BPS Solutions in AdS/CFT

0.09 MB·English

by  E. Gava

BPS solutions in AdS/CFT
EdiGava1,2,GiuseppeMilanesi2,3∗,KumarS.Narain1,andMartinO’Loughlin1,4
1 HighEnergySection,TheAbdusSalamInternationalCentreforTheoreticalPhysics,StradaCostiera11,
34014Trieste,Italy
2 IstitutoNazionalediFisicaNucleare,sez.diTrieste
3 ScuolaInternazionaleSuperiorediStudiAvanzati,ViaBeirut2-4,34014Trieste,Italy
7 4 UniverzavNoviGorici,Vipavska13,P.P.301,Rozˇnadolina,SI-5000NovaGorica,Slovenia
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2 We study a class of exact supersymmetric solutions of type IIB Supergravity. They have an SO(4) ×
SU(2)×U(1)isometryandpreservegenerically4ofthe32supersymmetriesofthetheory.Asymptotically
n AdS5×S5solutionsinthisclassaredualto1/8BPSchiraloperatorswhichpreservethesamesymmetries
a intheN =4SYMtheory. Theyareparametrizedbyasetoffourfunctionsthatsatisfycertaindifferential
J
equations. Weanalyzethesolutionstotheseequationsinalargeradiusasymptoticexpansion: theycarry
2 chargeswithrespecttotwoU(1)KKgaugefieldsandtheirmasssaturatestheexpectedBPSbound.Wealso
1 showhowthesameformalismissuitableforthedescriptionofexcitationsoftheAdS5×Yp,qgeometries.

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v ToappearintheproceedingsoftheRTNworkshop”ForcesUniverse”,Naples,October9-132006
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1 Introduction
1
0
7 OneofthefirststepsinunderstandingtheAdS/CFT correspondenceistosetupaprecisedictionarybe-
0 tweenthestatesofthetheoriesonthetwosidesofthecorrespondence.Itiswellknownthattheparameters
h/ oftheN = 4SU(N)SYM,namelyλ ≡ gY2MN andN shouldbeidentifiedwiththeparametersoftype
-t IIBStringTheoryonAdS5×S5,namelyLAdS,ℓs,gs,asL4AdS =4πλℓ4s,gs =λ/N.
p WemayconsidersolutionstothesupergravityequationsofmotionwhichareasymptoticallyAdS5 S5
×
e as good candidates for dual of states in the CFT, provided that N λ 1 and any dimension four
h ≫ ≫
curvatureinvariantofthesolutions, ,satisfies
: R4
v
i L4 λ N. (1.1)
X R4 AdS ≪ ≪
r Here and in [1] we address the problem of finding BPS supergravity solutions which are dual to chiral
a
primaryoperatorsoftheform
c Tr Zq Tr Zq Tr Zri , (1.2)
i 1 2 3
X (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
whereZ ,Z andZ arethethreecomplexscalarsofthe = 4CFTandthecoefficientsc arechosen
1 2 3 i
N
togivea(anti)-symmetrizationandtracestructuresuchthatthes-wavemodeonS3oftheseoperatorsare
BPS.ThecorrespondingstatessaturatetheBPSbound:
∆=2q+r, (1.3)
where∆istheconformaldimensionoftheoperator. Wewillalsoassumethattheindexstructureissuch
thatthestatespreserveanSU(2) U(1) subgroupoftheSO(6)R-symmetrygroup. Thefullamount
L R
×
ofbosonicsymmetrypreservedbythesestatesis:
R SU(2) U(1) SO(4) . (1.4)
BPS × L× R R−charge× KK
(cid:0) (cid:1)
∗ Correspondingauthor E-mail:[email protected],
2 E.Gava,G.Milanesi,K.S.Narain,andM.O’Loughlin:BPSsolutionsinAdS/CFT
ThefirstRcorrespondstothetransformationsgeneratedbyD′ D 2J3 J,whereJ =J generates
≡ − R− 3
phaseshiftsonZ , JR generatessimultaneousequalphaseshiftsonZ ,Z , D isthedilatationoperator
3 3 1 3
andthelastSO(4)factorrepresentsthefactthatweareconsiderings-wavemodesonS3inthereduction
of SYM theory on R S3 [4]. Consequently, we start from an Ansatz for the metric and the self-dual
×
RR5-formwhichpreservesthisamountofsymmetry. Wealsorequirethatthebackgroundpossessesthe
requiredamountofsupersymmetrybydemandingthatitpossessesaKillingspinor.
2 Supergravity analysis
Welookforgeometriesdualtothestatesin(1.2). TheywillthusbeinvariantunderSU(2) SO(4)
L KK
×
as definedin the previoussection and invariantbutchargedunderthe remainingU(1) . Theextra non-
R
compacttime-likesymmetry(R oftheprevioussection)isassociatedtoinvarianceunderthetransfor-
BPS
mationsgeneratedbyD′inthegaugetheoryandwillemergeasadirectconsequenceofsupersymmetry.
ThemostgenericAnsatzconsistentwiththesesymmetriesisgivenby
ds2 =g dxµdxν +ρ2 σˆ2+σˆ2 +ρ2(σˆ A dxµ)2+ρ˜2dΩ˜2. (2.1)
µν 1 1 2 3 3− µ 3
where ρ , ρ , ρ˜, A and g are(cid:0)function(cid:1)s of the four coordinates xµ. The space is a fibration of a
1 3 µ µν
squashed3-sphere(onwhichtheSU(2)left-invariant1-formsσaˆ aredefined)andaround3-sphereover
afourdimensionalmanifold.
Sincewearelookingforthegeometricdualtooperatorswhichinvolveonlyscalarfieldsinthegauge
theory,theonlypossiblenon-zeroRamond-RamondfieldstrengthisthefiveformF . Themostgeneric
(5)
Ansatzforthefiveformwhichisinvariantunderthegivensymmetriesis:
F = +⋆ (2.2)
(5) 5 5
F F
with
1
= G˜ dxµ dxν +V˜ dxµ (σˆ A dxµ)+g˜σˆ σˆ ρ˜3dΩ˜ (2.3)
5 µν µ 3 µ 1 2 3
2F ∧ ∧ − ∧ ∧
h i
Supersymmetryisguaranteedbytheexistenceofasupersymmetryparameterψsuchthatthegravitino
variationitgeneratesvanishes:
i
δχ = ψ+ F ΓM1M2M3M4M5Γ ψ =0. (2.4)
M ∇M 480 M1M2M3M4M5 M
ThevectorK = ψ¯Γµ∂ ψ isKilling andtimelike anditgeneratesthe extranon-compacttimelikeU(1).
µ
We may thus choose the time coordinate t such that ∂ = K. The Bianchi identity dF = 0 and the
t (5)
existence of the spinor ψ are sufficient for our supergravity background to satisfy the full equations of
motionoftypeIIBSupergravityandtoexpressthecompletesolutioninthefollowingform[1]:
ρ2
ds2 = h−2(dt+V dxi)2+h2 1(T2δ dxidxj +dy2)+ρ˜2dΩ˜2+
− i ρ2 ij 3
3
+ρ2 σˆ2+σˆ2 +ρ2(σˆ A dt A dxi)2 (2.5)
1 1 2 3 3− t − i
wherethecoordinateyistheproductoftworadii (cid:0) (cid:1)
y =ρ ρ˜>0. (2.6)
1
Alltheentriesofthemetricdependonlyon(x1,x2,y)andcanbeexpressedintermsoffourindependent
functionsm,n,p,T asfollows:
ρ4 = mp+n2y4 ρ4 = p2 ρ˜4 = m
1 m 3 m(mp+n2) mp+n2
h4 = mp2 A = n−p A =A V 1ǫ ∂ lnT
mp+n2 t p i t i− 2 ij j
dV = y⋆ [dn+(nD+2ym(n p)+2n/y)dy]
3
− −
3
where⋆ indicatestheHodgedualinthethreedimensionaldiagonalmetricds2 = T2δ dxidxj +dy2.
3 3 ij
Similarly,onecancompletelyexpressthe5-formintermsofm,n,p,T [1]. Thecompleteanalysisin[1]
showthatthesesolutionspreservegenerically4ofthe32supersymmetriesoftypeIIBSupergravity.
TheBianchiidentitiesandintegrabilityconditionsgiverisetoasetofnonlinearcoupledellipticdiffer-
entialequations
y3(∂2+∂2)n+∂ y3T2∂ n +y2∂ T2 yDn+2y2m(n p) +4y2DT2n=0
1 2 y y y −
y3(∂12+∂22)m+∂y(cid:0)y3T2∂ym(cid:1) +∂y (cid:2)y3T(cid:0)22mD =0 (cid:1)(cid:3)
(2.7)
y3(∂12+∂22)p+∂y (cid:0)y3T2∂yp +(cid:1) ∂y y(cid:0)3T24ny(n−(cid:1)p) =0
∂ylnT =D D(cid:0) 2y(m+(cid:1)n 1(cid:2)/y2). (cid:3)
≡ −
3 Solution tothe Supergravity equations
Asolutiontotheequationspresentedintheprevioussectionisdeterminedbyasetofboundaryconditions
at infinity (large values of y,xi) and on the plane y = 0; they should be chosen in such a way as to
give an asymptotically to AdS S5 non-singulargeometry. Due to the non-linearity of the equations
5
×
therelationshipbetweenboundaryconditionsandnon-singularsolutionswithAdS S5 asymptoticsis
5
×
difficulttocontrol.
Thesolutionsdescribedin[2],areasubsetofoursspecifiedbytheadditionalconstraints,
1 1/2 z
n=p= m= − T =1. (3.1)
y2 − y2
Inthiscasewehave
D =0 ρ =ρ =ρ A =0 T =1 (3.2)
1 3 t
and the three second order equationscollapse to one single linear equation. As this equationis linear it
hasbeenpossibletocompletelyidentifytheboundaryconditionsaty = 0andatinfinitythatgiveriseto
regularasymptoticallyAdS S5geometries[2,3].
5
×
4 Asymptotics andcharge
In this section we discuss asymptotic solutions to the differential equations in (2.7) for large values of
(x1,x2,y)whichgiveAdS S5asymptotics.
5
×
ThefirstcorrectionstotheAdS S5 geometrycapturetheglobalU(1)chargesundertwooutofthe
5
×
threeU(1)CartangaugefieldsarisingfromtheKKreductionofIIBsupergravityonS5tofivedimensional
maximalgaugedsupergravity.
Itisnothardtoseethatthefollowingexpressionsforourfunctions
1 1 1 1
m , n , p , T 1 (4.1)
∼ y2 − L4R4 ∼ L4R4 ∼ L4R4 ∼
satisfy the equations at leading order for large R, with (L2R,θ,φ) polar coordinates in the (x ,x ,y)
1 2
spaceandgivetheleadingasymptotics
dR˜2
ds2 =L2 R˜2dt2+ +R˜2dΩ˜2+dθ2+cos2θdφ˜2+sin2θdΩˆ2 (4.2)
− R˜2 3 3!
withφ˜=φ t,whichisAdS S5inPoincarecoordinates.
5
− ×
4 E.Gava,G.Milanesi,K.S.Narain,andM.O’Loughlin:BPSsolutionsinAdS/CFT
Wewillnowsolvethedifferentialequationstothenexttwoordersintheasymptoticexpansion.Forthe
sakeofsimplicitywewillassumethat∂ isalsoaKillingvectorofoursolutions.Despitethissimplifying
φ
assumption,ingeneralthesolutionswillstillbechargedunderthecorrespondingKKgaugefield.
Weassumethefollowingexpansionforourfunctions:
1 1 m (θ) m (θ)
2 3
m + +
∼ y2 − L4R4 R6 R8
1 n (θ) n (θ)
2 3
n + +
∼ L4R4 R6 R8
(4.3)
1 p (θ) p (θ)
2 3
p + +
∼ L4R4 R6 R8
t (θ)
2
T 1+ .
∼ R4
Ateachorder,thesolutionstotheequations(2.7)willdependgenericallyonsevenintegrationconstants.
For the first subleading order, regularity of the asymptotic solution and coordinate redefinitions reduce
thesetotwo,J andQ.
The global U(1) charges under the Cartan gauge fields arising in the Kaluza Klein reduction of IIB
supergravityoverS5canbereadofffromtheoffdiagonalcomponentsofthemetric
1
g g (4.4)
tω ∝ ωωR2
whereωisasuitableangleintheS5andtheconstantofproportionalityistheglobalcharge.Theconstants
J andQturnouttobetheglobalchargesundertheU(1)gaugefieldsdualtoJ3 andJ ofthegaugetheory
R
side. Fromtheg wecanreadoffthemassM ofourexcitationswhichcorrectlysatisfytheexpectedBPS
tt
bound
πL2
M = AdS(J +2Q). (4.5)
4G | | | |
5
5 Sasaki-Einstein manifolds
Intheprevioussectionweassumedthat∂ isaKillingvectorofourmetric.Thisliftstheinternalsymmetry
φ
toSU(2) U(1) U(1). We canfurtherrestrictoursolutionsrequiringthattheypreserve8insteadof
× ×
only 4 supersymmetries. This results in a set of constraints on the functions m,n,p,T and give rise to
AdS Yp,q geometries, where Yp,q are the Sasaki-Einstein manifolds constructed in [5]. Type IIB
5
×
StringTheoryonAdS Yp,q isconjecturedtobedualto an = 1 Quivergaugetheoryspecifiedby
5
× N
the 2 integers p,q. It is clear that, within our formalism, we can describe asymptotically AdS Yp,q
5
×
solutions which are dual to operators in the correspondingquiver theory which preserve one half of its
supersymmetry.
WecanshowanexplicitexampleforthecaseofT1,1 Y1,0[7]. Thecorrespondinggaugetheoryhas
≡
anSU(N) SU(N)gaugegroup,andtwoSU(2)doubletsA,Bofchiralsuperfieldstransforminginthe
(N,N¯)and×(N¯,N)respectivelyofthegaugegroup.TheSU(2)isaflavoursymmetry.InthecaseofT1,1
the SU(2) U(1) U(1) symmetryis lifted to SU(2) SU(2) U(1) as a reflection of this flavour
× × × ×
symmetry.Assumingthefollowingleadingbehaviourforthefunctionsm,n,p,T
3 1 2L4 4L4 y θ −1/6
m n p T √6sin(θ) tan
∼ 2y2 ∼ 3 y4 ∼ 9 y4 ∼ L2 2
(cid:18) (cid:19)
5
with y2 = 2L2ρ˜2 andx2 +x2 = L4 tanθ −1/3 anda properparametrisationof theleftinvariantone
3 1 2 2
forms,oneobtainsthefollowingexpressionfortheleadingordermetric
(cid:0) (cid:1)
L2dρ˜2
ds2 ρ˜2dt2+ +ρ˜2dΩ˜2 +
∼ − ρ˜2 3
(cid:18) (cid:19)
1 1 1
+L2 dθ2+sin2θdφ2 + dθˆ2+sin2θˆdφˆ2 + dψˆ+cosθˆdφˆ+cosθdφ 2 (5.1)
6 6 9
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
whereφisthe(properlyrescaled)polarangleinthex1,x2 plane. Wecansolvetheequationstothenext
orderassumingthebehaviour
3 1 m (θ) 2L4 n (θ) 4L4 p (θ)
2 3 3
m + n + p +
∼ 2y2 y4 ∼ 3 y4 y6 ∼ 9 y4 y6
−1/6
y θ t (θ)
T √6sin(θ) tan + 1
∼ L2 2 y
(cid:18) (cid:19)
Regularityandcoordinateredefinitionsnowreducethesevenintegrationsconstantstothree,J,Q,B. The
extraconstantofintegrationmeasuresthetotalfluxundertheU(1)gaugefieldsobtainedbythereduction
ofthe5-formF overanontrivial3-cycleofT1,1. Thisgaugefieldisassociatedtothebaryonchargein
(5)
thegaugetheory.Baryonscanbeconstructedfromoperatorsoftheform
N
ǫα1···αNǫβ1···βN k1···kN Bki (5.2)
Dlm αiβi
i=1
Y
where α ,β are gaugeindexes, k are flavour indexesand Dk1···kN is a properSU(2) Clebsch Gordon
i i i lm
coefficient.
6 Conclusions
WehavepresentedaparametrisationofexactsolutionsofTypeIIBSupergravitydualto(atleast)1/8BPS
chiralprimariesinthe = 4SCFTortohalfsupersymmetricstatesinacertainclassof = 1Quiver
N N
gauge theories. These realise to the fully backreacted geometry of giant gravitons and dibaryonsin the
correspondinggaugetheories.ThecorrectmatchingofchargesandsatisfactionoftheBPSboundisafirst
nontrivialcheckontheconsistencyofthederivation.Adeeperstudyofboundaryconditionsaty =0and
thegenericYp,q caseisdesirableandstillworkinprogress.
References
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[3] G.MilanesiandM.O’Loughlin,JHEP09(2005)00
[4] E.Witten,Adv.Theor.Math.Phys.2(1998)253–291
[5] J.P.Gauntlett,D.Martelli,J.SparksandD.Waldram,Adv.Theor.Math.Phys.8(2004)711
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