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

4.1ThecoordinateBetheansatz

WenowdiscusstheansatzthatenabledBethetodiagonalizetheHeisenbergmodelin1931[68]6.Forthiswewillforthemomentdropthecyclicityconstraintimposedonusfromtheunderlyingtracestructureofthegaugetheoryoperatorsandtreatageneralnon-cyclic,butperiodic,spinchain.ThevacuumstateoftheHeisenbergchainisthengivenby|↓...↓ .Let|x1,x2,...xJ withx1<x2<...<xJdenoteastateofthechainwithup-spins(magnons)locatedatsitesxi,i.e.|1,3,4 L=5=|↓↑↓↓↑ .Itisusefultothinkofthesespin ipsasparticleslocatedatthesitesxi.NotethattheHamiltonianpreservesthemagnonorparticlenumber.

TheonemagnonsectoristhentriviallydiagonalizedbyFouriertransformation

|ψ(p1) :=L x=1eip1x|x ,withQ2|ψ(p1) =4sin2(p1

withk∈Z.

Nextconsiderageneraltwo-magnonstateoftheform

|ψ(p1,p2) =ψ(x1,x2)|x1,x2 .L1≤x1<x2≤L(53)

withatwo-particlewave-functionψ(x1,x2).The“positionspace”Schr¨odingerequa- Ltionfollowingfromi=1(1 Pi,i+1)|ψ(p1,p2) =E2|ψ(p1,p2) thenleadstotwosetsofequations,dependingonwhethertheparticleslienexttoeachotherornot:x2>x1+1E2ψ(x1,x2)=2ψ(x1,x2) ψ(x1 1,x2) ψ(x1+1,x2)

+2ψ(x1,x2) ψ(x1,x2 1) ψ(x1,x2+1)

E2ψ(x1,x2)=2ψ(x1,x2) ψ(x1 1,x2) ψ(x1,x2 1).x2=x1+1(54)(55)

E2istheeigenvalueofQ2andrelatedtothegaugetheoryscalingdimensionsas =L+λ

Foraniceanddetailedreviewonthistopicsee[69].ThetechnologyofthealgebraicBetheansatzisreviewedin[70].6