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