  Integrability occupies an increasingly important role in direct tests of the
AdS/CFT correspondence. Integrable structures have appeared in both planar N=4
super Yang-Mills theory and type IIB superstring theory on AdS_5 x S^5. A
generalized statement of the AdS/CFT conjecture has therefore emerged in which,
in addition to string energies corresponding to gauge theory anomalous
dimensions, an infinite tower of higher charges on each side of the duality
should also be equated. Demonstrations of this larger equivalence have been
successful in certain regimes. To test this correspondence in a more stringent
setting, the bosonic sector of the fully quantized string theory on AdS_5 x S^5
is expanded about the pp-wave limit to sextic order in fields, or to O(1/J^2),
where J is the (large) angular momentum of string states boosted along an
equatorial geodesic in the S^5 subspace. To avoid issues of renormalization,
the analysis is restricted to zeroth order in the modified 't Hooft coupling
where consistency conditions demand that integrability be realized. The string
theory, however, fails to meet these conditions. This signals a potential
problem with higher-order corrections in the large-J expansion around the
pp-wave limit.
