Two-loop world-sheet corrections in AdS5 x S5 superstring

We initiate the computation of the 2-loop quantum AdS5 x S5 string corrections on the example of a certain string configuration in S5 related by an analytic continuation to a folded rotating string in AdS5 in the ''long string'' limit. The 2-loop term in the energy of the latter should represent the subleading strong-coupling correction to the cusp anomalous dimension and thus provide a further check of recent conjectures about the exact structure of the Bethe ansatz underlying the AdS/CFT duality. We use the conformal gauge and several choices of the κ-symmetry gauge. We present partial results: we compute the bosonic contribution to the effective action and also determine the transcendental form of the fermionic contribution but do not succeed in verifying the cancellation of all logarithmic divergences. The main obstacle appears to be an apparent non-renormalizability of the GS superstring (when expanded near a non-trivial background) action which first manifests itself at the 2-loop order and leads to subtleties in dealing with power divergent integrals