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.