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

quantumdataoftheAdS5×S5string:Thenearplane-wavespectrumofthesu-perstringof[48,49,50,51]aswellastheexpected[52]genericscalingofthestringenergieswithλ1/4inthestrongcouplinglimitagreewiththepredictionsofthequan-tumstringBetheequations.ButthereismorequantumdatafortheAdS5×S5stringavailable:InaseriesofpapersbyTseytlin,Frolovandcollaboratorsone-loopcorrectionsonthestringworldsheettotheenergiesofvariousspinningstringsolu-tionshavebeencomputed[16,53,54].TheoneloopcorrectionforacircularstringmovinginAdS3×S1 AdS5×S5obtainedin[54]wasrecentlycompared[55]totheresultobtainedfromtheproposedquantumstringBetheequationsof[47].Theauthorsof[55] ndagreementwhentheyexpandtheresultsinλ′(uptothirdorder),butdisagreementsemergeindi erentlimits(whereλ′isnotsmall).Theinterpretationofthisresultisunclearatpresent.FinallytheproposedquantumstringBetheequationsof[47]canalsobemicroscopicallyattributedtoas=1/2spinchainmodelwithlong-rangeinteractionsupto(atleast)order veinasmallλexpansion[56].

ThetechnicallyinvolvedconstructionofalgebraiccurvessolvingtheclassicalR×S3stringσ-modelhassubsequentlybeengeneralizedtolargersectors:In[57]toR×S5(orSO(6)ingaugetheorylanguage)con gurations,in[58]toAdS3×S1(orSL(2))stringcon gurationsand nallyin[59]tosuperstringspropagatinginthefullAdS5×S5space.

TherehasalsobeenprogressonanumberofpossiblepathstowardsthetruequantizationoftheclassicalintegrablemodeloftheAdS5×S5stringintheworks

[60,61,62,63,64,65],however,itisfairtosaythatthisproblemremainscurrentlyunsolved.

4Thedualgaugetheoryside

Letusnowturntotheidenti cationofthefoldedandcircularstringsolutionsinthedualgaugetheory.

OuraimistoreproducetheobtainedenergyfunctionsE1(J1,J2)plottedin gure3fromadualgaugetheorycomputationatone-loop.Forthisweneedtoidentifythegaugetheoryoperators,whicharedualtothespinningstringsonR×S3.AshereJ2=0=S1=S2therelevantoperatorswillbebuiltfromthetwocomplexscalarsZ:=φ1+iφ2andW:=φ3+iφ4withatotalnumberofJ1Z- eldsandJ2W- elds,i.e.J1,J2Oα=Tr[ZJ1WJ2]+...,(43)wherethedotsdenotesuitablepermutationsoftheZandWtobediscussed.Anoperatoroftheform(43)maybepicturedasaringofblack(“Z”)andred(“W”)beads–orequivalentlyasacon gurationofans=1/2quantumspinchain,where