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

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Figure3:Theone-loopenergiesofthefolded(dark)andcircular(light)stringsolutionsplottedagainstthe llingfractionJ2/J.ThedashedcurveisthemirroredfoldedstringsolutionwhereoneinterchangesJ1 J2.

stringwereconstructedbyreducingthesystemtotheso-calledNeumannintegrablesystemthroughasuitableansatz.Thesolutionsof[43,44]involvenonvanishingvaluesforallspinsandangularmomenta(S1,S2,J1,J2,J3).However,theydonotgenericallydisplayananalyticbehaviorinthee ectivecouplingconstantλ′.Thisistrueonlyforcon gurationswithatleastonelargechargeJiontheS5.

InthecontextoftheO(p,q)σ-modelsintegrabilityisbasedontheexistenceofaLaxpair,afamilyof atconnectionsonthe2dstringworldsheet,givingrisetoanin nitenumberofconservedcharges.Thesewere rstdiscussedinthecontextofthebosonicstringin[45]andforthefullsuperstringin[21].TheLaxpairforthestringwasputtousein[46]forstringcon gurationsonR×S3–thesectorwealsoconsideredintheabove.Theseinvestigationsledtotheconstructionofanunderlyingalgebraiccurveparametrizingthesolutions.Thisenabledtheauthorsof[46]towritedownanintegralequationofBethetypeyieldingtheassociatedenergiesofthesolutions.Verysimilarequationswillappearbelowinourdiscussioninsection4.2onthethermodynamiclimitofthegaugetheoryBetheequations.Theextractionoftheseintegralequationsfromthestringσ-modelthenallowsforadirectcomparisontothegaugetheoryBetheequations.Onthislevelofformalization,thereisnoneedtocompareexplicitsolutionsanylonger–aswearedoinghereforpedagogicalpurposes.ThisconstructionbasedonanunderlyingalgebraiccurvemakesfulluseofthetechnologyofintegrablesystemsandhasbeennicelyreviewedbyZaremboin[25].

Intheveryinterestingpaper[47]thesecontinuumstringBetheequationswereboldlydiscretizedleadingtoaconjecturedsetofBetheequationsforthequantumspectrumofthestring.Thisproposalhasbeenveri edbycomparingittoknown