Supergravity at the boundary of AdS supergravity

0.39 MB·English

by  Aaron J. Amsel

Supergravity at the boundary of AdS supergravity
Aaron J. Amsel and Geoffrey Comp`ere
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 super-
gravity 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 di-
mensions. 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 <d.
1
C. General asymptotic solutions
In order to describe the boundary fields and determine the allowedboundary conditions, we must first analyze the
behaviorofsolutionsto(2.10),(2.11)neartheAdSboundary. Evaluatingtheon-shellrenormalizedactionalsorequires
only the asymptotic solutions,since the Rarita-Schwingeraction is zero on-shell(see sectionIIE). Exactsolutions to
theRarita-SchwingerequationhavebeenfoundbyFourier-transformingtomomentumspace[11,12,13,14,15,16,17],
butfor ourpurposesitis usefultosolvethe equationsby performingaFefferman-Grahamtype expansionaroundthe
boundary.
To do this expansion, it will be convenient to use the conformal compactification of AdS spacetime. In Poincar´e
coordinates, the AdS metric takes the form
1
ds2 = dt2+dΩ2+dx2+...dx2 . (2.16)
Ω2 − 1 d−2
(cid:0) (cid:1)
One then defines an unphysical metric g˜ = Ω2g so that the unphysical spacetime is a manifold with boundary
µν µν
= Rd−1 at Ω = 0. In this spacetime, n˜ = D˜ Ω coincides with the unit normal to the boundary, where
I ∼ µ − µ
D˜ is the torsion-free covariant derivative compatible with g˜ . It is also useful to define the orthogonal projector
µ µν
h˜ =g˜ n˜ n˜ ,whichatΩ=0becomestheinducedmetricontheboundaryh˜ dxµdxν = dt2+dx2+...dx2 ,
µν µν− µ ν µν |I − 1 d−2
i.e., Minkowski space. Indices on all tensor fields with a tilde are raised and lowered with the unphysical metric g˜
µν
and its inverse g˜µν. Note that because we work in the Poincar´e patch, γ˜ Ωγ are now flat-space gamma matrices
µ µ
≡
and we may identify D˜ ∂ .
µ µ
Under the conformal tr→ansformation g˜ =Ω2g , the equation of motion (2.11) takes the form
µν µν
d 3
Ωγ˜ρD˜ ψ + − n˜ γ˜ρψ mψ +γ˜ n˜λψ =0. (2.17)
ρ σ ρ σ σ σ λ
2 −
By solving the equation (2.17) at leading order in Ω, we are lead to the existence of two independent solutions with
fall-off O(Ωd−23±m). We now distinguish between two general cases: 2m integer valued and 2m non-integer valued.
Without loss of generality, we take m 0 in the following.
≥
1. Case: 2m Z
6∈
Let k be the greatest integer strictly less than 2m, i.e., 0<2m k <1. We then expand the Rarita-Schwinger
−
field as
k
ψµ =Ωd−23−m αµ(n)Ωn+βµ(0)Ωd−23+m+O(Ωd−21+k−m), (2.18)
n=0
X
wherethe coefficients α(n),β(n) arevector-spinorsthatdonotdepend onΩ,but mayingeneraldepend ontime
µ µ
and the remaining spatial coordinates. Substituting this expansion into the equations of motion gives certain
5 WethankDonMarolffordiscussionsofthispoint.
7
constraintsonthe coefficients. The gamma-tracelessconstraintγµψ =0clearlyimpliesthatallthe coefficients
µ
arethemselvesgamma-traceless: γ˜µα(µj) =0=γ˜µβµ(j). FortermsuptoO(Ωd−23+m), wefindfromthe remaining
equations of motion
n˜µα(0) =0, P˜ α(0) =0, n˜µβ(0) =0, P˜ β(0) =0, (2.19)
µ − µ µ + λ
and for k =1,...,n
2
n˜µα(k) = h˜µν∂ α(k−1) (2.20)
µ −d 2k+2m µ ν
−
m+( 1)k(m k) 2
h˜ λα(k) = − − h˜ λh˜ρσ γ˜ ∂ α(k−1) γ˜ ∂ α(k−1) , (2.21)
µ λ k(2m k) µ ρ σ λ − d 2k+2m λ ρ σ
(cid:0) − (cid:1) (cid:20) − (cid:21)
where we have defined the radial gamma matrix projectors P˜ = 1(1 n˜ γ˜µ). We see that the solution is
± 2 ± µ
parameterized by the two vector-spinor boundary fields α(0) P˜ α(0) and β(0) P˜ β˜(0). Here we have
i,+ ≡ + i i,− ≡ − i
introduced the useful index notation µ = (Ω,i), i.e. i,j,k... denote indices tangent to the boundary. These
indices are then raisedand loweredwith the Minkowskimetric η and its inverse. The coefficients in the radial
ij
expansion at order O(Ωd−21−m+k) and beyond are completely determined in terms of these two independent
vector-spinorsand will not play any role in the analysis below.
2. Case: 2m Z
∈
The second general class of solutions occurs when m is an integer or half-integer. Note that the massless case
always lies in this category. Here, the spin-3/2 field can be expanded as
2m−1
ψµ =Ωd−23−m αµ(n)Ωn+αµ(2m)Ωd−23+mlogΩ+βµ(0)Ωd−23+m+... . (2.22)
n=0
X
The coefficients are once again all gamma-traceless and now satisfy
n˜µα(0) =0, mP˜ α(0) =0, (2.23)
µ − µ
d 1
m n˜µβ(0) = h˜µν∂ α(2m−1), mP˜ β(0) = P˜ ˜hρσγ˜ ∂ α(2m−1)+γ˜ n˜ρβ(0) , (2.24)
2 − µ − µ ν + λ 2 + σ ρ µ µ ρ
(cid:18) (cid:19) (cid:16) (cid:17)
while for k =1,...,2m 1
−
d
+m k n˜µα(k) = h˜µν∂ α(k−1) (2.25)
2 − µ − µ ν
(cid:18) (cid:19)
and
(m+(m k)n˜ργ˜ )
α(k) = − ρ ˜hρσγ˜ ∂ α(k−1)+γ˜ n˜ρα(k) . (2.26)
µ k(2m k) ρ σ µ µ ρ
− h i
The coefficient of the logarithmic term α(2m) is only nonzero when m Z+ 1, and in this case
µ ∈ 2
d
m n˜µα(2m) =0, (2.27)
2 − µ
(cid:18) (cid:19)
P˜ α(2m) =0, P˜ α(2m) = P˜ h˜ρσγ˜ ∂ α(2m−1)+γ˜ n˜ρβ(0) . (2.28)
+ µ − µ − − σ ρ µ µ ρ
(cid:16) (cid:17)
When m = 0, the linearly independent modes degenerate to a single unconstrained boundary field β(0), which
i
we then naturally split with P˜ into two independent parts that we call α(0), β(0).
± i,+ i,−
8
It is straightforwardto check that the right-hand side of (2.25) is also proportionalto (d +m k) for k 2 as
2 − ≥
a consequence of (2.26). Thus, the above relations indicate that for certain values of the mass and spacetime
dimension (i.e., when this factor vanishes), the radial components of certain coefficients are undetermined by
the equations of motion. This reflects the existence of an extra spin-1/2 degree of freedom which admits the
expansion(2.22) when m is half-integer or integer valued. This degree offreedom was not seenin the firstclass
ofsolutions becauseitfalls offasO(Ωd−3/2)andthe ansatz(2.18) didnotinclude integerorhalf-integerpowers
of Ω. The extra degree of freedom appears in the coefficients n˜µα(2m), n˜µβ(0) for m = d and in the coefficient
µ µ 2
n˜µα(m+d2) when m= d +j, j 1. In the following discussion of the on-shell action, the counterterms, and the
µ 2 ≥
normalizeability,it turns out that one only needs to solve the equations of motion up to order Ωd−23+m. Hence,
this phenomenon will only effect the counterterms if m d; we will not consider such cases below (note that
≥ 2
because m < d, this restriction does not exclude the interesting “massless” cases).
0 2
The occurrence of logarithmic modes for half integer masses hints at the appearance of conformal anomalies
that will be described below.
D. Standard normalizeability
Inthissection,weanalyzenormalizeabilityoftheabovesolutionswithrespecttothestandardsymplecticstructure.
Standard methods applied to the action (2.1) lead to
(ω) =N δ ψ Γµνλδ ψ δ ψ Γµνλδ ψ ǫ . (2.29)
µ1...µd−1 1 µ 2 λ− 2 µ 1 λ νµ1...µd−1
This expression simplifies after using the con(cid:0)straint γµψ =0, resulting in th(cid:1)e symplectic structure
µ
σ (δ ψ ,δ ψ )=N dd−1x√g t δ ψµγνδ ψ δ ψµγνδ ψ . (2.30)
Σ 1 µ 2 ν Σ ν 1 2 µ 2 1 µ
−
ZΣ
(cid:0) (cid:1)
Substituting the asymptotic form of the slow fall-off mode (i.e. ψµ O(Ωd−23−|m|)) into the symplectic structure, we
∼
find that the integrand near Ω 0 is O(Ω−|2m|). Therefore, the standard symplectic structure for the slow fall-off
→
mode is finite only for fields in the mass range m <1/2, where there is then a choice of boundary conditions. This
| |
mass range is thus analogous to the range at or slightly above the Breitenlohner-Freedman bound for scalar fields
[4]. However, normalizeability requires boundary conditions of Dirichlet-type for the limiting case m=1/2, where a
logarithmic mode appears. The corresponding results for spin-1/2 fields are essentially identical [8]. It was further
argued in [8] that for real values of the spin-1/2 fermion mass, there is no analogue of unstable modes below the
Breitenlohner-Freedman bound. We would similarly expect that there are no such unstable modes in the spin-3/2
case, though this has yet to be checked explicitly.
The massless spin-3/2 fields are notably outside this special mass range. Indeed, for all d 3, we have m 1/2.
0
≥ | |≥
Thus, taking m = m > 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<m 1, d 3. Following the generaldiscussionin sectionIA, we canvarythe counterterm
≤ ≥
action(fortheNeumanntheory)asδL = δLNeuδψ +dθ andobtaindθ =dd−1x√ h∂ N ψ γjδψi . Using
Neu δψi i ct ct − j 2m−1 i
the expansion (2.22), we find (cid:16) (cid:17)
2N
(ω ) = Ω1−2mδ α(0) γ˜lδ α(0),kǫ˜ . (2.39)
ct i1...id−2|I 2m 1 1 k,+ 2 + Ωli1...id−2
−
This divergent term is then recognized as the exact divergent term in the standard symplectic structure. The renor-
malized symplectic structure (1.6) takes the form
σ =2N dd−2xdΩ Ω3−dg˜µλδ ψ γ˜tδ ψ Ω−2mh˜µλδ α(0)γ˜tδ α(0) . (2.40)
Σ,ren 1 µ 2 λ− 1 µ 2 λ
ZΣ h i
ThefirsttermintheaboveexpressiondivergesasO(Ω−2m),butthisdivergenceisexactlycanceledbythesecondterm.
The remaining terms are then of order O(Ω0) and so σ is manifestly finite. After the divergences are removed,
Σ,ren
the renormalizedsymplectic fluxtakesthe same(finite)formasin(2.31). This explicitcheckofourmethodsincludes
in particular the special case m = 1, d = 4, which is relevant for = 1 minimal AdS supergravity. To have a
4
N
mixedboundaryconditioninthis case,we can,forexample,impose the linearrelationβ(0) =iqγ α(0) whichleadsto
i,− 5 i,+
vanishing flux and includes the Neumann boundary conditions as the case q =0. This corresponds to a deformation
of the Neumann theory given by
N
d3x√ hL = d3x√ h ψ ψi 2ψ γj∂ ψi 2iqΩ2ψ γ ψi . (2.41)
ct,q i i j i 5
− − 2 − − −
ZI ZI
(cid:0) (cid:1)
For q = 0, these mixed boundary conditions break conformal invariance, as can also be seen in the explicit radial
6
dependence of the counterterm.
F. Euclidean boundary propagator
In this section, we review the two-point correlation function of the CFT operator dual to the boundary field α(0).
i,+
Thiscorrelationfunctioncanbeobtainedfromtheon-shellactionandtheasymptoticsolutionsderivedintheprevious
sectionviastandardmethods [12, 13, 14, 15, 16,17]. We then discuss the correlationfunction for spin-3/2fields with
Neumannboundaryconditions,followingthe analysisof[6]performedfor scalarfieldsinthe Breitenlohner-Freedman
mass range.
TheAdS/CFTdictionarystatesthatthe(renormalized)EuclideanRarita-Schwingeraction(viewedasafunctional
of the boundary fields α(0)) is the generating functional for correlation functions of the CFT operator dual to α(0),
i,+ i,+