Generalized scaling function from light-cone gauge AdS 5 × S 5 superstring

We revisit the computation of the 2-loop correction to the energy of a folded spinning string in AdS ₅ with an angular momentum J in S ⁵ in the scaling limit ln S ≫ 1, J/{sqrt {λ ln S}} = {text{fixed}} . This correction gives the third term in the strong-coupling expansion of the generalized scaling function. The computation, using the AdS light-cone gauge approach developed in our previous paper, is done by expanding the AdS ₅ × S ⁵ superstring partition function near the generalized null cusp world surface associated to the spinning string solution. The result corrects and extends the previous conformal gauge result of arXiv:0712.2479 and is found to be in complete agreement with the corresponding terms in the generalized scaling function as obtained from the asymptotic Bethe ansatz in arXiv:0805.4615 (and also partially from the quantum O(6) model and the Bethe ansatz data in arXiv:0809.4952). This provides a highly nontrivial strong coupling comparison of the Bethe ansatz proposal with the quantum AdS ₅ × S ⁵ superstring theory, which goes beyond the leading semiclassical term effectively controlled by the underlying algebraic curve. The 2-loop computation we perform involves all the structures in the AdS light-cone gauge superstring action of hep-th/0009171 and thus tests its ultraviolet finiteness and, through the agreement with the Bethe ansatz, its quantum integrability. We do most of the computations for a generalized spinning string solution or the corresponding null cusp surface that involves both the orbital momentum and the winding in a large circle of S ⁵.