On the Fermionic Frequencies of Circular Strings

We revisit the semiclassical computation of the fluctuation spectrum around different circular string solutions in AdS_5xS^5 and AdS_4xCP^3, starting from the Green-Schwarz action. It has been known that the results for these frequencies obtained from the algebraic curve and from the worldsheet computations sometimes do not agree. In particular, different methods give different results for the half-integer shifts in the mode numbers of the frequencies. We find that these discrepancies can be removed if one carefully takes into account the transition matrices in the spin bundle over the target space.Comment: 13 pages, 1 figur

Three-point function of semiclassical states at weak coupling

We give the derivation of the previously announced analytic expression for the correlation function of three heavy non-BPS operators in N=4 super-Yang-Mills theory at weak coupling. The three operators belong to three different su(2) sectors and are dual to three classical strings moving on the sphere. Our computation is based on the reformulation of the problem in terms of the Bethe Ansatz for periodic XXX spin-1/2 chains. In these terms the three operators are described by long-wave-length excitations over the ferromagnetic vacuum, for which the number of the overturned spins is a finite fraction of the length of the chain, and the classical limit is known as the Sutherland limit. Technically our main result is a factorized operator expression for the scalar product of two Bethe states. The derivation is based on a fermionic representation of Slavnov's determinant formula, and a subsequent bosonisation.Comment: 28 pages, 5 figures, cosmetic changes and more typos corrected in v

Foundations of the AdS_5 x S^5 Superstring. Part I

We review the recent advances towards finding the spectrum of the AdS_5 x S^5 superstring. We thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem. In certain cases our exposition is original and cannot be found in the existing literature. The present Part I deals with foundations of classical string theory in AdS_5 x S^5, light-cone perturbative quantization and derivation of the exact light-cone world-sheet scattering matrix.Comment: 161 page

Asymptotic Bethe Ansatz S-matrix and Landau-Lifshitz type effective 2-d actions

Motivated by the desire to relate Bethe ansatz equations for anomalous dimensions found on the gauge theory side of the AdS/CFT correspondence to superstring theory on AdS_5 x S5 we explore a connection between the asymptotic S-matrix that enters the Bethe ansatz and an effective two-dimensional quantum field theory. The latter generalizes the standard ``non-relativistic'' Landau-Lifshitz (LL) model describing low-energy modes of ferromagnetic Heisenberg spin chain and should be related to a limit of superstring effective action. We find the exact form of the quartic interaction terms in the generalized LL type action whose quantum S-matrix matches the low-energy limit of the asymptotic S-matrix of the spin chain of Beisert, Dippel and Staudacher (BDS). This generalises to all orders in the `t Hooft coupling an earlier computation of Klose and Zarembo of the S-matrix of the standard LL model. We also consider a generalization to the case when the spin chain S-matrix contains an extra ``string'' phase and determine the exact form of the LL 4-vertex corresponding to the low-energy limit of the ansatz of Arutyunov, Frolov and Staudacher (AFS). We explain the relation between the resulting ``non-relativistic'' non-local action and the second-derivative string sigma model. We comment on modifications introduced by strong-coupling corrections to the AFS phase. We mostly discuss the SU(2) sector but also present generalizations to the SL(2) and SU(1|1) sectors, confirming universality of the dressing phase contribution by matching the low-energy limit of the AFS-type spin chain S-matrix with tree-level string-theory S-matrix.Comment: 52 pages, 4 figures, Imperial-TP-AT-6-2; v2: new sections 7.3 and 7.4 computing string tree-level S-matrix in SL(2) and SU(1|1) sectors, references adde

Interacting finite-size magnons

We explicitly construct a large class of finite-volume two-magnon string solutions moving on R x S^2. In particular, by making use of the relationship between the O(3) sigma model and sine-Gordon theory we are able to find solutions corresponding to the periodic analogues of magnon scattering and breather-like solutions. After semi-classically quantizing these solutions we invert the implicit expressions for the excitation energies in certain limits and find the corrections for the multi-magnon states. For the breather-like solutions we express the energies directly in terms of the action variable whereas for the scattering solution we express the result as a combination of corrections to the dispersion relation and to the scattering phase.Comment: 45pages, 7figures, v2: simplified expressions for periods and angular momenta when elliptic modulus is greater than one, references added, typos correcte

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010