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