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

5

2

g

Au

5

2

v

6

3

1

7

5

/h

t

-

p

e

:hv

i

X

r

aSpinningstringsandintegrablespinchainsintheAdS/CFTcorrespondenceJanPlefkaMax-Planck-Institutf¨urGravitationsphysik(Albert-Einstein-Institut)AmM¨uhlenberg1,14476Potsdam,Germanyjan.plefka@aei.mpg.deAEI-2005-124,hep-th/0507136AbstractInthisintroductoryreviewwediscussdynamicaltestsoftheAdS5S5string/N=4superYang-Millsduality.Afterabriefintroductionto×AdS/CFTwearguethatsemiclassicalstringenergiesyieldinformationonthequantumspectrumofthestringinthelimitoflargeangularmomentaontheS5.TheenergiesofthefoldedandcircularspinningstringsolutionsrotatingonaS3withintheS5arederived,whichyieldalllooppredictionsforthedualgaugetheoryscalingdimensions.ThesefollowfromtheeigenvaluesofthedilatationoperatorofN=4superYang-MillsinaminimalSU(2)subsectorandwedisplayitsreformulationintermsofaHeisenbergs=1/2spinchainalongwiththecoordinateBetheansatzforitsexplicitdiagonal-ization.Inordertomakecontacttothespinningstringenergieswethen

studythethermodynamiclimitoftheone-loopgaugetheoryBetheequationsanddemonstratethematchingwiththefoldedandclosedstringresultatthislooporder.Finallytheknowngaugetheoryresultsathigher-loopordersarereviewedandtheassociatedlong-rangespinchainBetheansatzisintro-duced,leadingtoanasymptoticall-loopconjectureforthegaugetheoryBetheequations.Thisuncoversdiscrepanciesatthethree-looporderbetweengaugetheoryscalingdimensionsandstringtheoryenergiesandtheimplicationsofthisarediscussed.Alongthewaywecommentonfurtherdevelopmentsandgeneralizationsofthesubjectandpointtotherelevantliterature.