Anomalous conformal currents, shadow fields and massive AdS fields

0.36 MB·English

by  R. R. Metsaev

FIAN-TD-2011-12;arXiv:1110.3749[hep-th]
Anomalousconformal currents, shadow fields andmassiveAdS fields
R.R. Metsaev∗
DepartmentofTheoreticalPhysics,P.N.LebedevPhysicalInstitute,Leninskyprospect53,Moscow119991,Russia
IntheframeworkofgaugeinvariantapproachinvolvingStueckelbergandauxiliaryfields,totallysymmetric
arbitraryspin anomalous conformal current and shadow fieldinflat space-time of dimension greater than or
equal to four arestudied. Gauge invariant differential constraintsfor such anomalous conformal current and
shadow fieldandrealization of global conformal symmetries areobtained. Gauge invariant two-point vertex
of the arbitrary spin anomalous shadow field is also obtained. In Stueckelberg gauge frame, the two-point
gaugeinvariantvertexbecomesthestandardtwo-pointvertexofCFT.Light-conegaugetwo-pointvertexofthe
arbitraryspinanomalousshadowfieldisderived. TheAdS/CFTcorrespondenceforarbitraryspinanomalous
conformal current andshadow fieldandtherespectivenormalizableandnon-normalizable modesof massive
arbitraryspinAdSfieldisstudied. TheAdSfieldisconsideredinmodifieddeDondergaugewhichsimplifies
2
considerablythestudyofAdS/CFTcorrespondence. Weshowthaton-shellleftovergaugesymmetriesofbulk
1
0 massivefieldarerelatedtogaugesymmetriesofboundaryanomalousconformalcurrentandshadowfield,while
2 themodifieddeDondergaugeconditionforbulkmassivefieldisrelatedtodifferentialconstraintsforboundary
anomalousconformalcurrentandshadowfield.
n
a
J PACSnumbers:11.25.Tq,11.15.Kc,11.40.Dw

0
3
I. INTRODUCTION AdS/CFT correspondence, spin-s anomalousconformalcur-

] rentandshadowfieldarerelatedtospin-smassiveAdSfield.
h
t We start with a brief recall of some basic notionsof CFT. Massivetotallysymmetricspin-sAdSfieldswithevens 4
- ≥
p Inspace-timeofdimensiond 4,fieldsofCFTcanbesepa- form leading Regge trajectory of AdS string theory. There-
e ≥ fore, extension of our approach to the case of arbitrary s is
ratedintotwogroups: conformalcurrentsandshadowfields.
h
important.Ourapproachtoanomalousconformalcurrentand
[ ThisistosaythatfieldhavingLorentzalgebraspins,s 1,
≥ shadowfieldissummarizedasfollows.
and conformal dimension ∆ = s + d 2 is referred to as
3
− i)Startingwiththefieldcontentofanomalousconformalcur-
v conformalcurrentwithcanonicaldimension,whilefieldhav-
9 ingLorentzalgebraspin s, s 1, andconformaldimension rent (and anomalous shadow field) in the standard CFT, we
4 ≥ introduce Stueckelberg fields and auxiliary fields. In other
∆>s+d 2isreferredtoasanomalousconformalcurrent.
7 − words,weextendspaceoffieldsenteringthestandardCFT.
3 Accordingly,fieldhavingLorentzalgebraspins,s 1,and
. conformaldimension∆=2 sisreferredtoasshad≥owfield ii) On the extended space of fields entering our approach,
0
1 withcanonicaldimension1,w−hilefieldhavingLorentzalgebra we introduce differential constraints, gauge transformations,
1 andconformalalgebratransformations. Thedifferentialcon-
spins,s 1,andconformaldimension∆<2 sisreferred
1 ≥ − straintsarerequiredtobeinvariantunderthegaugetransfor-
toasanomalousshadowfield.
:
v mationsandtheconformalalgebratransformations.
In Refs.[3, 4], we developed the gauge invariant formu-
i
X iii) Thegaugesymmetriesandthe differentialconstraintsal-
lation of the conformal currents and shadow fields having
lowustomatchourapproachandthestandardCFT.Namely,
r
the canonical conformal dimensions. We recall that, in the
a by imposing gauge conditions to exclude the Stueckelberg
frameworkofAdS/CFTcorrespondence,suchconformalcur-
fields and by solving differential constraints to exclude the
rentsandshadowfieldsarerelatedtomasslessAdSfields. In
auxiliaryfields, we obtainformulationofanomalousconfor-
Ref.[5], weextendedourapproachtothecase oflow spin-s,
malcurrentandshadowfieldinthestandardCFT.
s = 1,2, anomalous conformal currents and shadow fields.
The purpose of this paper is to develop gauge invariant ap- Besides the gauge invariant approach, we discuss the
proach to bosonic totally symmetric arbitrary spin-s anoma- anomalous conformal current and shadow field by using
lousconformalcurrentandshadowfield. Intheframeworkof Stueckelberggaugeandlight-conegaugeconditions.Reasons
fordiscussingthesetwogaugeconditionsareasfollows.
i) The Stueckelberggaugereducesour approachto the stan-
∗Electronicaddress:[email protected] dardformulationofCFT.ThisistosaythatuseoftheStueck-
1 Shadowfieldshavingthecanonicaldimensionareusedtobuildconformal elberg gauge allows us to demonstrate how the standard ap-
invariantequationsofmotionandLagrangianformulationsforconformal
proach to anomalous conformal current and shadow field is
fieldsinRefs.[1]. Interestingdiscussionofshadowfielddualitiesmaybe
foundinRef.[2]. connectedwithourgaugeinvariantapproach.
2
ii)StudyingofCFTinthelight-conegaugeframeismotivated iv) The modified de Donder gauge for massive bulk field is
bytheconjectureddualityofthesupersymmetricYang-Mills related to the differentialconstraintfor boundaryanomalous
theory and AdS superstring theory [6]. We expect, by anal- conformalcurrent(oranomalousshadowfield).
ogywithflatspace,thataquantizationofthetypeIIBGreen- v) On-shell leftover gauge symmetries of massive bulk field
Schwarz AdS superstring [7] will be straightforwardonly in are related to the gauge symmetries of boundaryanomalous
the light-cone gauge [8–10]. Therefore we think that, from conformalcurrent(oranomalousshadowfield).
thestringyperspectiveofAdS/CFTcorrespondence,thelight- Ourpaperisorganizedasfollows.
cone approach to CFT deserves to be understood better. In InSec. II,wesummarizeournotationandconventions.
thisrespect,wenotethatourapproachprovidesquickaccess SectionIIIisdevotedtogaugeinvariantformulationofarbi-
tothelight-conegaugeformulationofCFT.Thisimpliesthat traryspin-sanomalousconformalcurrent. We discussgauge
our approach gives easy access to the studying of AdS/CFT symmetriesandrealizationofglobalconformalalgebrassym-
correspondenceinlight-conegauge. Thisseemstobeimpor- metriesonspaceofgaugefieldsweuseforthedescriptionof
tantforfutureapplicationofourapproachtothestudyingof the anomalous conformalcurrent. We demonstrate how our
string/gaugetheoryduality. gaugeinvariantapproachisrelatedtothestandardCFT.Also,
We use our gauge invariant formulation of CFT for the usingourapproach,weobtainlight-conegaugedescriptionof
studying AdS/CFT correspondence between arbitrary spin thearbitraryspin-sanomalousconformalcurrent.
massiveAdSfieldandthecorrespondingarbitraryspinbound- In Sec. IV, we extend results in Sec. III to the case of
aryanomalousconformalcurrentandshadowfield. Namely, arbitraryspin-sanomalousshadowfield. Also,wefindgauge
we show that non-normalizable modes of arbitrary spin-s invarianttwo-pointvertexforthe arbitraryspin-sanomalous
massive AdS field are related to arbitrary spin-s anomalous shadowfield.Wediscussthetwo-pointvertexinStueckelberg
shadow field, while normalizable modes of arbitrary spin-s gaugeframeandinlight-conegaugeframe.
massive AdS field are related to arbitrary spin-s anomalous In Sec. V, we discuss the two-point current-shadow field
conformal current. We recall that, in earlier literature, the interactionvertex.
AdS/CFT correspondence between non-normalizable modes InSec. VI,wereviewtheCFTadaptedgaugeinvariantap-
of massivespin-1andspin-2AdSfields andthe correspond- proachtomassivearbitraryspinAdSfield.Becausetheuseof
ing spin-1 and spin-2 anomalous shadow fields was studied themodifieddeDondergaugemakesourstudyofAdS/CFT
in Refs.[11, 12]. The AdS/CFT correspondence for spin-s correspondenceforarbitraryspinfieldssimilartotheonefor
massive AdS field with s > 2 and the correspondingspin-s scalar field, we briefly review the AdS/CFT correspondence
anomalousconformalcurrentand shadow field has not been forthescalarfield.
consideredintheearlierliterature.OurtreatmentofAdS/CFT Section VII is devoted to the study of AdS/CFT corre-
correspondenceissummarizedasfollows. spondence between normalizable modes of massive spin-s
i) We exploit the CFT adapted gauge invariant approach to AdS field and spin-s anomalous conformal current, while,
massive AdS fields and modified de Donder gauge obtained inSec.VIII, westudythe AdS/CFT correspondencebetween
inRefs.[13,14]. ThemodifieddeDondergaugeleadstothe non-normalizable modes of massive spin-s AdS field and
simple decoupledbulk equationsof motion which are easily spin-sanomalousshadowfield.
solved. We show that the two-point gauge invariant vertex SectionIXsummarizesourconclusionsandsuggestsdirec-
of the arbitrary spin-s anomalous shadow field does indeed tionsforfutureresearch.
emergefrom massivearbitraryspin-sAdS field action when In Appendix, we present some details of matching of the
itisevaluatedonsolutionoftheDirichletproblem. Through- bulkandboundaryconformalboostsymmetries.
out this paper the AdS field action evaluated on the solution
oftheDirichletproblemisreferredtoaseffectiveaction.
ii) The numberof boundarygaugefields involvedin ourap- II. PRELIMINARIES
proach to the anomalous conformal current (or anomalous
shadow field) coincides with the number of gauge fields in- A. Notation
volvedintheCFT adaptedformulationofmassiveAdSfield
inRef.[14]. We use the following conventions. The Cartesian coor-
iii)Thenumberofgaugetransformationparametersinvolved dinates in d-dimensional flat space-time are denoted by xa,
in our approach to the anomalous conformal current (or whilederivativeswithrespecttoxa aredenotedby∂ ,∂
a a
≡
anomalousshadowfield)coincideswiththenumberofgauge ∂/∂xa. ThevectorindicesoftheLorentzalgebraso(d 1,1)
−
transformationparametersinvolvedintheCFTadaptedgauge takethevaluesa,b,c,e=0,1,...,d 1. Weusethemostly
−
invariantformulationofmassiveAdSfieldinRef.[14]. positiveflatmetrictensorηaband,tosimplifyourexpressions,
3
wedropη inscalarproducts:XaYa η XaYb. Creation α∂ αa∂a, α¯∂ α¯a∂a, (2.5)
ab ab
≡ ≡ ≡
operatorsαa, αz, ζ andthe respectiveannihilationoperators
α¯a, α¯z, ζ¯are referred to as oscillators.2 Commutation rela- α2 ≡αaαa, α¯2 ≡α¯aα¯a, (2.6)
tionsoftheoscillators,thevacuum|0i,andhermitianconju- Nα αaα¯a, Nz αzα¯z, Nζ ζζ¯, (2.7)
gationrulesaredefinedas ≡ ≡ ≡
Mab αaα¯b αbα¯a, (2.8)
[α¯a,αb]=ηab, [α¯z,αz]=1, [ζ¯,ζ]=1, (2.1) ≡ −
1
Π[1,2] 1 α2 α¯2, (2.9)
α¯a 0 =0, α¯z 0 =0, ζ¯0 =0, (2.2) ≡ − 2(2N +d)
α
| i | i | i
αa† =α¯a, αz† =α¯z, ζ† =ζ¯. (2.3) µ 1 1α2α¯2, (2.10)
≡ − 4
The oscillators αa, α¯a and αz, ζ, α¯z, ζ¯transform in the re-
1
spective vector and scalar representations of the Lorentz al- Ca αa α2 α¯a, (2.11)
≡ − 2N +d 2
α
gebra so(d 1,1). Throughoutthis paper we use operators −
− e 1
constructedoutofthederivatives,coordinates,andtheoscil- C¯a α¯a αaα¯2, (2.12)
⊥ ≡ − 2
lators,
✷ ∂a∂a, x∂ xa∂a, x2 xaxa, (2.4)
≡ ≡ ≡
2 Weuseoscillatorstohandlethemanyindicesappearingfortensorfields
(discussionofoscillatorformulationmaybefoundinRefs.[15,16].)
1/2
(s+ d−4 N )(κ s d−4 +N )(κ+1+N )
r 2 − ζ − − 2 ζ ζ ,
ζ ≡ 2(s+ d−24 −Nζ −Nz)(κ+Nζ −Nz)(κ+Nζ −Nz +1)!
(2.13)
1/2
(s+ d−4 N )(κ+s+ d−4 N )(κ 1 N )
r 2 − z 2 − z − − z ,
z ≡ 2(s+ d−24 −Nζ −Nz)(κ+Nζ −Nz)(κ+Nζ −Nz −1)!
where parameter κ appearing in (2.13) is defined below in (3.4). Throughout the paper the notation λ [n] implies that
2
∈
λ= n, n+2, n+4,...,n 4,n 2,n:
− − − − −
λ [n] = λ= n, n+2, n+4,...,n 4,n 2,n. (2.14)
2
∈ ⇒ − − − − −
Often,weusethefollowingsetofscalar,vector,andtotallysymmetrictensorfieldsoftheLorentzalgebraso(d 1,1):
−
φa1...as′ , s′ =0,1,...,s, λ [s s′] , (2.15)
λ ∈ − 2
whereφa1...as′ isrank-s′totallysymmetrictracefultensorfieldoftheLorentzalgebraso(d 1,1).Toillustratethefieldcontent
giveninλ(2.15),weuseshortcutφs′ forthefieldφa1...as′ andnotethatfieldsin(2.15)canb−erepresentedas
λ λ
φs
0
φs−1 φs−1
−1 1
... ... ... (2.16)
φ1 φ1 ... φ1 φ1
1−s 3−s s−3 s−1
φ0 φ0 ... φ0 φ0
−s 2−s s−2 s
Our conventionsfor light-cone frame are as follows. The where the coordinates in directions are defined as x± =
±
space-timecoordinatesare decomposedasxa = x+,x−,xi,
4
(xd−1 x0)/√2 and x+ is taken to be a light-cone time. is equal to zero, while, in the gauge invariant approach to
±
Vector indices of the so(d 2) algebra take values i,j = anomalousconformalcurrentandshadowfieldwedevelopin
−
1,...,d 2.Weusethefollowingconventionsforthederiva- this paper, the operator Ra is nontrivial. This is to say that,
−
tives: in the framework of our gauge invariantapproach, the com-
plete description of the conformal current and shadow field
∂i =∂ ∂/∂xi, ∂± =∂ ∂/∂x∓. (2.17)
i ≡ ∓ ≡ requires,amongotherthings,findingtheoperatorRa.
B. Globalconformalsymmetries
III. ARBITRARYSPINANOMALOUSCONFORMAL
CURRENT
In d-dimensional flat space-time, the conformal algebra
so(d,2)consistsoftranslationgeneratorsPa,dilatationgen-
A. Gaugeinvariantformulation
erator D, conformalboostgeneratorsKa, and generatorsof
the so(d 1,1) Lorentz algebra Jab. We use the following
Field content. To developgaugeinvariantformulationof
−
nontrivialcommutatorsoftheconformalalgebra:
arbitraryspin-sanomalousconformalcurrentinflatspaceof
[D,Pa]= Pa, [Pa,Jbc]=ηabPc ηacPb, dimensiond 4weusethefollowingfields:
≥
− −
[D,Ka]=Ka, [Ka,Jbc]=ηabKc ηacKb, φacu1r..,.λas′ , s′ =0,1,...,s, λ∈[s−s′]2. (3.1)
−
(2.18)
Wenotethat:
[Pa,Kb]=ηabD Jab,
− i) In (3.1), the fields φ and φa are the respective
cur,λ cur,λ
[Jab,Jce]=ηbcJae+3terms. scalarandvectorfieldsoftheLorentzalgebra,whilethefield
φa1...as′, s′ > 1, is rank-s′ totally symmetric tracefultensor
cur,λ
Letφdenotesanomalousconformalcurrent(oranomalous fieldofthe Lorentzalgebraso(d 1,1). Usingshortcutφs′
shadowfield)inthed-dimensionalflatspace-time. Underthe for the field φa1...as′, fields in (3.−1) can be representedas iλn
actionofconformalalgebra,theφtransformsas cur,λ
(2.16).
δGˆφ=Gˆφ, (2.19) ii) The tensor fields φacu1r..,.λas′ with s′ ≥ 4 satisfy the double-
tracelessnessconstraint
where the realization of the conformalalgebra generatorsGˆ
onspaceofφisgivenby φaabba5...as′ =0, s′ =4,5,...,s. (3.2)
cur,λ
Pa =∂a, (2.20)
iii) The fields φa1...as′ have the followingconformaldimen-
Jab =xa∂b xb∂a+Mab, (2.21) cur,λ
− sions:
D =x∂+∆, (2.22) d
∆(φa1...as′)= +κ+λ. (3.3)
cur,λ 2
Ka =Ka +Ra, (2.23)
∆,M
iv)IntheframeworkofAdS/CFTcorrespondence,κisrelated
1 tothemassparametermofspin-smassivefieldinAdS as
Ka x2∂a+xaD+Mabxb. (2.24) d+1
∆,M ≡−2
d 4 2
In relations (2.21)-(2.23), ∆ is an operator of conformaldi- κ m2+ s+ − . (3.4)
≡ 2
mension,whileMabisaspinoperatoroftheLorentzalgebra, r
(cid:16) (cid:17)
In order to obtain the gauge invariant description in an
[Mab,Mce]=ηbcMae+3terms. (2.25)
easy–to–useformwe use the oscillatorsandintroducea ket-
The spin operator of the Lorentz algebra is well known for vector φcur definedby
| i
arbitraryspinanomalousconformalcurrentandshadowfield
s
(see (2.8)). In general, operator Ra appearing in (2.23) de- φ = φs′ , (3.5)
| curi | curi
pends on the derivatives and does not depend on the space- s′=0
X
time coordinates.3 In the standard CFT, the operator Ra
thispaper,theoperatorRaisindependentofthederivatives. Dependence
of the operator Ra onthe derivatives appears in the ordinary-derivative
3 Forthe anomalous conformal currents andshadow fields considered in approachtoconformalfields(seeRefs.[17]-[19]).
5
φs′ ii)Thegaugetransformationparametersξa1...as′ withs′ 2
| curi cur,λ ≥
satisfythetracelessnessconstraint,
s−s′+λ s−s′−λ
= ζ 2 αz 2 αa1...αas′ φa1...as′ 0 . ξaaa3...as′ =0, s′ =2,3,...,s 1. (3.13)
λ∈[Xs−s′]2 s′! (s−s2′+λ)!(s−s2′−λ)! cur,λ | i iii) Thecugra,λuge transformation parameters ξ−a1...as′ have the
q cur,λ
conformaldimensions
From (3.5), we see that the ket-vector φ is degree-sho-
cur
| i
mogeneouspolynomialintheoscillatorsαa,αz,ζ,whilethe ∆(ξa1...as′)= d +κ+λ 1. (3.14)
ket-vector φs′ isdegree-s′homogeneouspolynomialinthe cur,λ 2 −
| curi
oscillatorsαa,i.e.,theseket-vectorssatisfytherelations Now, as usually, we collect the gauge transformation pa-
rametersinaket-vector ξ definedby
cur
| i
(N +N +N s)φ =0, (3.6)
α z ζ cur
− | i s−1
(Nα−s′)|φscu′ri=0. (3.7) |ξcuri=s′=0|ξcsu′ri, (3.15)
X
In terms of the ket-vector φcur , double-tracelessness con- ξs′
straint(3.2)takestheform4| i | curi
s−1−s′+λ s−1−s′−λ
(α¯2)2 φ =0. (3.8) = ζ 2 αz 2 αa1...αas′ ξa1...as′ 0 .
| curi λ∈[sX−1−s′]2 s′! (s−1−2s′+λ)!(s−1−2s′−λ)! cur,λ | i
Differentialconstraint. Wefindthefollowingdifferential q
Theket-vectors ξ , ξs′ satisfythealgebraicconstraints
constraintfortheanomalousconformalcurrent: | curi | curi
(N +N +N s+1)ξ =0, (3.16)
α z ζ cur
C¯ φ =0, (3.9) − | i
cur cur
| i (N s′)ξs′ =0, (3.17)
1 1 α− | curi
C¯ =α¯∂ α∂α¯2 e¯ Π[1,2]+ e α¯2, (3.10)
cur − 2 − 1,cur 2 1,cur whichtellusthat ξ isadegree-(s 1)homogeneouspoly-
cur
e1,cur =ζrζ✷+αzrz, nomialintheosci|llatoirsαa,αz,ζ,w−hiletheket-vector|ξcsu′ri
(3.11) isdegree-s′homogeneouspolynomialintheoscillatorsαa.In
e¯1,cur =−rζζ¯−rzα¯z✷, terms of the ket-vector |ξcuri, tracelessness constraint(3.13)
takestheform
wherethe operatorsΠ[1,2], r , r aredefinedin (2.9), (2.13).
ζ z α¯2 ξ =0. (3.18)
One can make sure that constraint (3.9) is invariant under | curi
gaugetransformationand conformalalgebratransformations Gauge transformation can entirely be written in terms of
whichwediscussbelow. φ and ξ . This is to say that gauge transformation
cur cur
| i | i
Gaugesymmetries. Wenowdiscussgaugesymmetriesof takestheform
the anomalous conformal current. To this end we introduce
δ φ =G ξ , (3.19)
thefollowinggaugetransformationparameters: | curi cur| curi
1
ξcau1r.,.λ.as′ , s′ =0,1,...,s−1, λ∈[s−1−s′]2. Gcur ≡α∂−e1,cur−α22Nα+d−2e¯1,cur, (3.20)
(3.12)
where e , e¯ are given in (3.11). As we have already
1,cur 1,cur
Wenotethat
said, constraint(3.9) is invariantundergaugetransformation
i) In (3.12), the gauge transformation parameters ξ and
cur,λ (3.19).
ξa aretherespectivescalarandvectorfieldsoftheLorentz
cur,λ Realizationof conformalalgebra symmetries. To com-
algebra, while the gauge transformation parameter ξa1...as′,
cur,λ pletethegaugeinvariantformulationofthespin-sanomalous
s′ >1,isrank-s′totallysymmetrictensorfieldoftheLorentz
conformalcurrentweproviderealizationoftheconformalal-
algebraso(d 1,1).
− gebrasymmetriesonspaceoftheket-vector φcur . Allthatis
| i
neededistofixtheoperatorsMab,∆,andRaandinsertthen
theseoperatorsinto(2.20)-(2.23). Realizationofthespinop-
4 Inthispaperweadapttheformulationintermsofthedoubletracelless eratorMab onket-vector φcur (3.5)isgivenin(2.8), while
| i
gaugefields [20]. Todevelop thegaugeinvariant approach onecanuse realizationoftheoperator∆,
unconstrainedgaugefieldsstudiedinRefs.[21].Discussionofothergauge
fieldswhichseemtobemostsuitable forthetheoryofinteracting fields d
∆ = +ν, ν κ+N N , (3.21)
maybefound,e.g.,in[22]. cur 2 ≡ ζ − z
6
can be read from (3.3). In the gauge invariant formulation, ( )s−s′ 2s−s′Γ(κ s d−4)Γ(κ+s s′) 1/2
X − − − 2 − .
finding the operator Ra providesthe real difficulty. We find cur≡(s s′)! Γ(κ s′ d−4)Γ(κ)
thefollowingrealizationoftheoperatorRaonspaceof φ : − (cid:16) − − 2 (cid:17)
| curi Relation (3.25) tells us that all fields φa1...as′ with s′ 2
cur,λ ≥
Ra = 2ζr (ν+1)α¯a C¯a becometraceless.Fromrelation(3.26),welearnthatthefields
cur − ζ − ⊥ φa1...as′ withλ=s s′becomeequaltozero,whilethefields
(cid:16) (cid:17) cur,λ 6 −
1 φa1...as′ withλ=s s′ands′ =0,1,...s 1areexpressed
− 2 νCa+α22N +d 2C¯⊥a rzα¯z, (3.22) incuter,rλmsoftherank-−stracelesstensorfieldφ−a1...as,
(cid:16) α − (cid:17) cur,0
wheretheoperatoresCa,C¯a aregivenin(2.11),(2.12),while φa1...as′ =0, for λ=s s′, (3.27)
⊥ cur,λ 6 −
theoperatorsr ,r aredefinedin(2.13).
ζ z
We derived the diefferential constraint, gauge transforma- φacu1r..,.sa−s′s′ = (s−s′)!Xcur∂b1...∂bs−s′φbc1u.r.,.0bs−s′a1...as′ .
tion,andtherealizationoftheoperatorRcaur bygeneralizing p (3.28)
ourresultsforthespin-sconformalcurrentwiththecanonical
We recall that, in the standard CFT, the spin-s anomalous
dimensionwhichwe obtainedin Ref.[3]. Itisworthwhileto
conformalcurrentis describedby the tracelessfield φa1...as.
notethatresultsinthissectioncanalsobeobtainedbyusing cur,0
Thus we see that making use of the gauge symmetry and
thetractorapproachinRefs.[23,24]5.Ourconstraint(3.9)and
differentialconstraint we reduce the field content of our ap-
gaugetransformation(3.19)canbematchedwiththeonesin
proachtotheoneinthestandardCFT.Inotherwords,useof
Ref.[23]byusingappropriatefieldredefinitions. Thebasisof
the gauge symmetry and differential constraint allows us to
thefieldsweuseinthispaperturnsouttobemoreconvenient
matchourapproachandthestandardformulationofthespin-
forthestudyofAdS/CFTcorrespondence.Tosummarize,our
s anomalous conformal current6. To summarize, our gauge
fields(3.1)canbewrittenasatractorrank-stensorfieldsub-
invariantapproachisequivalenttothestandardone.
jecttoaThomas-DdivergencetypeconstraintinRef.[23]. A
similarconstructionwasusedtodescribespin-smassivebulk
fieldinRef.[23]. Notethat,inourapproach,weuseourfields
C. Light-conegaugeframe
(3.1) for the discussion of spin-s anomalous conformal cur-
rent.
Wenowdiscussthespin-sanomalousconformalcurrentin
thelight-conegaugeframe. Tothisendwenotethat,forthe
anomalous conformalcurrent, the light-cone gauge frame is
B. Stueckelberggaugeframe
achieved through the use of differential constraint (3.9) and
light-cone gauge condition. Using gauge symmetry of the
We proceedwith discussion of the spin-s anomalouscon-
spin-s anomalous conformal current (3.19), we impose the
formalcurrentin the Stueckelberggaugeframe. To this end
light-conegaugeonthe φ ,
wenotethattheStueckelberggaugeframeisachievedthrough | curi
the use of differential constraint (3.9) and the Stueckelberg α¯+Π[1,2] φ =0, (3.29)
cur
| i
gaugecondition.From(3.19),weseethatafielddefinedby
whereΠ[1,2] isgivenin(2.9). Usinggauge(3.29)anddiffer-
α¯zΠ[1,2] φ (3.23) entialconstraint(3.9),wefind
cur
| i
α+
transformsas Stueckelbergfield. Thereforethis field can be φ =exp (α¯i∂i e¯ ) φl.c. , (3.30)
| curi −∂+ − 1cur | curi
gaugedawayviaStueckelberggaugefixing,
(cid:16) (cid:17)
α¯iα¯i φl.c. =0, (3.31)
α¯zΠ[1,2] φ =0, (3.24) | curi
cur
| i
where a light-cone ket-vector φl.c. is obtained from φ
where Π[1,2] is given in (2.9). Using gauge condition (3.24) (3.5)byequatingα+ =α− =|0,curi | curi
anddifferentialconstraint(3.9),wefindtherelations
φl.c. φ . (3.32)
α¯2 φs′ =0, (3.25) | curi≡| curi α+=α−=0
| curi (cid:12)
(cid:12)
φs′ =X ζs−s′(α¯∂)s−s′ φs , (3.26) (cid:12)
| curi cur | curi
6 Wenotethat,asinthestandardCFT,ourcurrentscanbeconsideredeither
ascompositeoperatorsorasfundamentalfielddegreesoffreedom.Atthe
group theoretical level, we study in this paper, this distinction does not
5 Inmathematicalliterature, interestingdiscussionofthetractorapproach matter. Methodsforbuildingconformalcurrents ascompositeoperators
maybefoundinRef.[25]. arediscussedinRefs.[26].
7
Weseethatweareleftwithlight-conefields From (4.4), we see that the ket-vector φ is degree-s ho-
sh
| i
mogeneouspolynomialintheoscillatorsαa,αz,ζ,whilethe
φic1u.r.,.λis′ , s′ =0,1,...,s, λ∈[s−s′]2, (3.33) ket-vector φs′ isdegree-s′ homogeneouspolynomialinthe
| shi
which are traceless tensor fields of so(d 2) algebra, oscillatorsαa,i.e.,theseket-vectorssatisfytherelations
−
φiii3...is′ = 0. Thesefieldsconstitutethefieldcontentofthe
cur,λ (N +N +N s)φ =0, (4.5)
α z ζ sh
light-conegaugeframe. Note that, in contrasttothe Stueck- − | i
elberg gauge frame, all fields φi1...is′ are not equal to zero. (N s′)φs′ =0. (4.6)
cur,λ α− | shi
Alsonotethat,incontrasttothegaugeinvariantapproach,the
In terms of the ket-vector φ , double-tracelessness con-
fields(3.33)arenotsubjecttoanydifferentialconstraint. | shi
straint(4.2)takestheform
(α¯2)2 φ =0. (4.7)
IV. ARBITRARYSPINANOMALOUSSHADOWFIELD | shi
Differentialconstraint. Wefindthefollowingdifferential
A. Gaugeinvariantformulation constraintfortheanomalousshadowfield:
Field content. To discuss gauge invariant formulation of C¯sh φsh =0, (4.8)
| i
arbitrary spin-s anomalous shadow field in flat space of di-
1 1
C¯ =α¯∂ α∂α¯2 e¯ Π[1,2]+ e α¯2, (4.9)
mensiond 4weusethefollowingfields: sh 1,sh 1,sh
− 2 − 2
≥
φash1,.λ..as′ , s′ =0,1,...,s, λ∈[s−s′]2. (4.1) e1,sh =ζrζ +αzrz✷,
(4.10)
Wenotethat: e¯ = r ζ¯✷ r α¯z,
1,sh ζ z
i)In(4.1),thefieldsφ andφa aretherespectivescalar − −
sh,λ sh,λ
and vector fields of the Lorentz algebra, while the field wheretheoperatorsΠ[1,2], rζ, rz aredefinedin(2.9), (2.13).
φa1...as′, s′ > 1, is rank-s′ totally symmetrictracefultensor One can make sure that constraint (4.8) is invariant under
sh,λ
field oftheLorentzalgebraso(d 1,1). Usingshortcutφs′ gaugetransformationand conformalalgebratransformations
for the field φa1...as′, fields in (4.−1) can be representedas iλn whichwediscussbelow.
sh,λ
Gaugesymmetries. Wenowdiscussgaugesymmetriesof
(2.16).
ii) The tensorfields φa1...as′ with s′ 4 satisfy the double- the anomalous shadow field. To this end we introduce the
sh,λ ≥ followinggaugetransformationparameters:
tracelessnessconstraint
φsaha,bλba5...as′ =0, s′ =4,5,...,s. (4.2) ξsah1,.λ..as′ , s′ =0,1,...,s−1, λ∈[s−1−(s4′].211.)
Wenotethat
iii) The fields φa1...as′ have the followingconformaldimen- i) In (4.11), the gauge transformation parameters ξ and
sh,λ sh,λ
sions: ξa aretherespectivescalarandvectorfieldsoftheLorentz
sh,λ
∆(φash1,.λ..as′)= d2 −κ+λ, (4.3) sal′g>eb1ra,,iswrahnilke-sth′etogtaalulygesytmramnseftorricmtaetniosonrpfiaerladmoefttehreξLsah1o,.rλ.e.anst′z,
algebraso(d 1,1).
iv)IntheframeworkofAdS/CFTcorrespondence,κisrelated −
ii)Thegaugetransformationparametersξa1...as′ withs′ 2
tothemassparametermofspin-smassivefieldinAdSd+1as sh,λ ≥
satisfythetracelessnessconstraint
in(3.4).
In order to obtain the gauge invariant description in an ξaaa3...as′ =0, s′ =2,3,...,s 1. (4.12)
sh,λ −
easy–to–useformwe use the oscillatorsandintroducea ket-
vector φsh definedby iii) The gauge transformation parameters ξsah1,.λ..as′ have the
| i conformaldimensions
s
|φshi=s′=0|φssh′i, (4.4) ∆(ξsah1,.λ..as′)= d2 −κ+λ−1. (4.13)
X
φs′ Now, as usually, we collect the gauge transformation pa-
| shi rametersinaket-vector ξ definedby
sh
s−s′−λ s−s′+λ | i
=λ∈[Xs−s′]2ζs′!2 (sα−zs2′+2λ)!α(sa−1s.2′.−.λα)a!s′ φash1,.λ..as′|0i. |ξshi=ss′−=10|ξssh′i, (4.14)
X
q
8
ξs′ to say that we find the following gauge invariant two-point
| shi
vertex:
s−1−s′−λ s−1−s′+λ
= ζ 2 αz 2 αa1...αas′ ξa1...as′ 0 . Γ= ddx ddx Γ , (4.22)
λ∈[sX−1−s′]2 s′! (s−1−2s′+λ)!(s−1−2s′−λ)! sh,λ | i Z 1 2 12
q 1 µf
Theket-vectors|ξshi,|ξssh′isatisfythealgebraicconstraints Γ12 = 2hφsh(x1)||x12|2νν+d|φsh(x2)i, (4.23)
(N +N +N s+1)ξ =0, (4.15) Γ(ν+ d)Γ(ν +1)
α z ζ − | shi fν = 4κ−νΓ(κ+2 d)Γ(κ+1), (4.24)
(N s′)ξs′ =0, (4.16) 2
α− | shi
ν =κ+N N , (4.25)
ζ z
−
whichtellusthat ξ isadegree-(s 1)homogeneouspoly-
sh
nomialintheosci|llatoirsαa,αz,ζ,w−hiletheket-vector|ξssh′i |x12|2 ≡xa12xa12, xa12 =xa1 −xa2, (4.26)
is degree-s′ homogeneous polynomial in the oscillators αa. where µ is given in (2.10). Vertex Γ (4.22) is invariant un-
Intermsoftheket-vector|ξshi,tracelessnessconstraint(4.12) dergaugetransformationoftheanomalousshadowfield|φshi
takestheform (4.18) provided this anomalous shadow field satisfies differ-
entialconstraint(4.8).Thevertexisobviouslyinvariantunder
α¯2 ξ =0. (4.17)
sh thePoincare´algebraanddilatationsymmetries.Wemakesure
| i
thatvertex(4.22)isinvariantundertheconformalboosttrans-
Gauge transformation can entirely be written in terms of
formations.
φ and ξ . Thisistosaythatgaugetransformationtakes
| shi | shi To illustrate structure of the vertex Γ12 we note that, in
theform terms of the tensor fields φa1...as′, vertex Γ (4.23) can be
sh,λ 12
representedas
δ φ =G ξ , (4.18)
sh sh sh
| i | i s
1 Γ = Γs′ , (4.27)
G =α∂ e α2 e¯ , (4.19) 12 12,λ
sh − 1,sh− 2Nα+d−2 1,sh sX′=0λ∈[Xs−s′]2
where e1,sh, e¯1,sh are givenin (4.10). Constraint(4.8) is in- Γs′ = wλ φa1...as′(x )φa1...as′(x )
variantundergaugetransformation(4.18). 12,λ 2s′!x 2κ−2λ+d sh,λ 1 sh,λ 2
12
Realizationof conformalalgebra symmetries. To com- | | (cid:16)
pletethegaugeinvariantformulationofthespin-sanomalous s′(s′ 1)
− φaaa3...as′(x )φbba3...as′(x ) , (4.28)
shadowfieldweproviderealizationoftheconformalalgebra − 4 sh,λ 1 sh,λ 2
(cid:17)
symmetries on space of the ket-vector φ . All that is re-
sh
| i
quiredistofixtheoperatorsMab,∆,andRa andinsertthen Γ(κ λ+ d)Γ(κ λ+1)
theseoperatorsinto(2.20)-(2.23). Realizationofthespinop- wλ ≡ 4λ−Γ(κ+2d)Γ(κ−+1) . (4.29)
erator Mab on ket-vector φ (4.4) is given in (2.8), while 2
sh
| i We note that the kernel of the vertex Γ is connected with
realizationoftheoperator∆,
a two-pointcorrelationfunctionof theanomalousconformal
d current. In the framework of our approach, the anomalous
∆ = ν, ν =κ+N N , (4.20)
sh 2 − ζ − z conformal current is described by gauge fields (3.1) subject
to differential constraints. In order to discuss the correla-
can be read from (4.3). Realization of the operator Ra on
tion function of the spin-s anomalous conformal current in
spaceof φ ,whichwefind,isgivenby
sh
| i aproperway,wecanimposeagaugeconditiononthegauge
fieldsgivenin(3.1). Recallthatwehaveconsideredthespin-
Ra = 2αzr (ν 1)α¯a+C¯a
sh z − ⊥ sanomalousconformalcurrentbyusingtheStueckelbergand
(cid:16) (cid:17)
1 light-conegaugeframes.Obviously,thetwo-pointcorrelation
+ 2 νCa α2 C¯a r ζ¯, (4.21)
− 2N +d 2 ⊥ ζ functionoftheanomalousconformalcurrentintheStueckel-
α
(cid:16) − (cid:17) berg and light-cone gauge frames is obtained from the two-
e
wheretheoperatorsCa,C¯a aregivenin(2.11),(2.12),while pointvertexΓtakenintherespectiveStueckelbergandlight-
⊥
theoperatorsr ,r aredefinedin(2.13). conegaugeframes. Tothisendweproceedbydiscussingthe
ζ z
Two-pointgaugeeinvariantvertex. We nowdiscusstwo- anomalous shadow field in the Stueckelberg and light-cone
point vertex for the spin-s anomalous shadow field. This is gaugeframes.
9
B. Stueckelberggaugeframe Γstand =s! φs (x )O φs (x ) , (4.37)
12 h sh 1 | 12| sh 2 i
We now discuss the spin-s anomalousshadow field in the s ( )n2n(αx )n(α¯x )n
Stueckelberg gauge frame. We note that the Stueckelberg O12 ≡ −n! x122κ+d+122n , (4.38)
12
gauge frame can be achieved through the use of differential nX=0 | |
constraint given in (4.8) and the Stueckelberg gauge condi-
2κ+2s+d 2
tion. From(4.18),weseethatafielddefinedby k − , (4.39)
s
≡ 2s!(2κ+d 2)
−
ζ¯Π[1,2] φ (4.30)
sh
| i where αx = αaxa , α¯x = α¯axa and Γstand in (4.36),
12 12 12 12 12
transformsas Stueckelbergfield. Thereforethis field can be (4.37)standsforthetwo-pointvertexofthespin-sanomalous
gaugedawayviaStueckelberggaugefixing, shadow field in the standard CFT. Relation (4.37) provides
oscillatorrepresentationfortheΓstand. Intermsofthetensor
ζ¯Π[1,2] φ =0. (4.31) 12
| shi field φa1...as, vertex Γstand (4.37) can be represented in the
sh,0 12
Using this gauge condition and differential constraint (4.8), commonlyusedform,
wefindthefollowingrelations:
Oa1b1...Oasbs
α¯2 φs′ =0, (4.32) Γs1t2and =φsah1,.0..as(x1) 12x 2κ+d12 φbsh1.,.0.bs(x2), (4.40)
| shi | 12|
φs′ =X αs−s′(α¯∂)s−s′ φs , (4.33) 2xa xb
| shi sh z | shi Oab ηab 12 12 . (4.41)
12 ≡ − x 2
( )s−s′ 2s−s′Γ(κ+d−2+s′)Γ(κ+1) 1/2 | 12|
X − 2 .
sh≡(s s′)! Γ(κ+s+d−2)Γ(κ s+1+s′) From(4.36),weseethatourgaugeinvariantvertexΓ12 con-
− (cid:16) 2 − (cid:17) sideredintheStueckelberggaugeframecoincides,uptonor-
Relation (4.32) tells us that all fields φash1,.λ..as′ with s′ ≥ 2 malization factor ks, with the two-point vertex in the stan-
becometraceless.Fromrelation(4.33),welearnthatthefields dardCFT.InsectionIIIB,wehavedemonstratedthat,inthe
φash1,.λ..as′ withλ6=s−s′becomeequaltozero,whilethefields Stueckelberg gauge frame, we are left with rank-s traceless
φash1,.λ..as′ withλ=s−s′ands′ =0,1,...,s−1areexpressed tensor field φcau1r..,.0as. Two-point correlation function of this
intermsoftherank-stracelesstensorfieldφsah1,.0..as, tensorfieldisdefinedbythekernelofvertexΓstand (4.40).
φa1...as′ =0, for λ=s s′, (4.34)
sh,λ 6 −
C. Light-conegaugeframe
φash1,.s..−ass′′ = (s−s′)!Xsh∂b1...∂bs−s′φbsh1.,.0.bs−s′a1...as′ .
p (4.35) Weproceedwithdiscussionoftheanomalousshadowfield
in the light-cone gauge frame. For the anomalous shadow
We recall that, in the standard CFT, the spin-s anomalous
field,thelight-conegaugeframecanbeachievedthroughthe
shadow field is describedby the traceless rank-stensor field
φa1...as. Thusweseethatmakinguseofthegaugesymmetry useofdifferentialconstraint(4.8)andlight-conegaugecondi-
sh,0
tion. Usinggaugesymmetryofthespin-sanomalousshadow
and differentialconstraintwe reduce the field contentof our
field(4.18),weimposethelight-conegaugeonthe φ ,
approachtotheoneinthestandardCFT.Inotherwords,use | shi
of the gauge symmetry and differential constraint allows us
α¯+Π[1,2] φ =0, (4.42)
sh
to match our approach and the standard formulation of the | i
anomalousshadowfield. To summarize, ourgaugeinvariant
where Π[1,2] is given in (2.9). Using gauge condition (4.42)
approachisequivalenttothestandardone.
anddifferentialconstraint(4.8),weobtain
WenowdiscussStueckelberggauge-fixedtwo-pointvertex
oftheanomalousshadowfield. Inotherwords,wearegoing α+
φ =exp (α¯i∂i e¯ ) φl.c. , (4.43)
toconnectourvertex(4.22)withtheoneinthestandardCFT. | shi −∂+ − 1sh | sh i
(cid:16) (cid:17)
TodothatwenotethatvertexofthestandardCFTisobtained
α¯iα¯i φl.c. =0, (4.44)
fromourgaugeinvariantvertex(4.22)bypluggingsolutionto | sh i
thedifferentialconstraint(4.33)into(4.22).Doingso,wefind
where a light-cone ket-vector φl.c. is obtained from φ
thefollowingtwo-pointdensity(uptototalderivative)inthe | sh i | shi
(4.4)byequatingα+ =α− =0,
Stueckelberggaugeframe:
ΓS12tuck.g.fram =ksΓs1t2and, (4.36) |φls.hc.i≡|φshi α+=α−=0. (4.45)
(cid:12)
(cid:12)
(cid:12)
10
Weseethatweareleftwithlight-conefields V. TWO-POINTCURRENT-SHADOWFIELD
INTERACTIONVERTEX
φi1...is′ , s′ =0,1,...,s, λ [s s′] , (4.46)
sh,λ ∈ − 2
Wenowbrieflydiscussthetwo-pointcurrent-shadowfield
which are traceless tensor fields of so(d 2) algebra,
φiii3...is′ = 0. Thesefieldsconstitutethefield−contentofthe interactionvertex. In ourapproach,thisinteractionvertexis
sh,λ determinedbyrequiringthat:
light-conegaugeframe. Note that, in contrasttothe Stueck-
elberg gauge frame, all fields φi1...is′ are not equal to zero. i)thevertexisinvariantunderbothgaugetransformationsof
sh,λ anomalousconformalcurrentandshadowfield;
Also note that, in contrast to the gauge invariant approach,
ii) vertex is invariant under conformal algebra transforma-
fields(4.46)arenotsubjecttoanydifferentialconstraint. Us-
tions.
ing(4.43)in(4.23)leadstolight-conegauge-fixedvertex
Wefindthefollowingvertex:
1 f
Γl.c. = φl.c.(x ) ν φl.c.(x ) , (4.47)
12 2h sh 1 ||x12|2ν+d| sh 2 i L=hφcur|µ|φshi, (5.1)
wheref isdefinedin(4.24). whereµisgivenin(2.10).Wenotethat,undergaugetransfor-
ν
ToillustratethestructureofvertexΓl.c.(4.47)wenotethat, mationoftheanomalousconformalcurrent(3.19),thevaria-
12
intermsofthefieldsφi1...is′,thevertexcanberepresentedas tionofvertex(5.1)takestheform(uptototalderivative)
sh,λ
Γl.c. = s Γs′l.c., (4.48) δξcurL=−hξcur|C¯sh|φshi. (5.2)
12 12,λ
sX′=0λ∈[Xs−s′]2 From(5.2),weseethatthevertex isinvariantundergauge
L
transformationof the anomalousconformalcurrentprovided
Γs′l.c.= wλ φi1...is′(x)φi1...is′(x), (4.49) the anomalous shadow field satisfies differential constraint
12,λ 2s′!x 2κ−2λ+d sh,λ 1 sh,λ 2
| 12| (4.8). Next, we note that under gauge transformation of the
where w is given in (4.29). We see that, as in the case of anomalousshadow field (4.18) the gauge variation of vertex
λ
gauge invariant vertex, light-cone vertex (4.48) is diagonal (5.1)takestheform(uptototalderivative)
with respect to the light-cone fields φi1...is′. Note however
that, in contrast to the gauge invariantsvh,eλrtex, the light-cone δξshL=−hξsh|C¯cur|φcuri. (5.3)
vertexisconstructedoutofthelight-conefieldswhicharenot
From(5.3),weseethatthevertex isinvariantundergauge
subjecttoanydifferentialconstraints. L
transformation of the anomalous shadow field provided the
Thus, we see that our gauge invariant vertex does indeed
anomalous conformal current satisfies differential constraint
provideeasyandquickaccesstothelight-conegaugevertex.
(3.9).
Namely, all that is needed to obtain light-cone gauge vertex
Usingtherealizationoftheconformalalgebrasymmetries
(4.48)istoremovetracesofthetensorfieldsφa1...as′ andre-
sh,λ obtainedinSectionsIII,IV,wemakesurethatvertex (5.1)
placetheso(d 1,1)Lorentzalgebravectorindicesappear- L
− isinvariantundertheconformalalgebratransformations.
ing in gauge invariant vertex (4.23) by the respective vector
indicesoftheso(d 2)algebra.
−
Thekernelofthelight-conevertexgivesthetwo-pointcor- VI. ADS/CFTCORRESPONDENCE.PRELIMINARIES
relation function of the spin-s anomalous conformal current
takentobeinthelight-conegauge.Definingtwo-pointcorre- We now study the AdS/CFT correspondence for free ar-
lationfunctionsofthefieldsφi1...is′ asthesecondfunctional bitrary spin massive AdS field and boundary arbitrary spin
cur,λ
derivativeofΓ with respectto theshadowfieldsφi1...is′, we anomalousconformalcurrentandshadowfield. Tostudythe
sh,−λ
obtainthefollowingcorrelationfunctions: AdS/CFT correspondence we use the gauge invariant CFT
adapted formulation of massive AdS field and modified de
hφic1u.r.,.λis′(x1),φcj1u.r.,.λjs′(x2)i Donder gauge condition found in Ref.[14].7 We emphasize
w
= −λ Πi1...is′;j1...js′ , (4.50)
x 2κ+2λ+d
12
| |
7 ApplicationsofthestandarddeDondergaugetothevariousproblemsof
where wλ is defined in (4.29) and Πi1...is′;j1...js′ stands for masslessfieldsmaybefoundinRefs.[28].Recentinterestingdiscussionof
theprojectorontracelessrank-s′tensorfieldoftheso(d 2) modifieddeDondergaugemaybefoundinRef.[29]. Webelievethatour
− modifieddeDondergaugewillalsobeusefulforbetterunderstandingof
algebra. Explicitformoftheprojectormaybefound,e.g.,in
variousaspectsofAdS/QCDcorrespondencewhicharediscussed,e.g.,in
Ref.[27]. Refs.[30].