above way. References [1] G. Festuccia and N. Seiberg,
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