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