Spinning strings and integrable spin chains in the AdSCFT correspondence

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

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aSpinningstringsandintegrablespinchainsintheAdS/CFTcorrespondenceJanPlefkaMax-Planck-Institutf¨urGravitationsphysik(Albert-Einstein-Institut)AmM¨uhlenberg1,14476Potsdam,Germanyjan.plefka@aei.mpg.deAEI-2005-124,hep-th/0507136AbstractInthisintroductoryreviewwediscussdynamicaltestsoftheAdS5S5string/N=4superYang-Millsduality.Afterabriefintroductionto×AdS/CFTwearguethatsemiclassicalstringenergiesyieldinformationonthequantumspectrumofthestringinthelimitoflargeangularmomentaontheS5.TheenergiesofthefoldedandcircularspinningstringsolutionsrotatingonaS3withintheS5arederived,whichyieldalllooppredictionsforthedualgaugetheoryscalingdimensions.ThesefollowfromtheeigenvaluesofthedilatationoperatorofN=4superYang-MillsinaminimalSU(2)subsectorandwedisplayitsreformulationintermsofaHeisenbergs=1/2spinchainalongwiththecoordinateBetheansatzforitsexplicitdiagonal-ization.Inordertomakecontacttothespinningstringenergieswethen

studythethermodynamiclimitoftheone-loopgaugetheoryBetheequationsanddemonstratethematchingwiththefoldedandclosedstringresultatthislooporder.Finallytheknowngaugetheoryresultsathigher-loopordersarereviewedandtheassociatedlong-rangespinchainBetheansatzisintro-duced,leadingtoanasymptoticall-loopconjectureforthegaugetheoryBetheequations.Thisuncoversdiscrepanciesatthethree-looporderbetweengaugetheoryscalingdimensionsandstringtheoryenergiesandtheimplicationsofthisarediscussed.Alongthewaywecommentonfurtherdevelopmentsandgeneralizationsofthesubjectandpointtotherelevantliterature.

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

1Introduction

Stringtheorywasinitiallydiscoveredinanattempttodescribethephysicsofthestronginteractionspriortotheadventofgauge eldtheoriesandQCD.Todayithasmaturedtoaverypromisingcandidateforauni edquantumtheoryofgravityandalltheotherforcesofnature.Inthisinterpretationgauge eldsariseasthelowenergyexcitationsoffundamentalopenstringsandarethereforederived,non-fundamentalobjects,justasthetheoryofgravityitself.Ironicallythough,advancesinourunderstandingofnon-perturbativestringtheoryandofD-braneshasledtoaresurrectionofgauge lyitisnowgenerallybelievedthatstringtheoryinsuitablespace-timebackgroundscanhaveadual,holographicdescriptionintermsofgauge eldtheoriesandthusthequestionofwhichofthetwoisthefundamentalonebecomesredundant.ThisbeliefbuildsonaremarkableproposalduetoMaldacena

[1]knownastheAnti-de-Sitter/ConformalFieldTheory(AdS/CFT)correspondence(forreviewssee[2,3]).

Theinitialideaofastring/gaugedualityisdueto’tHooft[4],whorealizedthattheperturbativeexpansionofSU(N)gauge eldtheoryinthelargeNlimitcanbereinterpretedasagenusexpansionofdiscretizedtwodimensionalsurfacesbuiltfromthe eldtheoryFeynmandiagrams.Here1/NcountsthegenusoftheFeynman

2diagram,whilethe’tHooftcouplingλ:=gYMN(withgYMdenotingthegauge

theorycouplingconstant)enumeratesquantumloops.E.g.thegenusexpansionofthefreeenergyFofaSU(N)gaugetheoryinthe’tHooftlimit(N→∞withλ xed)takesthepictorialform

F=N2+1+1

N2g 2∞

l=0cg,lλl(1)

withsuitablecoe cientscg,ldenotingthecontributionsatgenusgandlooporderl.Obviouslythis1/NexpansionresemblestheperturbativeexpansionofastringtheoryinthestringcouplingconstantgS.

TheAdS/CFTcorrespondenceisthe rstconcreterealizationofthisideaforfourdimensionalgaugetheories.Initspurestform–whichshallalsobethesettingwewillbeinterestedin–itidenti esthe‘fundamental’typeIIBsuperstringinatendimensionalanti-de-Sittercrosssphere(AdS5×S5)space-timebackgroundwiththemaximallysupersymmetricYang-MillstheorywithgaugegroupSU(N)(N=4SYM)infourdimensions.TheN=4superYang-Millsmodelisaquantumconfor-mal eldtheory,asitsβ-functionvanishesexactly.Thestringmodeliscontrolledbytwoparameters:ThestringcouplingconstantgSandthe‘e ective’stringtensionR2/α′whereRisthecommonradiusoftheAdS5andS5geometries.Thegaugetheory,ontheotherhand,isparametrizedbytherankNofthegaugegroupandthe

2couplingconstantgYM,orequivalentlythe’tHooftcouplingλ:=gYMN.According

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

totheAdS/CFTproposal,thesetwosetsofparametersaretobeidenti edas

4πλλ=R2

(x y)2 A(λ,1α′,gS)|OA (3)

,gS).with (λ,1

α′

AzerothordertestoftheconjectureisthentheagreementoftheunderlyingsymmetrysupergroupPSU(2,2|4)ofthetwotheories,whichfurnishestherepre-sentationsunderwhichOA(x)and|OA transform.Thisthenyieldsahintonhowonecouldsetupanexplicitstringstate/gaugeoperatordictionary.

Clearlythereislittlehopeofdeterminingeithertheallgenus(allordersingS)stringspectrumorthecomplete1/Ndependenceofthegaugetheoryscalingdimensions .Buttheidenti cationoftheplanargaugetheorywiththefree(gS=0)stringseemsfeasibleandfascinating:FreeAdS5×S5stringtheoryshouldgivetheexactallloopgaugetheoryscalingdimensionsinthelargeNlimit!Unfortunatelythough,ourknowledgeofthestringspectrumincurvedbackgrounds,eveninsuchahighlysymmetriconeasAdS5×S5,remainsscarce.Thereforeuntilveryrecentlyinvestigationsonthestringsideofthecorrespondencewerelimitedtothedomainofthelowenergye ective eldtheorydescriptionofAdS5×S5stringsintermsoftypeIIBsupergravity.This,however,isnecessarilylimitedtoweaklycurvedgeometries√instringunits,i.e.tothedomainof

λ 1.Hence

oneisfacingastrong/weakcouplingduality,inwhichstronglycoupledgauge elds

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

aredescribedbyclassicalsupergravityandweaklycoupledgauge eldscorrespondtostringspropagatinginahighlycurvedbackgroundgeometry.Thisinsightiscertainlyfascinating,butatthesametimestronglyhindersanydynamicaltests(orevenaproof)oftheAdS/CFTconjectureinregimeswhicharenotprotectedbythelargeamountofsymmetryintheproblem.

Thissituationhasprofoundlychangedsince2002byperformingstudiesofthecorrespondenceinnovellimitswherequantumnumbers(suchasspinsorangularmomentainthegeometricAdS5

N→∞.Thiswasinitiated×inS5language)becomelargeinacorrelatedfashionastheworkofBerenstein,MaldacenaandNastase

[5]whoconsideredthequantum uctuationexpansionofthestringaroundade-generatedpoint-likecon guration,correspondingtoaparticlerotatingwithalargeangularmomentumJonagreatcircleoftheS5space.InthelimitofJ→∞withJ2/Nheld xed(the‘BMNlimit’)thegeometryseenbythefastlymovingparticleisagravitationalplane-wave,whichallowsforanexactquantizationofthefreestringinthelight-cone-gauge[6,7].Theresultingstringspectrumleadstoaformidablepredictionfortheallloopscalingdimensionsofthedualgauge

inthecorrespondinglimit,i.e.thefamousformula n=J+2 theoryoperatorsJ2forthe

simplesttwostringoscillatormodeexcitation.Thekeypointhereistheemergenceofthee ectivegaugetheoryloopcountingparameterλ/J2intheBMNlimit.Thesescalingdimensionshavebynowbeen rmlyreproduceduptothethreelooporderingaugetheory[8,9,10].Thishasalsoledtoimportantstructuralinformationforhigher(orallloop)attemptsingaugetheory,whichmaximallyemploytheuncov-eredintegrablestructurestobediscussedbelow.Moreover,theplanewavestringtheory/N=4SYMdualitycouldbeextendedtotheinteractingstring(gS

respectivelynon-planargaugetheoryregimeprovidinguswiththemostconcrete=0)realizationofastring/gaugedualitytodate(forreviewssee[11,12,13,14]).

InthisreviewweshalldiscussdevelopmentsbeyondtheBMNplane-wavecor-respondencewhichemploymoregeneralsectorsoflargequantumnumbersintheAdS/CFTduality.Thekeypointfromthestringperspectiveisthatsuchalimitcanmakethesemiclassical(intheplane-wavecase)orevenclassical(inthe‘spin-ningstring’case)computationofthestringenergiesalsoquantumexact[15,16],i.e.higherσ-modelloopsaresuppressedbyinversepowersofthetotalangularmomentumJonthe vesphere2.Theseconsiderationsonthestringsidethen(arguably)yieldalllooppredictionsforthedualgaugetheory.Additionallytheperturbativegaugetheoreticstudiesatthe rstfewordersinλledtothediscoverythatthespectrumofscalingdimensionsoftheplanargaugetheoryisidenticaltothatofanintegrablelong-rangespinchain[19,8,20].ConsistentlytheAdS5

stringisaclassicallyintegrablemodel[21],whichhasbeenheavilyexploitedin×theS5

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

constructionofspinningstringsolutions.

Thisreviewaimsatamoreelementaryintroductiontothisveryactiveareaofresearch,whichinprincipleholdsthepromiseof ndingtheexactquantumspectrumoftheAdS5×S5stringorequivalentlytheallloopscalingdimensionsofplanarN=4superYang-Mills.Itisintendedasa rstguidetothe eldforstudentsandinterested‘newcomers’andpointstotherelevantliteraturefordeeperstudies.WewilldiscussthesimplestsolutionsoftheAdS5×S5stringcorrespondingtofoldedandcircularstringcon gurationspropagatinginaR×S3subspace,withtheS3lyingwithintheS5.OnthegaugetheorysidewewillmotivatetheemergenceofthespinchainpictureattheleadingonelooporderanddiscusstheemergingHeisenbergXXX1/2modelanditsdiagonalizationusingthecoordinateBetheansatztechnique.Thisthenenablesustoperformacomparisonbetweentheclassicalstringpredictionsinthelimitoflargeangularmomentaandthedualthermodynamiclimitofthespinchainspectrum.Finallyweturntohigher-loopcalculationsinthegauge-theoryanddiscussconjecturesfortheall-loopformoftheBetheequations,parisonwiththeobtainedstringresultsuncoversadiscrepancyfromlooporderthreeonwardsandtheinterpretationofthisresultisalsodiscussed.

Therealreadyexistanumberofmoredetailedreviewsonspinningstrings,in-tegrabilityandspinchainsintheAdS/CFTcorrespondence:Tseytlin’sreview[22]mostlyfocusesonthestringsidethecorrespondence,whereasBeisert’sPhysicsRe-port[23]concentratesprimarilyonthegaugeside.SeealsoTseytlin’ssecondreview

[24],ontheso-calledcoherent-statee ectiveactionapproach,whichwewillnotdis-cussinthisreview.RecommendedisalsotheshorterreviewbyZarembo[25]ontheSU(2)respectivelyR×S3subsector,discussingtheintegrablestructureappearingontheclassicalstring–notcoveredinthisreview.Foradetailedaccountofthenearplane-wavesuperstring,itsquantumspectroscopyandintegrabilitystructuresseeSwanson’sthesis[26].

2Thesetup

wherewehavesuppressedthefermionictermsintheaction,astheywillnotberelevantinourdiscussionofclassicalsolutions(thefullfermionicactionisstatedin[27,28]).AnaturalchoiceofcoordinatesfortheAdS5×S5space(“global

WiththeembeddingcoordinatesXm(τ,σ)andYm(τ,σ)thePolyakovactionoftheAdS5×S5stringinconformalgauge(ηab=diag( 1,1))takestheform(m,n=1,2,3,4,5)√ (AdS5)man(S5)manI= dτdσGmn aX X+Gmn aY Y+fermions,(4)4π

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

3×

Figure1:CartoonoftheAdS5(bulkcylinderwithboundaryR×S3)space-timeandtheS5(sphere)space.

coordinates”)is3

ds2AdS2 cosh2ρdt2+sinh2ρ(dψ¯2+cos2

5=dρψ¯d 12+sin2ψ¯d 22)

ds2S5=dγ2+cos2γdφ32+sin2γ(dψ2+cos2ψdφ12+sin2ψdφ22).(5)

Moreover,wehave

stringtension√directlywrittenthestringactionwiththehelpofthee ective

λ.Forthisoneexpandsaroundaclassicalsolutionof

(4)andintegratesoutthe uctuationsinthepath-integrallooporderbyorder.Thisistheroutewewillfollow.Ofcourseindoingsoonewillonlyhaveapatch-wiseaccesstothefullspectrumofthetheory,witheachpatchgivenbythesolutionexpandedaround.

2.1Therotatingpoint-particle

Itisinstructivetosketchthisprocedurebyconsideringtheperhapssimplestsolutiontotheequationsofmotionof(4):therotatingpoint-particleonS5,whichisa

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

degeneratedstringcon guration.

πt=κτρ=0γ=

˙ t=√

=

=√√2π˙=coshρt222π√2˙1 φ

˙2 φ˙1=sinγcosψφ√J:=J1+J2.2π˙2=0,sin2γsin2ψφ

HenceclassicallyE=J.Onemaynowconsiderquantum uctuationsaroundthisµsolution,i.e.Xµ=Xsolution(τ)+1

λ

E=√√

λheld xed.Thislimitofalargequantum

numbersuppressesallthehigherloopcontributionsbeyondone-loop,i.e.

E J=E2(κ)+1 J→∞E4(κ)+... →E2(κ).J(9)

Hencethequadraticapproximationbecomesexact!Thisquadratic uctuationac-tion(includingthefermions)isnothingbuttheIIBsuperstringinaplanewavebackground

[6],whicharisesfromtheAdS5×S5geometrythroughaso-calledPen-roselimit(see[29,30]forthisconstruction).Thequantizationofthisstringmodel

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

isstraightforwardinthelight-conegauge[6,7]andleadstoafree,massivetwodimensionaltheoryforthetransversedegreesoffreedom(i=1,...,8)

I2= dτdσ(1

2xixi+fermions)(10)

withacompactexpressionforthespectrum

∞

E12=λ′2(11)

n √= ∞√J

whereN

matching n:=α condition withniαi[αnistheexcitationnumberoperatorfortransversestringexcita-tionsα ni|0im ,α nj]=δnmδij.TheVirasoroconstraints(6)reducetothelevel

nnN n=0knownfromstringtheoryin atMinkowskispace-

time.Hencethe rststringyexcitationisα

Foramoredetailedtreatmentoftheplanewavenα superstringn|0 with√1+λ′n2.

see[11,12,13,14].

2.2N=4SuperYang-Mills

TheconjectureddualgaugetheoryoftheAdS5×S5superstringisthemaximallysupersymmetric(N=4)Yang-Millstheoryinfourdimensions[31,32].Its eldcontentisgivenbyagluon eldAµ(x),sixscalarsφi(x)(i=1,...,6)aswellas4Majoranagluinos,whichwechoosetowriteasa16component10dMajorana-Weylspinorχα(x)(α=1,...16).All eldsareintheadjointrepresentationofSU(N).TheactionofN=4SuperYang-Millsisuniquelydeterminedbytwoparameters,thecouplingconstantgYMandtherankofthegaugegroupSU(N)

S=2

4(F2+1

4[φ[φ1µν)i,φj]i,φj]+2χ¯Γi[φi,χ]

(12)

withthecovariantderivativeDµ= µ

Diracmatrices. i[Aµ,].Furthermore,(Γµ,Γi)aretheten

dimensional

Duetothelargeamountofsupersymmetrypresent,theconformalinvarianceoftheclassical eldtheorysurvivesthequantizationprocedure:ThecouplingconstantgYMisnotrenormalizedanditsβ-functionvanishestoallordersinperturbationtheory[33,34,35].ThisiswhyoneoftenreferstoN=4SuperYang-Millsasa“ nite”quantum eldtheory.Neverthelesscompositegaugeinvariantoperators,i.e.tracesofproductsoffundamental eldsandtheircovariantderivativesatthesamespace-point,e.g.Oi1...ik(x)=Tr[φi1(x)φi2(x)...φik(x)],arerenormalizedandacquireanomalousdimensions.Thesemaybereado fromthetwopointfunctions(statedhereforthecaseofscalaroperators)

OA(x)OB(y) =δAB

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

where OAisthescalingdimensionofthecompositeoperatorOA.Classicallythesescalingdimensionsaresimplythesumoftheindividualdimensionsoftheconstituent elds([φi]=[Aµ]=1and[χ]=3/2).Inquantumtheorythescalingdimensions

2receiveanomalouscorrections,organizedinadoubleexpansioninλ=gYMN(loops)

and1/N2(genera)∞∞ 1λl = 0+

l=1g=0

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

ψ=0

ψ=ψ0

ψ= ψ0

Figure2:Thefoldedstingextendingfromψ= ψ0toψ=ψ0,wheresin2ψ0:=q.withthetimecoordinateoftheAdS5space-time.Thiswas rstdiscussedbyFrolovandTseytlinin[15,36].For

this

let

us

consider

thefollowingansatzintheglobalcoordinatesof(5)

t=κτρ=0γ=π

λ

dσ=ω21

ω212(ω212=0),(20)

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

relatingtheintegrationconstantqtotheparametersofouransatz4.OurgoalistocomputetheenergyEofthesetwosolutionsasafunctionofthecommutingangularmomentaJ1andJ2onthethreespherewithinS5.Uponusingeqs.(7)andinsertingtheansatz(16)thesearegivenby

√E= 2π 2πdσdσλω1λω2

00

λ=J1

ω2.(23)

3.1Thespecialcase:ω1=ω2

Itisinstructiveto rstdiscusstheparticularlysimplespecialcaseofacircularstring,whereω1=ω2and(20)doesnotapply.Thiswillturnouttobealimitingcaseoftheq>1scenario.Forω21=0theequationofmotionforψ(σ)(18)immediatelyyields

ψ′′=0 ψ(σ)=nσ(24)

withnbeingtheintegerwindingnumberofthisciruclarstringψ(σ+2π)=ψ(σ)+√2πn.InthiscasetheVirasoroconstraints(6)yieldκ=

1+

J12n2λ!Thisamountstoanalllooppredictionforthedualgauge

theoryscalingdimensionintheBMNlimitJ1→∞withλ/J12 xed,quitesimilartotheresultfortheplane-wavesuperstringdiscussedabove.

Letusnowdiscussthefoldedandcircularstringsolutionsinturn.

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

3.2Thefoldedstring:q≤1

InthefoldedcaseJ1maybeexpressedintermsofanellipticintegral5bysubstituting(using(19))dψ

(26)dσ=

2q

sin

ψinto(22)andperformingsomeelementarytransformationsto nd(q=sin2ψ0)√√ ψ02cosψ224dψE(sinψ0),(27)J1==222ππω21sinψ0 sinψ0

whereweonlyneedtointegrateoveronequarterofthefoldedstringduetosymmetryconsiderations(see gure2).Additionallywehave

2π ψ0dψ42π=dσ=4=sin2ψ0 sin2ψ00

λ)ω1

2√

ω2E(q),κ2√K(q),(29)anduse(23)todeduce√.(30)K(q) E(q)2

ThentheVirasoroconstraintequation(20)andtheidentity1=(ω22 ω12)/ω212yieldthetwofoldedstringequations

4qλ

K(q)2 J12

π2=J22

E(q)2,(31)

whichimplicitlyde nethesoughtafterenergyfunctionE=E(J1,J2)uponfurthereliminationofq.ThisisachievedbyassumingananalyticbehaviorofqandEintheBMNtypelimitoflargetotalangularmomentumJ:=J1+J2→∞withλ/J2held xed

λq=q0+q2+...J4 λE2+....E=JE0+J4

1 xsinψ2(32)K(x):= π/2

dψ1

1 xsin2ψ.

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

Pluggingtheseexpansionsinto(31)

one

cansolvefortheqiandEiiteratively.AtleadingorderE0=1(asitshouldfromthedualgaugetheoryperspective)andq0isimplicitlydeterminedthroughthe“ llingfraction”J2/J

J2

K(q0).(33)

The rstnon-trivialtermintheenergyisthenexpressedintermsofq0through

E=21

λλ2

832

λω1

ω21

λω1√qE(q 1)

Analogously(28)nowbecomes .

= 2π

dσ=4 π/2

2πdψ(q 1).

00sin2ψ0

Thecorrespondingrelationsto(29)and(30)then sin2ψ=4qKtaketheform

1√ω1qJ1

2K(q 1),2√

πJ

ω=2

21λq[K(q 1) E(q 1)].

(37)(38)(39)

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

Fromtheseonededucesincompleteanalogyto(31)thetwocircularstringequations

4λJ12

K(q 1)2

22

π2=J

[(1 q)K(q 1)+qE(q 1)]2,(40)

whichencodetheenergyrelationE=E(J1,J2)uponeliminationofq.Inordertodothisweagainmakeananalyticansatzinλ/J2forqandEaswedidin(32).Thisyieldsthefollowingimplicitexpressionforq0intermsofthe llingfractionJ2/J

J2

K(q 1(41)

0) ,

The rsttwoenergytermsintheλ/J2expansionthentaketheform

E0=1,E1=2

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

1.6

1.4

1.2

1

E10.8

0.6

0.4

0.2

00.20.40.60.81J2 J

Figure3:Theone-loopenergiesofthefolded(dark)andcircular(light)stringsolutionsplottedagainstthe llingfractionJ2/J.ThedashedcurveisthemirroredfoldedstringsolutionwhereoneinterchangesJ1 J2.

stringwereconstructedbyreducingthesystemtotheso-calledNeumannintegrablesystemthroughasuitableansatz.Thesolutionsof[43,44]involvenonvanishingvaluesforallspinsandangularmomenta(S1,S2,J1,J2,J3).However,theydonotgenericallydisplayananalyticbehaviorinthee ectivecouplingconstantλ′.Thisistrueonlyforcon gurationswithatleastonelargechargeJiontheS5.

InthecontextoftheO(p,q)σ-modelsintegrabilityisbasedontheexistenceofaLaxpair,afamilyof atconnectionsonthe2dstringworldsheet,givingrisetoanin nitenumberofconservedcharges.Thesewere rstdiscussedinthecontextofthebosonicstringin[45]andforthefullsuperstringin[21].TheLaxpairforthestringwasputtousein[46]forstringcon gurationsonR×S3–thesectorwealsoconsideredintheabove.Theseinvestigationsledtotheconstructionofanunderlyingalgebraiccurveparametrizingthesolutions.Thisenabledtheauthorsof[46]towritedownanintegralequationofBethetypeyieldingtheassociatedenergiesofthesolutions.Verysimilarequationswillappearbelowinourdiscussioninsection4.2onthethermodynamiclimitofthegaugetheoryBetheequations.Theextractionoftheseintegralequationsfromthestringσ-modelthenallowsforadirectcomparisontothegaugetheoryBetheequations.Onthislevelofformalization,thereisnoneedtocompareexplicitsolutionsanylonger–aswearedoinghereforpedagogicalpurposes.ThisconstructionbasedonanunderlyingalgebraiccurvemakesfulluseofthetechnologyofintegrablesystemsandhasbeennicelyreviewedbyZaremboin[25].

Intheveryinterestingpaper[47]thesecontinuumstringBetheequationswereboldlydiscretizedleadingtoaconjecturedsetofBetheequationsforthequantumspectrumofthestring.Thisproposalhasbeenveri edbycomparingittoknown

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

quantumdataoftheAdS5×S5string:Thenearplane-wavespectrumofthesu-perstringof[48,49,50,51]aswellastheexpected[52]genericscalingofthestringenergieswithλ1/4inthestrongcouplinglimitagreewiththepredictionsofthequan-tumstringBetheequations.ButthereismorequantumdatafortheAdS5×S5stringavailable:InaseriesofpapersbyTseytlin,Frolovandcollaboratorsone-loopcorrectionsonthestringworldsheettotheenergiesofvariousspinningstringsolu-tionshavebeencomputed[16,53,54].TheoneloopcorrectionforacircularstringmovinginAdS3×S1 AdS5×S5obtainedin[54]wasrecentlycompared[55]totheresultobtainedfromtheproposedquantumstringBetheequationsof[47].Theauthorsof[55] ndagreementwhentheyexpandtheresultsinλ′(uptothirdorder),butdisagreementsemergeindi erentlimits(whereλ′isnotsmall).Theinterpretationofthisresultisunclearatpresent.FinallytheproposedquantumstringBetheequationsof[47]canalsobemicroscopicallyattributedtoas=1/2spinchainmodelwithlong-rangeinteractionsupto(atleast)order veinasmallλexpansion[56].

ThetechnicallyinvolvedconstructionofalgebraiccurvessolvingtheclassicalR×S3stringσ-modelhassubsequentlybeengeneralizedtolargersectors:In[57]toR×S5(orSO(6)ingaugetheorylanguage)con gurations,in[58]toAdS3×S1(orSL(2))stringcon gurationsand nallyin[59]tosuperstringspropagatinginthefullAdS5×S5space.

TherehasalsobeenprogressonanumberofpossiblepathstowardsthetruequantizationoftheclassicalintegrablemodeloftheAdS5×S5stringintheworks

[60,61,62,63,64,65],however,itisfairtosaythatthisproblemremainscurrentlyunsolved.

4Thedualgaugetheoryside

Letusnowturntotheidenti cationofthefoldedandcircularstringsolutionsinthedualgaugetheory.

OuraimistoreproducetheobtainedenergyfunctionsE1(J1,J2)plottedin gure3fromadualgaugetheorycomputationatone-loop.Forthisweneedtoidentifythegaugetheoryoperators,whicharedualtothespinningstringsonR×S3.AshereJ2=0=S1=S2therelevantoperatorswillbebuiltfromthetwocomplexscalarsZ:=φ1+iφ2andW:=φ3+iφ4withatotalnumberofJ1Z- eldsandJ2W- elds,i.e.J1,J2Oα=Tr[ZJ1WJ2]+...,(43)wherethedotsdenotesuitablepermutationsoftheZandWtobediscussed.Anoperatoroftheform(43)maybepicturedasaringofblack(“Z”)andred(“W”)beads–orequivalentlyasacon gurationofans=1/2quantumspinchain,where

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

D

=

+

++...= pNc

pplanard

pnonplanar

Figure4:Theactionofthedilatationoperatoronatraceoperator.

Wcorrespondstothestate|↑ andZto|↓ .

Tr[ZW2ZW4]

|↓↑↑↓↑↑↑↑ cyclic=J1+J2.

Averye cienttooltodealwiththisproblemisthedilatationoperatorD,which

J1,J2wasintroducedin[66,8].ItactsonthetraceoperatorsOαata xedspace-time

pointxanditseigenvaluesarethescalingdimensions

J1,J2J1,J2DαβOβ(x).(44)D Oα(x)=

βHowdoesonecomputetheassociatedscalingdimensionsat(say)oneloopor-derforJ1,J2→∞?ClearlyoneisfacingahugeoperatormixingproblemasallJ1,J2OiwitharbitrarypermutationsofZ’sandW’saredegenerateattreelevelwhereOi1 0J,J2

Thedilatationoperatorisconstructedinsuchafashionastoattachtherelevantdiagramstotheopenlegsofthe“incoming”traceoperatorsasdepictedin gure4andmaybecomputedinperturbationtheory

D=∞ n=0D(n),(45)

2nwhereD(n)isofordergYM.Fortheexplicitcomputationoftheone-looppieceD(1)

seee.g.thereview[12],wheretheconcreterelationtotwo-pointfunctionsisalsoexplained.Inour‘minimal’SU(2)sectorofcomplexscalar eldsZandWittakestherathersimpleform

.dZji(46)

Notethatthetree-levelpieceD(0)simplymeasuresthelengthoftheincidentoper-ator(orspinchain)J1+J2.Theeigenvaluesofthedilatationoperatorthenyieldthescalingdimensionswearelookingfor–diagonalizationofDsolvesthemixingproblem.

D(0)ˇ+WWˇ),=Tr(ZZD(1)= 2gYM

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

WeshallbeexclusivelyinterestedintheplanarcontributiontoD,asthissectorofthegaugetheorycorrespondstothe“free”(inthesenseofgs=0)AdS5×S5string.ForthisitisimportanttorealizethattheplanarpieceofD(1)onlyactsontwoneighboring eldsinthechainofZ’sandW’s.ThismaybeseenbyevaluatingexplicitlytheactionofD(1)ontwo eldsZandWseparatedbyarbitrarymatricesAandB

ˇWˇ] Tr(ZAWB)= Tr(A)Tr([Z,W]B)+Tr(B)Tr([Z,W]A).(47)Tr[Z,W][Z,

ClearlythereisanenhancedcontributionwhenA=1orB=1,i.e.ZandWarenearestneighboursonthespinchain.Fromtheabovecomputationwelearnthat

Dplanar=(1)λ

8πH=2XXX1/2λ4 σi· σi+1).(49)

Duetothepositivesigninfrontofthesum,thespinchainisferromagneticanditsgroundstateis|↓↓...↓ cyclic Tr(ZL):Thegaugedualoftherotatingpointparticleofsection

2.1.Excitationsofthegroundstatearegivenbyspin ipsor“magnons”.Notethataone-magnonexcitation|↓...↓↑↓...↓ cyclichasvanishingenergyduetothecyclicpropertyofthetrace,itcorrespondstoa zeromodeplane-wavestringexcitationα0|0 .Two-magnonexcitationsarethe rst stringyexcitationswhicharedualtotheαnα n|0 stateoftheplane-wavestringintheBMNlimit.

Theintegrabilityofthespin-chainamountstotheexistenceofL 1higherchargesQkwhichcommutewiththeHamiltonian(aliasdilatationoperator)andamongstthemselves,i.e.[Qk,Ql]=0.Explicitlythe rsttwochargesoftheHeisen-bergchainaregivenby

Q2:=HXXX1/2Q3=L i=1( σi× σi+1)· σi+2.(50)

TheexplicitformofallthehigherQkmaybefoundin[67].NotethatQkwillinvolveuptokneighboringspininteractions.

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

4.1ThecoordinateBetheansatz

WenowdiscusstheansatzthatenabledBethetodiagonalizetheHeisenbergmodelin1931[68]6.Forthiswewillforthemomentdropthecyclicityconstraintimposedonusfromtheunderlyingtracestructureofthegaugetheoryoperatorsandtreatageneralnon-cyclic,butperiodic,spinchain.ThevacuumstateoftheHeisenbergchainisthengivenby|↓...↓ .Let|x1,x2,...xJ withx1<x2<...<xJdenoteastateofthechainwithup-spins(magnons)locatedatsitesxi,i.e.|1,3,4 L=5=|↓↑↓↓↑ .Itisusefultothinkofthesespin ipsasparticleslocatedatthesitesxi.NotethattheHamiltonianpreservesthemagnonorparticlenumber.

TheonemagnonsectoristhentriviallydiagonalizedbyFouriertransformation

|ψ(p1) :=L x=1eip1x|x ,withQ2|ψ(p1) =4sin2(p1

withk∈Z.

Nextconsiderageneraltwo-magnonstateoftheform

|ψ(p1,p2) =ψ(x1,x2)|x1,x2 .L1≤x1<x2≤L(53)

withatwo-particlewave-functionψ(x1,x2).The“positionspace”Schr¨odingerequa- Ltionfollowingfromi=1(1 Pi,i+1)|ψ(p1,p2) =E2|ψ(p1,p2) thenleadstotwosetsofequations,dependingonwhethertheparticleslienexttoeachotherornot:x2>x1+1E2ψ(x1,x2)=2ψ(x1,x2) ψ(x1 1,x2) ψ(x1+1,x2)

+2ψ(x1,x2) ψ(x1,x2 1) ψ(x1,x2+1)

E2ψ(x1,x2)=2ψ(x1,x2) ψ(x1 1,x2) ψ(x1,x2 1).x2=x1+1(54)(55)

E2istheeigenvalueofQ2andrelatedtothegaugetheoryscalingdimensionsas =L+λ

Foraniceanddetailedreviewonthistopicsee[69].ThetechnologyofthealgebraicBetheansatzisreviewedin[70].6

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

exchangedtheirmomenta.Oneeasilyseesthat(54)is

ful lled

for

an

arbitraryS(p2,p1)yieldingtheenergyasasumofone-particleenergies

E2=4sin2(p1

2).(57)

Eq.(55)thendeterminestheS-matrixtobeoftheform

S(p1,p2)= (p1) (p2)+ipcot(2

2 i<jθP(i)P(j) (61)

wherethesumisoverallM!permutationsofthelabels{1,2,...,M}andthephaseshiftsθij= θjiarerelatedtotheS-matrix(58)by

S(pi,pj)=exp[iθij].

TheM-magnonBetheansatzthenyieldsthesetofMBetheequations

eipkL=M (62)S(pk,pi)(63)

i=1,i=k

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

withthetwo-bodyS-matrixof(58)

and

theadditiveenergyexpressionE2=M i=14sin2(pi

2

2+i=eip i eip(L 1)=1

p=2πnL→∞

L 1

dimension (1)=λ→8π8π22n2

L2

J p=0yieldstheone-loopscaling)ofthetwo-magnonoperators[71,19] 2p+1cosπn(J,2)On=

1+n2λ/J2[5].

Hencefromtheviewpointofthespinchaintheplane-wavelimitcorrespondstoachainofdiverginglengthL>>1carryinga nitenumberofmagnonsM,whicharenothingbutthegaugedualsoftheoscillatorexcitationsoftheplane-wavesuperstring.

4.2Thethermodynamiclimitofthespin-chain

Inordertomakecontactwiththespinningstringsolutiondiscussedinsection3wewillnowconsiderthethermodynamiclimitofthespinchaininwhichthelengthLandthenumberofmagnonsMbecomelarge.ThisisnecessaryastheclassicalstringsolutionsonlylimittothetruequantumresultintheBMNtype

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