Integrability in AdS/CFT correspondence: quasi-classical analysis

Abstract

In this review, we consider a quasi-classical method applicable to integrable field theories which is based on a classical integrable structure—the algebraic curve. We apply it to the Green–Schwarz superstring on the AdS5 × S5 space. We show that the proposed method reproduces perfectly the earlier results obtained by expanding the string action for some simple classical solutions. The construction is explicitly covariant and is not based on a particular parameterization of the fields and as a result is free from ambiguities. On the other hand, the finite size corrections in some particularly important scaling limit are studied in this paper for a system of Bethe equations. For the general superalgebra , the result for the 1/L corrections is obtained. We find an integral equation which describes these corrections in a closed form. As an application, we consider the conjectured Beisert–Staudacher (BS) equations with the Hernandez–Lopez dressing factor where the finite size corrections should reproduce quasi-classical results around a general classical solution. Indeed, we show that our integral equation can be interpreted as a sum of all physical fluctuations and thus prove the complete one-loop consistency of the BS equations. We demonstrate that any local conserved charge (including the AdS energy) computed from the BS equations is indeed given at one loop by the sum of the charges of fluctuations with an exponential precision for large S5 angular momentum of the string. As an independent result, the BS equations in an sub-sector were derived from Zamolodchikovs's S-matrix. The paper is based on the author's PhD thesis.