Spinning strings and integrable spin chains in the AdSCFT co(8)

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

isstraightforwardinthelight-conegauge[6,7]andleadstoafree,massivetwodimensionaltheoryforthetransversedegreesoffreedom(i=1,...,8)

I2= dτdσ(1

2xixi+fermions)(10)

withacompactexpressionforthespectrum

∞

E12=λ′2(11)

n √= ∞√J

whereN

matching n:=α condition withniαi[αnistheexcitationnumberoperatorfortransversestringexcita-tionsα ni|0im ,α nj]=δnmδij.TheVirasoroconstraints(6)reducetothelevel

nnN n=0knownfromstringtheoryin atMinkowskispace-

time.Hencethe rststringyexcitationisα

Foramoredetailedtreatmentoftheplanewavenα superstringn|0 with√1+λ′n2.

see[11,12,13,14].

2.2N=4SuperYang-Mills

TheconjectureddualgaugetheoryoftheAdS5×S5superstringisthemaximallysupersymmetric(N=4)Yang-Millstheoryinfourdimensions[31,32].Its eldcontentisgivenbyagluon eldAµ(x),sixscalarsφi(x)(i=1,...,6)aswellas4Majoranagluinos,whichwechoosetowriteasa16component10dMajorana-Weylspinorχα(x)(α=1,...16).All eldsareintheadjointrepresentationofSU(N).TheactionofN=4SuperYang-Millsisuniquelydeterminedbytwoparameters,thecouplingconstantgYMandtherankofthegaugegroupSU(N)

S=2

4(F2+1

4[φ[φ1µν)i,φj]i,φj]+2χ¯Γi[φi,χ]

(12)

withthecovariantderivativeDµ= µ

Diracmatrices. i[Aµ,].Furthermore,(Γµ,Γi)aretheten

dimensional

Duetothelargeamountofsupersymmetrypresent,theconformalinvarianceoftheclassical eldtheorysurvivesthequantizationprocedure:ThecouplingconstantgYMisnotrenormalizedanditsβ-functionvanishestoallordersinperturbationtheory[33,34,35].ThisiswhyoneoftenreferstoN=4SuperYang-Millsasa“ nite”quantum eldtheory.Neverthelesscompositegaugeinvariantoperators,i.e.tracesofproductsoffundamental eldsandtheircovariantderivativesatthesamespace-point,e.g.Oi1...ik(x)=Tr[φi1(x)φi2(x)...φik(x)],arerenormalizedandacquireanomalousdimensions.Thesemaybereado fromthetwopointfunctions(statedhereforthecaseofscalaroperators)

OA(x)OB(y) =δAB