Supergravity at the boundary of AdS supergravity Aaron J. Amsel and Geoffrey Compère Department of Physics University of California, Santa Barbara Santa Barbara, CA 93106, USA We give a general analysis of AdS boundary conditions for spin-3/2 Rarita-Schwinger fields and investigateboundaryconditionspreservingsupersymmetryforagravitonmultipletinAdS4. Linear Rarita-SchwingerfieldsinAdSd areshowntoadmitmixedDirichlet-Neumannboundaryconditions when their mass is in the range 0 ≤ |m| < 1/2lAdS. We also demonstrate that mixed boundary conditionsareallowed forlargermasses whentheinnerproductis“renormalized” accordingly with theaction. Wethenusetheresultsobtainedfor |m|=1/lAdS toexploresupersymmetricboundary conditionsforN =1AdS4 supergravityinwhichthemetricandRarita-Schwingerfieldsarefluctu- 9 ating at theboundary. Weclassify boundaryconditions that preserveboundarysupersymmetry or 0 superconformalsymmetry. UndertheAdS/CFTdictionary,Neumannboundaryconditionsind=4 0 supergravitycorrespond togauging thesuperconformal group ofthe3-dimensional CFT describing 2 M2-branes, while N = 1 supersymmetric mixed boundary conditions couple the CFT to N = 1 n superconformal topologically massive gravity. a J 3 2 Contents ] h I. Introduction 1 t - A. Preliminaries 3 p e h II. Rarita-Schwinger fields in AdSd background 4 [ A. “Massive” Rarita-Schwinger fields 4 B. “Massless” Rarita-Schwinger fields 5 1 C. General asymptotic solutions 6 v D. Standard normalizeability 8 9 0 E. Renormalized modes 8 6 F. Euclidean boundary propagator 10 3 . III. Boundary analysis of d=4, =1 AdS supergravity 12 1 N 0 A. Preliminaries 12 9 B. Asymptotic solutions 12 0 C. Asymptotic gauge transformations 14 : D. Conjugate fields and regularizationof the action 16 v i E. Boundary gauge invariance 16 X F. Boundary conditions 17 r a IV. Discussion 19 Acknowledgments 20 A. Conventions 21 B. Asymptotic expansions 21 References 22 I. INTRODUCTION Supergravity theories can be understood as low energy approximations to string theory. This has led via the AdS/CFTcorrespondence(seethereview[1])totherecognitionthatsupergravitytheoriesaredualtoconformalfield theories in the planar limit and at strong coupling. In the usual formulation of the AdS/CFT correspondence, the 2 leading order coefficients in the radial expansion of bulk fields near the boundary are held fixed. These coefficients, hereafter referred to as “boundary fields,” are then interpreted as sources for dual operators in the boundary CFT. Deforming the boundary conditions corresponds on the field theory side to multi-trace deformations of the CFT action [2, 3]. A systematic analysis of boundary conditions for linear matter fields around AdS has been performed for spin-0 fields [4, 5, 6, 7], spin-1/2 fields [4, 5, 8], and spin-1 fields [4, 5, 7, 9]. For bulk scalar supermultiplets in AdS,these resultswereusedto classifythe supersymmetricmulti-tracedeformationsofthecorrespondingdualCFTs in[8,10]. Inthiswork,wewillbeinterestedinsupersymmetricboundaryconditionsforthegravitonmultiplet,which contains a spin-3/2 Rarita-Schwinger fermion. Rarita-Schwinger fields in AdS with the standard Dirichlet boundary conditions have been extensively studied [11, 12, 13, 14, 15, 16, 17], though no complete treatment including more general boundary conditions has been developed. Hence, the first aim of this paper is to fill this gap. Using the standard definition of the symplectic structure, it is generally accepted that the leading mode of the gravitonclose to the conformalAdS boundary is “non-normalizeable.” However,a detailed examination of linearized gravitonsaroundAdS[7]indicatedthatamoreinterestingtreatmentispossibleforsomegravitonmodesind=4,5,6 spacetime dimensions. This hint motivated the work [18] where, perhaps surprisingly, it was shown that alternative “Neumann-type”boundaryconditions(inwhichtheboundarymetricisdynamical)existforthefullnon-lineargravity theory in any dimension (see also related comments in [9, 19, 20, 21]). Varying the “non-normalizeable” part of the bulkmetricturnsouttobeallowedoncethesymplecticstructureisrenormalizedaccordinglywiththeaction. Inshort, all infinities appearing in the standard symplectic structure are canceled by the contributions of the counterterms. The second aim of this paper is to analyze boundary conditions for Rarita-Schwingerfields using the renormalized symplectic structure. Once the renormalizationhas been performed, modes that were previously consideredas “non- normalizeable” are now allowed to fluctuate. This step will allow us later in the paper to extend the Neumann boundary conditions for gravityto supergravity. For minimal supergravityin AdS , the standard Dirichlet boundary 4 conditions were shown to preserve supersymmetry on the boundary in [10]. Allowing the boundary metric to fluctuate corresponds to varying the metric on which the CFT lives, i.e., to considering the induced gravity of the dual CFT. The resulting boundary gravity theory is non-local in the metric or, equivalently, admits an infinite expansion in higher curvature terms. In the simple example of two dimensions, conformalinvariancedictatesthattheeffectiveactionofanyCFTisthePolyakovaction[22]. Higher-derivativegravity theories are genericallyplaguedwith a lack of unitarity [23]. However,quite remarkably,the induced gravitytheories obtained in odd boundary dimensions (d 1) were shown to be ghost and tachyonfree around a flat boundary1 [18]. − Thesimplestsuchtheoryisthedualtofour-dimensionalEinsteingravitywithnegativecosmologicalconstant. Asthe precise mapping between strongly coupled CFTs and gravitationaltheories generally requires a large amount of bulk supersymmetry, it is a natural step to extend the pure gravity results of [18] to = 1 AdS supergravity [26, 27]2. 4 N To analyze boundary conditions for this theory, we follow the holographic renormalization procedure [31, 32, 33]. The bulk fields of supergravity theories with negative cosmological constant generically induce conformal supergravity multiplets at the conformal boundary. In particular, it was noticed soon after the emergence of AdS/CFT that the gauged d = 5, = 8 supermultiplet in the bulk induces the d= 4, = 4 conformal supermultiplet at the N N boundary[34,35]. Ind=3,(p,q)-multipletsgeneratetwo-dimensional(p,q)-conformalsupermultipletsatthebound- ary (for p,q 2) [36], while in the less studied gauged d = 6 F(4) supergravity, the boundary fields form a = 2 ≤ N superconformal multiplet [37, 38]. One can also argue that gauged d=4, =8 supergravity will induce the d= 3, N = 8 conformal supermultiplet at the boundary. Indeed, it has been shown recently that = 8 superconformal N N gravity can be coupled to Bagger-Lambert-Gustavsson theory [39], which is closely related [40] to the CFT dual of M-theory on AdS C4/Z [41]. In this paper, we will show that at least the =1 conformalsupermultiplet [42] is 4 k × N induced at the conformal boundary of minimal supergravity. The outline of this paper is as follows. Section II is devoted to linear Rarita-Schwinger fields in d spacetime dimensions. After reviewing general properties of spin-3/2 fields in AdS, we analyze admissible boundary conditions withrespecttoboththe standardsymplecticstructureandthe renormalizedsymplecticstructure. Inthe lastsubsec- tion, we review the standard Euclidean propagator for spin-3/2 fields with Dirichlet boundary conditions. We then determine the corresponding propagatorin the Neumann theory and discuss the presence of tachyons. In section III, we implement the holographic renormalizationprocedure for d=4 minimal supergravity. In particular, this requires solving the equations of motion asymptotically and finding the allowed asymptotic gauge transformations. We then discussrenormalizingtheactionbyaddingcertainboundarycountertermsandshowthattheactionisinvariantunder 1 Forevendimensionalboundaries,thetheoriescontainbothtachyons andghostsaroundflatspace. However,thetheorymaybestillbe stableinsomeparameterregionarounddeSitterspace, whichmayhaveconsequences forcertaininflationmodels[24,25]. 2 Boundaryconditionsimposedatafinitevalueoftheradialcoordinatehavebeenstudiedrecentlyinsupergravitywithzerocosmological constant[28,29,30]. Inthiscurrentwork,theboundaryisatinfinity. 3 supersymmetry, special conformalsupersymmetry, and Weyl transformationsat the boundary. Section IV contains a summary of our results and a discussion of unitarity in theories with Neumann boundary conditions. A. Preliminaries Beforerestrictingtoanyspecifictheory,wereviewthegeneraldefinitionofthesymplecticstructure. Thisformalism will be applied to the theory of spin-3/2 fields linearized about AdS in section II and to AdS supergravity in section III. Conventions used throughout this work are summarized in appendix A. InAdSspacetimes,onemustbecarefultochooseboundaryconditionsthatleadtoawell-definedbulktheory. This means in particular that the bulk symplectic structure is both finite and conserved. In general, the first variation of a Lagrangian density (written as a d-form) for a set of fields φ can be expressed as δL = E δφ+dθ, where the equations of motion of the theory are E = 0, and dθ corresponds to surface terms that would a·rise from integrating by parts. Now, let Σ be a constant time hypersurface with unit normal tµ. Then, for linearized solutions δ φ, the 1,2 standard symplectic structure on Σ is σ (φ;δ φ,δ φ)= ω(φ;δ φ,δ φ), (1.1) Σ 1 2 1 2 ZΣ where we have defined the standard symplectic current (a (d 1)-form ) as − ω(φ;δ φ,δ φ)=δ θ(φ;δ φ) δ θ(φ;δ φ). (1.2) 1 2 1 2 2 1 − For scalar fields, σ (φ;δ φ,δ φ) is just the familiar Klein-Gordon inner product. Σ 1 2 Itis well-knownthatthe symplectic potential, θ, admits anambiguityinits definitiongivenbyθ θ+dB, where → B is an arbitrary (d 2)-form. This leads directly to a corresponding ambiguity in the definition of the symplectic − current (1.2). However, it has been suggested in [18] that this ambiguity can be fixed once the action has been “renormalized.” This refers to the fact that the subtraction of divergences in the action by diffeomorphism-invariant boundary terms L [φ ] is required in order to render the action finite on-shell and to define the boundary stress- I ct ∗ tensor [31, 43, 44]. Here we denote the AdS conformal boundary by and the induced fields on the boundary R I collectivelyby φ . Now,these boundaryterms generallycontaintime derivativesofthe boundaryfields andtherefore ∗ shouldcontributetothesymplecticstructure3. Inaddition,non-minimal(finite)termsmaybeaddedtotheboundary actionandshouldalsocontributetotheboundarydynamics. Toseethis,wefirstdefineθ [φ ]bythevariationformula ct ∗ δL = δLctδφi +d θ [φ ;δφ ], where d is the induced exterior derivative on surfaces of constantradialcoordinate. ct δφi∗ ∗ ∗ ct ∗ ∗ ∗ We then define the corresponding renormalized symplectic current as ω =ω+dω , (1.3) ren ct where ω =δ θ [φ ;δ φ ] (1 2), and dω is an arbitrary smooth extension of d ω away from the boundary. ct 1 ct ∗ 2 ∗ ct ∗ ct − ↔ Forodddimensionalboundaries,thisdefinitionwillbeindependentofthechoiceofradialfoliation. However,foreven dimensional boundaries the renormalized symplectic structure may depend explicitly on the radial foliation because of the conformal anomaly. Ingeneral,varyingtherenormalizedactionyieldsafiniteexpressionoftheformδS = dd−1xπiδφ∗,whichdefines I ∗ i the fields πi that are conjugate to φ∗. The renormalized symplectic flux through the boundary is then given by the ∗ i R finite expression = ω = dd−1x δ πi δ φ∗ (1 2) . (1.4) Fren ren 1 ∗ 2 i − ↔ ZI ZI (cid:0) (cid:1) Note that since ω and ω differ only by a boundary term, it is possible to deduce the integrand in (1.4) from the ren originalLagrangianL. However,the counterterms are needed to remove the divergences occurring at the boundaries ∂ . In order to ensure that the symplectic structure is conserved, i.e. that = 0, we have to impose either ren DIirichlet(φ∗ fixed),Neumann(πi fixed),ormixed(πi = δW[φ∗i] forsomearbitraFryfunctionW)boundaryconditions. i ∗ ∗ δφ∗ i 3 InthecaseofEinsteingravity,theGibbons-Hawkingandcosmologicalboundarytermsdonotprovideacontributiontothesymplectic structure,whiletheboundaryEinsteinandhighercurvatureactions docontribute[18]. 4 Let us consider a spacetime region bounded by two constant time slices Σ ,Σ whose boundaries ∂Σ S , 1 2 1 1 ∂Σ S are Sd−2 spheres4. Since the symplectic current is closed on-shell, we can write the flux in the form≡ 2 2 ≡ =σ σ , (1.5) Fren Σ2,ren− Σ1,ren where the symplectic structure associated with a t=constant hypersurface Σ is given by σ (φ;δ φ,δ φ)= ω(φ;δ φ,δ φ) ω (φ ;δ φ ,δ φ ). (1.6) Σ,ren 1 2 1 2 ct ∗ 1 ∗ 2 ∗ − ZΣ Z∂Σ We see that the renormalized symplectic structure (1.6) differs from the standard symplectic structure (1.1) only when boundary fields are allowed to vary. Now, it was found in [18] that when φ is the metric field, this boundary contributioncancelsthedivergencesappearinginthestandardsymplecticstructure. Thisresultisexpectedtoholdin general. Below,we indeed verify explicitly the cancelationof divergencesfor a large rangeof parametersin the linear spin-3/2theory. Inthatsense,therenormalizedsymplectic structureextends thestandarddefinitionbytransforming “non-normalizeable”modes into “renormalized” modes. II. RARITA-SCHWINGER FIELDS IN ADSd BACKGROUND We now consider the linearized theory of spin-3/2 fermions in AdS , i.e., the fields propagate on a fixed AdS d background. Afterprovidinggeneralasymptoticsolutionstothe Rarita-Schwingerequation,wethenanalyzenormal- izeabilityofthesolutionswithrespecttothestandardsymplecticstructureandtherenormalizedsymplecticstructure. To conclude, we briefly discuss several interesting features of the spin-3/2 boundary propagator. A. “Massive” Rarita-Schwinger fields We now review the massive Rarita-Schwinger equation in AdS spacetime. The most general Rarita-Schwinger action in d spacetime dimensions is S =N ddx√ g ψ ΓµνλD ψ +m ψ Γµλψ +m ψ ψµ , (2.1) µ ν λ 1 µ λ 2 µ − Z (cid:2) (cid:3) whereN issomeconstantnormalizationthatfornowweleavearbitrary. Intheaboveaction,wetakethe(torsion-free) covariant derivative acting on a vector-spinor to be 1 D ψ =∂ ψ Cλ ψ + ω µˆνˆ(e)Γ ψ , (2.2) ρ σ ρ σ− ρσ λ 4 ρ µˆνˆ σ where ω µˆνˆ(e) are the rotation coefficients defined by the “vielbein postulate” ρ D eµˆ =∂ eµˆ Cλ eµˆ +ω µˆ (e)eνˆ =0 (2.3) ρ µ ρ µ− ρµ λ ρ νˆ µ and Cλ is the standard Christoffel symbol. Note, however, that torsion terms will be required in the nonlinear ρσ supergravity theory discussed in section III. By varying the action (2.1) with respect to the Rarita-Schwinger field, we obtain the equations of motion ΓµνλD ψ +m Γµλψ +m ψµ =0. (2.4) ν λ 1 λ 2 If we apply D to both sides of the Rarita-Schwinger equation (2.4) and use the gamma matrix identity Γµνλ = µ γµγνγλ gµνγλ gνλγµ+gµλγν, we obtain − − (cid:26)(cid:26)D(cid:26)(cid:26)D(γ ψ) D2(γ ψ) (cid:26)(cid:26)D(D ψ)+D ((cid:26)(cid:26)Dψµ)+m ΓµλD ψ +m D ψ =0, (2.5) µ 1 µ λ 2 · − · − · · where(cid:26)(cid:26)D =γµD . Similarly, applying γ to (2.4) yields µ µ (d 2)(cid:26)(cid:26)D(γ ψ)+(2 d)D ψ+((d 1)m +m )γ ψ =0. (2.6) 1 2 − · − · − · 4 Ourconventions fortheorientationofsurfacesintheapplicationofStokes’s TheoremmatchthosegiveninappendixBof[45]. 5 Now, in exact AdS the Riemann tensor takes the simple maximally symmetric form R = (g g g g ), ρσµν ρµ σν ρν σµ − − where we have set the AdS radius to one. Using the general relation 1 [D ,D ]ψλ =Rλ ψσ+ R Γρσψλ (2.7) µ ν σµν µνρσ 4 combined with the above expression for the AdS Riemann tensor, we find that eq. (2.5) becomes 1 (d 1)(d 2) D ψ = − − +m (cid:26)(cid:26)D γ ψ. (2.8) 1 · m m 4 · 1− 2 (cid:18) (cid:19) Substituting this result into (2.6) implies that m D ψ = Cγ ψ, where C is a constant depending on d,m ,m . In 2 1 2 · · the case m =0, this relation becomes 2 (d 2)2 m2 − γ ψ =0. (2.9) 1− 4 · (cid:18) (cid:19) Let us now define m2 to be the special mass value given by m2 (d 2)2/4. Hence, if m2 = m2, we arrive at the 0 0 ≡ − 1 6 0 gamma-traceless and divergence-free conditions γµψ =0=D ψµ. (2.10) µ µ Applying these conditions to the original Rarita-Schwinger equation (2.4) leads to the “Dirac equation” (γµD m )ψ =0. (2.11) µ 1 λ − Note that if m = 0 or m2 = m2, we do not obtain the divergence-free and gamma-traceless conditions from the equations of mo2ti6on. The c1ase m0=0, m2 =m2 deserves special attention and will be discussed in detail in section 2 1 0 IIB. For m = 0, the γ-trace of ψ becomes a dynamical Dirac spinor; we will not consider such cases any further 2 µ 6 (see [13, 16]). We refer to the case m m=m ,m =0 as the “massive” theory of the Rarita-Schwinger field. 1 0 2 ≡ 6 B. “Massless” Rarita-Schwinger fields Let us now discuss the special case m2 =m2 =(d 2)2/4 and m =0, where it was seen above that the equations 0 − 2 of motion do not automatically imply the gamma-traceless and divergence-free conditions. One may check that the “massless” spectrum of supergravity on, for example, AdS S7 [46] or AdS S5 [47] contains spin-3/2 fields with 4 5 × × exactly these mass values. The same statement holds for = 1 minimal AdS supergravity in d = 4 [27], which we N discuss in sectionIII below. Itis thereforenot surprisingthatfor this particularvalue of the mass,the theoryadmits anextragaugesymmetry whichis closelyrelatedto supersymmetry. This symmetrycanbe seenas follows. First,by defining the “superderivative” m =D γ , (2.12) ν ν ν D − d 2 − the Rarita-Schwinger equation can be expressed in the form Γµνλ ψ = 0. Acting on this relation with , we ν λ µ D D obtain the Bianchi-type identity (d 1)(d 2) 4m2 0= (Γµνλ ψ )= − − 1 γσψ . (2.13) Dµ Dν λ − 4 (d 2)2 − σ (cid:18) − (cid:19) Note thatthe righthandside vanishesidenticallyinthe “massless”casem2 =m2,while inthe massivecaseweagain 0 obtainthe constraintγµψ =0. Asthis suggests,inthe masslesscaseonecanshowthatthe actionisinvariantunder µ the gauge transformation δψ = ǫ, where ǫ is an arbitrary spinor (so this is essentially a supersymmetry-type µ µ D transformation). In particular, the variation of the Rarita-Schwinger Lagrangianunder these transformations is N(d 2)(d 1) 4m2 δL = − − 1 ǫγσψ ψ γσǫ , (2.14) RS 8 (d 2)2 − σ− σ (cid:18) − (cid:19) (cid:0) (cid:1) which vanishes (off-shell) in the massless case. 6 Infact,wecanusethisgaugefreedominthemasslesscasetofixthegaugeconditionγµψ =0. Thedivergence-free µ condition then directly follows from (2.8). To see this explicitly, let ψ′ =ψ + ǫ. Then we require µ µ Dµ d γµψ′ =γµψ + (cid:26)(cid:26)D m ǫ=0, (2.15) µ µ − d 2 (cid:18) − (cid:19) so we get the desired result by choosing ǫ= ((cid:26)(cid:26)D M)−1γµψ for M dm/(d 2). Note there is still some residual µ gauge freedom to take δψ′ = χ, for χ a so−lutio−n of ((cid:26)(cid:26)D M)χ=0.≡ − µ Dµ − In summary, we have seen that by choosing an appropriate gauge in the massless case, we can write the Rarita- Schwinger equation in the same form given earlier for the massive case: γµψ = 0,D ψµ = 0,((cid:26)(cid:26)D m)ψ = 0. We µ µ µ − can thus discuss in a unified way solutions to the equations of motion for both massive and massless fields. One can also learn from the existence of an extra gauge invariance in the “massless” case that massive fields (m =(d 2)/2, m arbitrary) should admit an extra spin-1/2 degree of freedom5. However,as further explained in 1 2 6 − section IIC, this extra spinorial degree of freedom will have no significant role in our analysis as long as 2m 0 and using the standard choice for the symplectic structure, we are required to choose 0 the “Dirichlet” boundary condition that fixes the slow fall-off mode to zero. This implies that the behavior near the boundary is ψµ =O(Ωd−25), which matches the boundary conditions imposed for d=4 in [10]. Normalizeabilityplacesnorestrictionontheboundaryfields,butwemuststillimposeconservationofthesymplectic structure. Using the expansion (2.18), we find that for m <1/2, the symplectic flux through the boundary is given | | by = ω = N dd−1x δ α(0)δ β(0),i δ β(0)δ α(0),i δ δ . (2.31) F − 1 i,+ 2 − − 1 i,− 2 + − 1 ↔ 2 ZI ZI h(cid:16) (cid:17) (cid:16) (cid:17)i Any boundary condition such that the integrand vanishes is sufficient to give well defined dynamics. For example, (0) (0) a non-trivial mixed condition in even dimensions is given by the linear relation β = iqγ α , where q is an i,− d+1 i,+ arbitrary real parameter and γ is the higher dimensional generalization of γ in four dimensions. Dirichlet or d+1 5 Neumann boundary conditions correspond to the particular choices q = or q =0, respectively. ∞ E. Renormalized modes We next turn to the case m 1/2. For these masses, the mode α(0) is non-normalizeable with respect to the ≥ standard symplectic structure. However, as in the case of gravitational perturbations [18], one can consider the 9 renormalized symplectic structure (1.3) for which we expect the slow fall-off modes to be normalizeable as well, allowing more general boundary conditions. In what follows, we will determine the boundary counterterms required to renormalize the symplectic structure associated with the action (2.1). We then verify normalizeability of the slow fall-off modes for a large class of parameters. For convenience, we will assume below that the spin-3/2 fields obey the Majorana “reality” condition, with the charge conjugation matrix satisfying the same properties as in d = 4 (see appendix A). Hence, our results will be strictly valid in d = 2,3,4 mod 8 dimensions, but could be easily generalized to other dimensions by introducing extra signs in certain places or working with complex Dirac spinors. In order to determine how to modify the symplectic structure, we must first discuss boundary counterterms in the action: S =S + dd−1xL , (2.32) RS ct ZI whereS isthestandardRarita-SchwingeractionandtheboundarycountertermLagrangianL istobedetermined. RS ct Note that because the Rarita-Schwinger action vanishes on-shell, S will be manifestly finite on-shell if we also take S finite. In other words, S is not a divergent counterterm, but is simply the finite expression required to provide ct ct a valid variational principle. Using the asymptotic solutions of ψ given in (2.18) and (2.22), we find in general that µ on-shell δS = N dd−1x (divergentterms)+α(0)δβ(0),i β(0),iδα(0) +Fi[α(0)]δα(0) (2.33) RS − i,+ − − − i,+ + i,+ ZI h i where Fi[α(0)] refers to possible additional finite terms involving at least3 derivatives of α(0) that may only arise for + + m 3/2 and m Z+ 1. By the argument above, the divergent terms must combine to form a total derivative and ≥ ∈ 2 canthus be dropped. We havecheckedthatthis is true explicitly for all m <3/2in any d 3 and m =3/2in any | | ≥ | | d>3. To impose Dirichlet-type boundary conditions (fixing α(0)), one has to add to the original action a counterterm of i,+ the form L =Nα(0)β(0),i+L [α(0)], (2.34) ct i,+ − fin + where L [α(0)] are non-minimal terms depending on α(0) and its derivatives. The total action then obeys fin + + δS =δS +δ dd−1xL = dd−1xπ(0),iδα(0) . (2.35) RS ct − i,+ ZI ZI Here,theconjugatefieldisfoundtobeπ(0),i 2Nβ(0),i NFi[α(0)]+δLfin. Thesymplecticfluxhasthegeneralform − ≡ − − + δα(0) i,+ (0),i (1.4). Neumann-typeboundaryconditions(fixingπ )arethenobtainedbyperformingtheLegendretransformation − S =S π(0),iα(0). Neu − I − i,+ We now make the observation that the first term in the right-hand side of (2.34) can be expressed in terms of the R original bulk fields as 1 α(0)β(0),i = √ hhijψ ψ +(divergent terms)+G [α(0)] (2.36) i,+ − 2 − i j fin + wherethenumberofdivergentterms(dependingonlyonα(0) anditsderivatives)increaseswiththevalueofthemass. + (0) (0) The various finite terms G [α ] (depending also only on α and its derivatives) appear for masses m 3/2,m fin + + ≥ ∈ Z+1. We conclude that the countertermof the form (2.34) containing the minimal number of terms whenexpressed 2 in terms of the bulk field ψ is given by the first term on the right-hand side of (2.36) plus a finite number of terms µ canceling the additional divergences. The resulting minimal counterterm LagrangianL for m 3/2 is found to be ct ≤ N L = √ h ψ ψi+2f (Ω)ψ γj∂ ψi ct 2 − " i 1/2 i j 4d +2f (Ω) ψ γj∂k∂ ∂ ψi ψiγj∂ ∂ ∂ ψk , (2.37) 3/2 (cid:18) i k j − (d−1)(d+1) i j k (cid:19)# 10 where 1 if m= 1 0 if m< 3 f (Ω)= 1−2m 6 2 f (Ω)= 2 (2.38) 1/2 logΩ if m= 1 3/2 1logΩ if m= 3,d>3, (cid:26) 2 (cid:26) 4 2 and where all indices are raised with hij = Ω2ηij. For 3/2 < m < 5/2, we expect the appearance of additional counterterms involving 3 derivatives, while for m 5/2 we would require additional counterterms involving more ≥ than 3 derivatives. The properties of the recursion relations for the coefficients α(n) found in section IIC imply that all such counterterms will involve an odd number of derivatives. This is consistent with the fact that acting withsupersymmetrytransformationsongravitycounterterms[43]generatesonlyfermioniccountertermswithanodd number of derivatives. Forhalf-integermasses,wehaveseenthatlogarithmicmodesalwaysariseintheasymptoticsolutionstotheRarita- Schwinger equation. Furthermore, we have found in the specific cases m = 1/2,3/2 that the required counterterms contain explicit logΩ dependence, and we would expect this to be true for larger half-integer masses as well. This suggests that for all m Z+ 1, the renormalization procedure explicitly breaks radial diffeomorphism invariance. ∈ 2 ThisisinterpretedintheAdS/CFTlanguageasadynamicalbreakingofconformalinvariance[48]. Thisphenomenon is partially explained by the analogousresults in gravity. Indeed, the m=1/2 field appears in d=3 supergravity,as does m = 3/2 for d = 5. In these theories, the gravitational counterterms contain a logarithmic term (proportional to the Einstein-Hilbert action for d = 3 or Weyl curvature squared for d= 5) that breaks conformal invariance. We point out though that the spin-3/2 conformal symmetry breaking is generic for any half-integer m in any dimension. We end this section by providing a non-trivial check of the claim that the renormalized symplectic structure is finite for the “non-normalizeable”mode and by providinga non-trivialmixedboundary condition. Let us consider in particularthe cases1/2

Supergravity at the boundary of AdS supergravity

Similar Supergravity at the boundary of AdS supergravity