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.