Table Of ContentFIAN-TD-2008-17
arXiv: 0808.3945[hep-th]
CFT adapted gauge invariant formulation of arbitrary spin
fields in AdS and modified de Donder gauge
9
0
0
R.R. Metsaev∗
2
n
a
J
Departmentof TheoreticalPhysics,P.N. LebedevPhysicalInstitute,
2
Leninskyprospect53, Moscow119991,Russia
2
]
h
t
-
p
e
h
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3
v
Abstract
5
4
9 UsingPoincare´ parametrizationofAdSspace,westudytotallysymmetricarbitraryspinmass-
3
less fields in AdS space of dimension greater than or equal to four. CFT adapted gauge invariant
.
8
formulation for such fields is developed. Gauge symmetries are realized similarly to the ones of
0
8 Stueckelberg formulation of massive fields. We demonstrate that the curvature and radial coordi-
0
nate contributionsto the gauge transformationand Lagrangian of the AdS fields can be expressed
:
v
in terms of ladder operators. Realization of the global AdS symmetries in the conformal algebra
i
X basis is obtained. Modified de Donder gauge leading to simple gauge fixed Lagrangian is found.
r ThemodifieddeDondergaugeleadstodecoupledequationsofmotionwhichcaneasilybesolved
a
in terms of the Bessel function. Interrelations between our approach to the massless AdS fields
andtheStueckelberg approach to massivefields inflat spaceare discussed.
∗ E-mail:[email protected]
1
1 Introduction
FurtherprogressinunderstandingAdS/CFTcorrespondence[1]requires,amongotherthings,bet-
ter understandingof field dynamics in AdS space. Conjectured dualityof conformal SYM theory
and superstring theory in AdS × S5 has lead to intensive and in-depth study of various aspects
5
of AdS field dynamics. Although many interesting approaches to AdS fields are known in the
literature (for review see [2]-[4]), analysis of concrete dynamical aspects of such fields is still a
challengingprocedure. OneofwaystosimplifyanalysisoffieldandstringdynamicsinAdS space
is based on use of the Poincare´ parametrization of AdS space1. Use of the Poincare´ coordinates
simplifies analysis of many aspect of AdS field dynamics and therefore these coordinates have
extensivelybeen used for studying the AdS/CFT correspondence. In this paper we develop a for-
mulationwhichisbasedonconsideringofAdS fielddynamicsinthePoincare´ coordinates. Thisis
tosaythatusingthePoincare´ parametrizationofAdS spacewediscussmasslesstotallysymmetric
arbitrary spin-s, s ≥ 1, bosonic field propagating in AdS space of dimension d+ 1 ≥ 4. Our
d+1
resultscan besummarizedas follows.
i) Using the Poincare´ parametrization of AdS, we obtain gauge invariant Lagrangian for free
masslessarbitrary spinAdS field. TheLagrangian is explicitlyinvariantwith respect toboundary
Poincare´ symmetries,i.e.,manifestsymmetriesofourLagrangianareadaptedtomanifestsymme-
triesofboundaryCFT.WeshowthatallthecurvatureandradialcoordinatecontributionstoourLa-
grangianand gaugetransformationare entirelyexpressed intermsofladderoperators that depend
on radial coordinate and radial derivative. Besides this, our Lagrangian and gauge transformation
are similar to the ones of Stueckelberg formulation of massive field in flat d-dimensional space.
General structure of the Lagrangian we obtained is valid for any theory that respects Poincare´
symmetries. Varioustheoriesare distinguishedby appropriateladderoperators.
ii)WefindmodifieddeDondergaugethatleadstosimplegaugefixedLagrangian. Thesurprise
is that this gauge gives decoupled equations of motion2. Note that the standard de Donder gauge
leads to coupled equations of motion whose solutions for s ≥ 2 are not known in closed form so
far. In contrast to this, our modified de Donder gauge leads to simple decoupled equations which
are easily solved in terms of the Bessel function. Application of our approach to studying the
AdS/CFT correspondencemay befoundinRef.[10].
Motivationforourstudyofhigher-spinAdS fieldsinPoincare´parametrizationwhichisbeyond
thescopeofthispapermay befound at theend ofSection 5.
2 Lagrangian and gauge symmetries
We begin with discussion of field content of our approach. In Ref.[11], the massless spin-s field
propagating in AdS space is described by double-traceless so(d,1) algebra totally symmetric
d+1
tensorfield ΦA1...As.3 This tensorfield can bedecomposedin scalar, vector, and totallysymmetric
1 StudyingAdS5×S5 superstringaction[5]inPoincare´ parametrizationmaybefoundinRef.[6]. Recentinter-
esting application of Poincare´ coordinates to studying AdS5 ×S5 string T−duality may be found in [7] (see also
[8]).
2 Our modified de Donder gauge seems to be unique first-derivativegauge that leads to decoupledequationsof
motion. Light-conegauge[9]alsoleadstodecoupledequationsofmotion,butthelight-conegaugebreaksboundary
Lorentzsymmetries.
3 A,B,C = 0,1,...,d and a,b,c = 0,1,...,d − 1 are the respective flat vector indices of the so(d,1) and
so(d − 1,1) algebras. In Poincare´ parametrization of AdSd+1 space, ds2 = (dxadxa + dzdz)/z2. We use the
conventions:∂a ≡∂/∂xa,∂z ≡∂/∂z.Vectorsofso(d,1)algebraaredecomposedasXA =(Xa,Xz).
2
tensorfields oftheso(d−1,1)algebra:
φa1...as′ , s′ = 0,1,...,s−1,s. (2.1)
s′
Thefieldsφa1...as′ withs′ > 3are double-traceless4
s′
φaabba5...as′ = 0, s′ = 4,5,...,s−1,s. (2.2)
s′
Thefieldsin(2.1)subjecttoconstraints(2.2)constituteafieldcontentofourapproach. Tosimplify
presentation we use a set of the creation operators αa, αz, and the respective set of annihilation
operators,α¯a,α¯z. Then,fields (2.1)can becollectedintoa ket-vector|φidefined by5
s
|φi ≡ (αz)s−s′|φs′i, (2.3)
s′=0
X
1
|φs′i ≡ s′! (s−s′)! αa1...αas′ φas′1...as′|0i. (2.4)
From (2.3),(2.4), we see that the ket-vector |φi is degree-s homogeneous polynomial in the oscil-
p
latorsαa,αz,whiletheket-vector|φs′iisdegree-s′ homogeneouspolynomialintheoscillatorsαa,
i.e.,theseket-vectorssatisfytherelations6
(Nα +Nz −s)|φi = 0, (Nα −s′)|φs′i = 0. (2.5)
In termsoftheket-vector|φi,double-tracelessnessconstraint(2.2)takes theform7
(α¯2)2|φi = 0. (2.6)
Actionand Lagrangianwe foundtaketheform
1
S = ddxdz L, L = hφ|E|φi, (2.7)
2
Z
hφ| ≡ (|φi)†, whereoperatorE is givenby
E = E +E +E , (2.8)
(2) (1) (0)
1 1 1 1
E ≡ 2−α∂α¯∂ + (α∂)2α¯2 + α2(α¯∂)2 − α22α¯2 − α2α∂α¯∂α¯2, (2.9)
(2) 2 2 2 4
E ≡ e¯ A+e A¯, (2.10)
(1) 1 1
E ≡ m +α2α¯2m +m¯ α2 +m α¯2, (2.11)
(0) 1 2 3 3
1
A ≡ α∂ −α2α¯∂ + α2α∂α¯2, (2.12)
4
1
A¯≡ α¯∂ −α∂α¯2 + α2α¯∂α¯2, (2.13)
4
4 Note that so(d − 1,1) tensorial components of the Fronsdal field ΦA1...As are not double-traceless. Using
appropriatetransformation(see(5.22))thosetensorialcomponentscanbetransformedtoourfieldsin(2.1).
5 Weuseoscillatorformulation[12]-[14]tohandlethemanyindicesappearingfortensorfields(seealso[15]). It
canalsobereformulatedasanalgebraactingonthesymmetric-spinorbundleonthemanifoldM [16].
6Throughoutthispaperweusethefollowingnotationforoperatorsconstructedouttheoscillatorsandderivatives:
Nα ≡αaα¯a,Nz ≡αzα¯z,α2 =αaαa,α¯2 =α¯aα¯a,2=∂a∂a,α∂ =αa∂a,α¯∂ =α¯a∂a.
7 Weadapttheformulationintermsofthedouble-tracelessgaugefields[11]. AdaptationofapproachinRef.[11]
tomassivefieldsmaybefoundinRefs.[17,18]. Discussionofvariousformulationsintermsofunconstrainedgauge
fieldsmaybefoundinRefs.[19]-[24]. Studyofotherinterestingapproacheswhichseemtobemostsuitableforthe
theoryofinteractingfieldsmaybefounde.g. inRefs.[25]-[27].
3
2s+d−5−2N
e = e ∂ + z , (2.14)
1 1,1 z
2z
(cid:16) (cid:17)
2s+d−5−2N
e¯ = ∂ − z e¯ , (2.15)
1 z 1,1
2z
(cid:16) (cid:17)
e = αzf , e¯ = fα¯z, (2.16)
1,1 1,1
2s+d−4−N 1/2
f ≡ ε z , ε = ±1, (2.17)
2s+d−4−2N
z
(cid:16) (cid:17)
2s+d−3−2N
m = e¯ e −2 ze e¯ , (2.18)
1 1 1 1 1
2s+d−4−2N
z
1 1 2s+d−2N
m = − e¯ e + z e e¯ , (2.19)
2 1 1 1 1
2 42s+d−4−2N
z
1 1
m = e e , m¯ = e¯ e¯ , (2.20)
3 1 1 3 1 1
2 2
and subscript n in E (2.8) tells us that E is degree-n homogeneous polynomial in the flat
(n) (n)
derivative ∂a. We note that gauge invariance requires ε2 = 1. Because ε depends on N , this
z
leavestwopossibilitiesε = ±1 atleast.
Thefollowingremarks areinorder.
i)Operator E (2.9) is thesymmetrizedFronsdal operator represented in terms of theoscillators.
(2)
This operator does not depend on the radial coordinate and derivative,z, ∂ , and it takes the same
z
formas theoneofmasslessfield in d-dimensionalflat space.
ii) Dependence of operator E (2.8) on the radial coordinate and derivative, z, ∂ , is entirely gov-
z
erned by the operators e and e¯ which are similar to ladder operators appearing in quantum me-
1 1
chanics. Sometimes,werefer totheoperators e and e¯ as ladderoperators8.
1 1
iii)RepresentationfortheLagrangianin(2.7)-(2.13)isuniversalandisvalidforarbitraryPoincare´
invarianttheory. VariousPoincare´ invarianttheoriesaredistinguishedbyladderoperatorsentering
theoperatorE. ThisistosaythattheoperatorsE ofmassiveandconformalfieldsinflatspacede-
pend ontheoscillatorsαa, α¯a and theflat derivative∂a in thesameway as theoperatorE ofAdS
fields(2.8). Inotherwords,theoperatorsE formasslessAdS fields,massiveandconformalfields
in flat space are distinguishedonly by the operators e and e¯ . For example, all that is required to
1 1
gettheoperatorE formassivespin-sfield ind-dimensionalflat space istomakethesubstitutions
e → mαzf , e¯ → −mfα¯z, (2.21)
1 1
where m is mass parameter of the massive field and f is given in (2.17). Note also that our field
content (2.1) is similar to the one of Stueckelberg formulation of massive field in d-dimensional
space[17]. Expressionsfore , e¯ appropriateforconformalfields maybefound inRefs.[31].
1 1
Gauge symmetries. We now discuss gauge symmetries of Lagrangian in (2.7). To this end we
introducethefollowingset ofgaugetransformationparameters:
ξa1...as′ , s′ = 0,1,...,s−1. (2.22)
s′
8InterestingapplicationofotherladderoperatorstostudyingAdS/QCDcorrespondencemaybefoundin[28]. We
believethatourapproachwillalsobeusefulforbetterunderstandingofvariousaspectsofAdS/QCDcorrespondence
whicharediscussede.g. in[28]-[30].
4
The gauge parameters ξ , ξa, and ξa1...as′, s′ ≥ 2 in (2.22), are the respective scalar, vector, and
0 1 s′
rank-s′ totally symmetric tensor fields of the so(d − 1,1) algebra. The gauge parameters ξa1...as′
s′
withs′ ≥ 2 aresubjectedto thetracelessness constraint
ξaaa3...as′ = 0, s′ ≥ 2. (2.23)
s′
Wenow,as usually,collect gaugetransformationparameters inket-vector|ξidefined by
s−1
|ξi ≡ (αz)s−1−s′|ξs′i, (2.24)
s′=0
X
1
|ξs′i ≡ s′! (s−1−s′)! αa1...αas′ ξsa′1...as′|0i. (2.25)
Theket-vectors|ξi,|ξs′isatisfytheaplgebraicconstraints
(Nα +Nz −s+1)|ξi = 0, (Nα −s′)|ξs′i = 0, (2.26)
which tellus that |ξiis adegree-(s−1)homogeneouspolynomialin theoscillatorsαa, αz, while
|ξs′i is degree-s′ homogeneous polynomial in the oscillators αa. In terms of the ket-vector |ξi,
tracelessnessconstraint(2.23)takestheform
α¯2|ξi = 0. (2.27)
Gaugetransformationcan entirely bewritten in terms of|φi and |ξi. We find the followinggauge
transformation:
1
δ|φi = (α∂ −e − α2e¯ )|ξi, (2.28)
1 1
2s+d−6−2N
z
where e , e¯ are given in (2.14),(2.15). From (2.28), we see that the flat derivative∂a enters only
1 1
in α∂-term in (2.28), while the radial coordinate and derivative, z, ∂ , enter only in the operators
z
e , e¯ . Thus, all radial coordinate and derivative contributions to gauge transformation (2.28) are
1 1
entirelyexpressedin termsoftheladderoperators e and e¯ 9.
1 1
Wefinish thisSection withthefollowingremark. Introducingnewmass-likeoperator
2s+d−2−2N
M2 ≡ −e¯ e + ze e¯ , (2.29)
1 1 1 1
2s+d−4−2N
z
andusingexplicitexpressionsforoperators e and e¯ (2.14),(2.15)wefind
1 1
1 1 d−4
M2 = −∂2 + (ν2 − ), ν ≡ s+ −N . (2.30)
z z2 4 2 z
WemakesurethattheoperatorsM2, e e¯ satisfythefollowingcommutators:
1 1
[e ,M2] = 0, [e¯ ,M2] = 0. (2.31)
1 1
Because the operators e , e¯ enter gaugetransformation (2.28), relations (2.31) can be considered
1 1
as requirement for gauge invariance of the operator M2. Therefore, M2 in (2.29) can be consid-
ered as a definition of gauge invariant mass operator. We note that making substitutions (2.21)
in (2.29) gives M2 = m2. Thus, we see that our definition of mass operator M2 (2.29) gives
desired result for massive field in flat space and provides interesting generalization of notion of
massoperatortothecaseofmasslessAdS field (2.30).
9Makingsubstitutions(2.21)in(2.8)and(2.28)onecanmakesurethatourLagrangianandgaugetransformation
matchwiththoseofflatlimitofAdS massivefieldtheoryinRef.[17].
5
3 Global so(d,2) symmetries
Relativistic symmetries of AdS space are described by the so(d,2) algebra. In our approach,
d+1
the massless spin-s AdS field is described by the set of the so(d − 1,1) algebra fields (2.1).
d+1
Therefore it is reasonable to represent the so(d,2)algebra so that to respect manifest so(d−1,1)
symmetries. ForapplicationtotheAdS/CFTcorrespondence,mostconvenientformoftheso(d,2)
algebra that respects the manifest so(d − 1,1) symmetries is provided by nomenclature of the
conformal algebra. This is to say that the so(d,2) algebra consists of translation generators Pa,
conformalboostgeneratorsKa,dilatationgeneratorD,andgeneratorsJab whichspanso(d−1,1)
algebra. Weusethefollowingnormalizationfor commutatorsoftheso(d,2)algebragenerators:
[D,Pa] = −Pa, [Pa,Jbc] = ηabPc −ηacPb, (3.1)
[D,Ka] = Ka, [Ka,Jbc] = ηabKc −ηacKb, (3.2)
[Pa,Kb] = ηabD −Jab, (3.3)
[Jab,Jce] = ηbcJae +3 terms. (3.4)
Requiring so(d,2) symmetries implies that the action is invariant with respect to transformation
δ |φi = Gˆ|φi, where the realization of so(d,2) algebra generators Gˆ in terms of differential
ˆ
G
operatorstakes theform
Pa = ∂a, Jab = xa∂b −xb∂a +Mab, (3.5)
d−1
D = x∂ +∆, ∆ ≡ z∂ + , (3.6)
z
2
1
Ka = − x2∂a +xaD +Mabxb +Ra, (3.7)
2
x∂ ≡ xa∂a, x2 ≡ xaxa. In (3.5),(3.7), Mab is spin operator of the so(d − 1,1) algebra. Com-
mutation relations for Mab and representation of Mab on space of ket-vector |φi (2.3) take the
form
[Mab,Mce] = ηbcMae +3 terms, Mab = αaα¯b −αbα¯a. (3.8)
OperatorRa appearing inKa (3.7)isgivenby
1
Ra = −zCae¯ +ze α¯a − z2∂a, (3.9)
1,1 1,1
2
1
Cea ≡ αa −α2 α¯a, (3.10)
2N +d−2
α
where e , e¯ are given in (2.16). Weesee that realization of Poincare´ symmetries on bulk AdS
1,1 1,1
fields (3.5) coincide with realization of Poincare´ symmetries on boundary CFT operators. Note
that realization of D- and Ka-symmetries on bulk AdS fields (3.6),(3.7) coincides, by module of
contributions of operators ∆ and Ra, with the realization of D- and Ka-symmetries on boundary
CFT operators. Realizations of the so(d,2)algebra on bulk AdS fields and boundary CFT opera-
torsaredistinguishedby ∆and Ra. Therealizationoftheso(d,2)symmetriesonbulkAdS fields
givenin(3.5)-(3.7)turnsout tobeveryconvenientforstudyingAdS/CFT correspondence[10].
6
4 Modified de Donder gauge
We begin with discussion of gauge-fixing procedure at the level of Lagrangian (2.7). We find that
useofthefollowingmodifieddeDondergauge-fixingterm
1
L = hφ|E |φi, E ≡ CC¯, (4.1)
g.fix g.fix g.fix
2
1 1
C ≡ α∂ − α2α¯∂ −e Π[1,2] + e¯ α2, (4.2)
1 1
2 2
1 1
C¯ ≡ α¯∂ − α∂α¯2 + e α¯2 −e¯ Π[1,2], (4.3)
1 1
2 2
1
Π[1,2] ≡ 1−α2 α¯2, (4.4)
2(2N +d)
α
leadsto thesurprisinglysimplegaugefixed Lagrangian L :
total
L ≡ L+L , (4.5)
total g.fix
1
L = hφ|E |φi, (4.6)
total total
2
1
E = (1− α2α¯2)(2−M2), (4.7)
total
4
where M2 is given in (2.30).10 We note that our gauge-fixing term (4.1) respects the Poincare´
and dilatationsymmetriesbut breaks theconformalboostKa-symmetries,i.e., thesimpleformof
gaugefixed Lagrangian(4.5)isachievedat thecostoftheKa-symmetries.
We now discuss gauge-fixing procedure at the level of equations of motion. To this end we
notethat gaugeinvariantLagrangian (2.7)leadsto thefollowingequationsofmotion:11
E +Π[2,3](E +E ) |φi = 0, (4.8)
(2) (1) (0)
(cid:16) (cid:17)
1
Π[2,3] ≡ 1−(α2)2 (α¯2)2. (4.9)
8(2N +d)(2N +d+2)
α α
Theseequationscan berepresented as
(E +E +E )|φi = 0, (4.10)
(2) (1) (0)
1
E ≡ 2−α∂α¯∂ + (α∂)2α¯2, (4.11)
(2) 2
2
E ≡ e (α¯∂ −α∂α¯ )
(1) 1
1 1
+ e¯ α∂ +α2 α¯∂ −α2α∂ α¯2 , (4.12)
1
2N +d−2 2N +d
α α
(cid:16) (cid:17)
1 2
E = m −(m +4m )α2 α¯2 +m α¯2 −m¯ α2 Π[1,2]. (4.13)
(0) 1 1 2 2(2N +d−2) 3 3 2N +d−2
α α
10 Makingsubstitutions(2.21)in(4.2),(4.3)givesgaugefixedLagrangianformassivefieldoftheform(4.5)-(4.7)
withM2 =m2.
11 AppearanceoftheprojectorΠ[2,3] inequationsofmotion(4.8)isrelatedtothefactthattheoperatorsE ,E ,
(1) (0)
incontrasttothesymmetrizedFronsdaloperatorE ,donotrespectdoubletracelessnessconstraint(2.6). Notethat
(2)
theket-vectorsE |φi,E |φiaretriple-traceless,(α¯2)3E |φi=0,(α¯2)3E |φi=0.
(1) (0) (1) (0)
7
ModifieddeDondergaugeconditionisthen defined tobe
C¯|φi = 0, (4.14)
where the operator C¯ is given in (4.3). Because of double-tracelessness of |φi (2.6), operator C¯
(4.3) satisfies the relation α¯2C¯|φi = 0, i.e., gauge condition (4.14) respects constraint for gauge
transformation parameter |ξi, (2.27). Using the modified de Donder gauge condition in gauge
invariantequationsofmotion(4.10)leadsto thefollowinggaugefixedequationsofmotion:
(2−M2)|φi = 0, (4.15)
whereM2 isdefined in(2.30). In terms offields(2.1), equation(4.15)can berepresented as
1 1 d−4
2+∂z2 − z2(νs2′ − 4) φas′1...as′ = 0, νs′ ≡ s′ + 2 , (4.16)
(cid:16) (cid:17)
s′ = 0,1,...,s. Thus, our modified de Donder gauge condition (4.14) leads to decoupled equa-
tions of motion (4.16) which can easily be solved in terms of the Bessel function 12. For spin-1
field, gaugecondition(4.14), foundin [9], turnsouttobeamodificationoftheLorentzgauge.
We note that equations of motion (4.15) have on-shell leftover gauge symmetries. These on-
shell leftovergauge symmetriescan simplybe obtainedfrom generic gaugesymmetries(2.28) by
thesubstituting|ξi → |ξ i, wherethe|ξ i satisfiesthefollowingequationsofmotion:
lfov lfov
(2−M2)|ξ i = 0. (4.17)
lfov
5 Comparison of standard and modified de Donder gauges
Ourapproachtothemasslessspin-sfieldinAdS isbasedonuseofdouble-tracelessso(d−1,1)
d+1
algebrafields (2.1). Oneof popularapproaches to the masslessspin-s field in AdS is based on
d+1
useofdouble-tracelessso(d,1)algebrafieldΦA1...As [11]. TheaimofthisSectionistwofold. First
weexplainhowourmodifieddeDondergaugeisrepresentedintermsofthecommonlyusedfield
ΦA1...As. Also,wecomparethemodifieddeDondergaugeandcommonlyusedstandarddeDonder
gauge13. Second weshowexplicitlyhowourfields (2.1) arerelated tothefield ΦA1...As.
We begin with discussion of modified de Donder gauge-fixing procedure at the level of La-
grangian. First we present gaugeinvariant Lagrangian for the field ΦA1...As. To simplifypresenta-
tionweintroduce,as before, thefollowingket-vector
1
|Φi = ΦA1...AsαA1...αAs|0i, (5.1)
s!
(α¯2)2|Φi = 0, (5.2)
α2 ≡ αAαA, α¯2 ≡ α¯Aα¯A, (5.3)
where (5.2) tells us that the ΦA1...As is double-traceless, and the scalar products like αAαA are
decomposed as αAαA = αaαa + αzαz. In terms of |Φi, gauge invariant Lagrangian takes the
12InterestingmethodofsolvingAdS fieldequationsofmotionwhichisbasedonstaralgebraproductsinauxiliary
spinorvariablesisdiscussedinRef.[32].
13RecentapplicationsofthestandarddeDondergaugetothevariousproblemsofhigher-spinfieldsmaybefound
inRefs.[33,34].
8
form14
1 1
L = ehΦ|(1− α2α¯2)E|Φi, (5.4)
2 4
1
E ≡ 2 −αDα¯D+ (αD)2α¯2 −s(s+d−5)+2d−4−α2α¯2, (5.5)
AdS
2
2 ≡ DADA +ωAABDB, α¯D ≡ α¯ADA, αD ≡ αADA, (5.6)
AdS
where e = deteA, eA stands for vielbein of AdS space, and DA are covariant derivatives (for
µ µ d+1
detailsofnotation,seeAppendix). Lagrangian (5.4)can berepresented as
1
L = ehΦ|E|Φi, (5.7)
2
1 1 1 1
E ≡ 2 −αDα¯D+ (αD)2α¯2 + α2(α¯D)2 − α22 α¯2 − α2αDα¯Dα¯2
AdS AdS
2 2 2 4
1
− s(s+d−5)+2d−4+ (s(s+d−3)−d)α2α¯2. (5.8)
2
We now ready to discuss the modified de Donder gauge. To make our study more useful we
discuss both the modified and standard de Donder gauges. Note that our formulas for standard
de Donder gauge are valid for arbitrary parametrization of AdS, while the ones for modified de
Dondergaugeareadapted tothePoincare´ parametrization. Gauge-fixingterm isdefined to be
1
L = ehΦ|E |Φi, (5.9)
g.fix g.fix
2
where operator E corresponding to the standard de Donder gauge fixing and the modified de
g.fix
Dondergaugefixingis givenby
C C¯ , standardgauge;
stand stand
E = (5.10)
g.fix ( C C¯ , modifiedgauge,
mod mod
andwe usethenotation
1 1
C ≡ αD− α2α¯D, C¯ ≡ α¯D− αDα¯2, (5.11)
stand stand
2 2
C ≡ C −2Cz , C¯ ≡ C¯ +2C¯z , (5.12)
mod stand ⊥ mod stand ⊥
1 1
Cz ≡ αz − α2α¯z, C¯z ≡ α¯z − αzα¯2. (5.13)
⊥ 2 ⊥ 2
Wenowmakesurethat thegaugefixedLagrangian L takes theform
total
L ≡ L+L , (5.14)
total g.fix
1 1
L = ehΦ|(1− α2α¯2)E |Φi, (5.15)
total total
2 4
14 Since Ref.[11], various approaches to massless totally symmetric AdS fields were developed in the literature
(seee.g. [12,9,35,16]). WeusesetupdiscussedinRef.[9]. FormulasinRef.[11]areadaptedtoAdS4 withmostly
negativemetrictensor,whileourformulasareadaptedtoAdSd+1withmostlypositivemetrictensor. Takingthisinto
accountandpluggingd = 3 in (5.5) we make sure that ouroperatorE matcheswith the operatorL0 in Eq.(2.7)in
Ref.[11].
9
2 −s(s+d−5)+2d−4−α2α¯2, standardgauge;
AdS
E =
total (20AdS −s(s+d−4)+2d−4−α2α¯z2 +(2s+d−5)Nz, modifiedgauge,
(5.16)
where 2 ≡ z2(2+∂2)+(1−d)z∂ . Alternatively, the operator E corresponding to the
0AdS z z total
modifieddeDondergaugein(5.16)can berepresented as
d2
E = 2 −α2α¯zα¯z −ν2 + , (5.17)
total 0AdS
4
whereν is givenin(2.30).
We proceed with discussion of gauge-fixing procedure at the level of equations of motion. To
thisendwenotethatgaugeinvariantLagrangian (5.4) leadsto thefollowingequationsofmotion:
E|Φi = 0. (5.18)
Wenowdefinethestandardand modifieddeDondergaugeconditionsas
C¯ |Φi = 0, standarddeDondergauge; (5.19)
stand
C¯ |Φi = 0, modifieddeDondergauge, (5.20)
mod
where C¯ , C¯ are given in (5.11),(5.12). Using (5.19),(5.20) in (5.18) we get gauge fixed
stand mod
equationsofmotion
E |Φi = 0, (5.21)
total
where E is given in (5.16). We note that, because of Cz - and C¯z -terms, the modified de
total ⊥ ⊥
Donder gauge breaks some of the so(d,2) symmetries. In the conformal algebra nomenclature,
thesebroken symmetriescorrespond tobroken conformalboostKa-symmetries.
FromE (5.16),weseethat,becauseofα2α¯zα¯z-term,themodifieddeDondergaugefor|Φi
total
doesnotleadtodecoupledequationsfortheket-vector|Φiwhen15 s ≥ 2. Itturnsoutthatinorder
toobtain decoupledequationsofmotionweshouldintroduceourset offields in(2.1). Weremind
that|Φiisadouble-tracelessfield(5.2)oftheso(d,1)algebra,while|φidescribesdouble-traceless
fields(2.6)oftheso(d−1,1)algebra. Thisistosaythattogetdecoupledequationsofmotionwe
have to make transformation from the so(d,1) ket-vector |Φi to so(d − 1,1) ket-vector |φi. We
find thefollowingtransformationfromtheket-vector|Φi toourket-vector|φi:
|φi = z1−2dNΠφΦ|Φi, (5.22)
1 2N +d
ΠφΦ ≡ Π[1] +α2 Π[1](α¯2 + α α¯zα¯z), (5.23)
α 2(2N +d) α 2N +d−2
α α
Π[1] ≡ Π[1](α,0,N ,α¯,0,d), (5.24)
α α
2NzΓ(N +N + d−3)Γ(2N +d−3) 1/2
N ≡ α z 2 α , (5.25)
Γ(N + d−3)Γ(2N +N +d−3)
α 2 α z
(cid:16) (cid:17)
N = αaα¯a, N = αzα¯z, (5.26)
α z
15Forspin-1field,gaugecondition(5.20)andthecorrespondingdecoupledequationsofmotionwerefoundin[9].
10
