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

constructionofspinningstringsolutions.

Thisreviewaimsatamoreelementaryintroductiontothisveryactiveareaofresearch,whichinprincipleholdsthepromiseof ndingtheexactquantumspectrumoftheAdS5×S5stringorequivalentlytheallloopscalingdimensionsofplanarN=4superYang-Mills.Itisintendedasa rstguidetothe eldforstudentsandinterested‘newcomers’andpointstotherelevantliteraturefordeeperstudies.WewilldiscussthesimplestsolutionsoftheAdS5×S5stringcorrespondingtofoldedandcircularstringcon gurationspropagatinginaR×S3subspace,withtheS3lyingwithintheS5.OnthegaugetheorysidewewillmotivatetheemergenceofthespinchainpictureattheleadingonelooporderanddiscusstheemergingHeisenbergXXX1/2modelanditsdiagonalizationusingthecoordinateBetheansatztechnique.Thisthenenablesustoperformacomparisonbetweentheclassicalstringpredictionsinthelimitoflargeangularmomentaandthedualthermodynamiclimitofthespinchainspectrum.Finallyweturntohigher-loopcalculationsinthegauge-theoryanddiscussconjecturesfortheall-loopformoftheBetheequations,parisonwiththeobtainedstringresultsuncoversadiscrepancyfromlooporderthreeonwardsandtheinterpretationofthisresultisalsodiscussed.

Therealreadyexistanumberofmoredetailedreviewsonspinningstrings,in-tegrabilityandspinchainsintheAdS/CFTcorrespondence:Tseytlin’sreview[22]mostlyfocusesonthestringsidethecorrespondence,whereasBeisert’sPhysicsRe-port[23]concentratesprimarilyonthegaugeside.SeealsoTseytlin’ssecondreview

[24],ontheso-calledcoherent-statee ectiveactionapproach,whichwewillnotdis-cussinthisreview.RecommendedisalsotheshorterreviewbyZarembo[25]ontheSU(2)respectivelyR×S3subsector,discussingtheintegrablestructureappearingontheclassicalstring–notcoveredinthisreview.Foradetailedaccountofthenearplane-wavesuperstring,itsquantumspectroscopyandintegrabilitystructuresseeSwanson’sthesis[26].

2Thesetup

wherewehavesuppressedthefermionictermsintheaction,astheywillnotberelevantinourdiscussionofclassicalsolutions(thefullfermionicactionisstatedin[27,28]).AnaturalchoiceofcoordinatesfortheAdS5×S5space(“global

WiththeembeddingcoordinatesXm(τ,σ)andYm(τ,σ)thePolyakovactionoftheAdS5×S5stringinconformalgauge(ηab=diag( 1,1))takestheform(m,n=1,2,3,4,5)√ (AdS5)man(S5)manI= dτdσGmn aX X+Gmn aY Y+fermions,(4)4π