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We demonstrate that the recently found agreement between one-loop scalingdimensions of large dimension operators in N=4 gauge theory and energies ofspinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinitenumber of hidden higher commuting charges. This dynamical agreement is of amathematically highly intricate and non-trivial nature. In particular, on thegauge side the generating function for the commuting charges is obtained byintegrable quantum spin chain techniques from the thermodynamic densitydistribution function of Bethe roots. On the string side the generatingfunction, containing information to arbitrary loop order, is constructed bysolving exactly the Backlund equations of the integrable classical string sigmamodel. Our finding should be an important step towards matching the integrablestructures on the string and gauge side of the AdS/CFT correspondence.Comment: Latex, 33 pages, v2: new section added (completing the analytic proof  that the entire infinite towers of commuting gauge and string charges match);  references adde

Matching Higher Conserved Charges for Strings and Spins