Semiclassical short strings in AdS_5 x S^5

We present results for the one-loop correction to the energy of a class of string solutions in AdS_5 x S^5 in the short string limit. The computation is based on the observation that, as for rigid spinning string elliptic solutions, the fluctuation operators can be put into the single-gap Lame' form. Our computation reveals a remarkable universality of the form of the energy of short semiclassical strings. This may help to understand better the structure of the strong coupling expansion of the anomalous dimensions of dual gauge theory operators.Comment: 12 pages, one pdf figure. Invited Talk at 'Nonlinear Physics. Theory and Experiment VI', Gallipoli (Italy) - June 23 - July 3, 201

Slow-string limit and "antiferromagnetic" state in AdS/CFT

We discuss a slow-moving limit of a rigid circular equal-spin solution on R x S^3. We suggest that the solution with the winding number equal to the total spin approximates the quantum string state dual to the maximal-dimension ``antiferromagnetic'' state of the SU(2) spin chain on the gauge theory side. An expansion of the string action near this solution leads to a weakly coupled system of a sine-Gordon model and a free field. We show that a similar effective Hamiltonian appears in a certain continuum limit from the half-filled Hubbard model that was recently suggested to describe the all-order dilatation operator of the dual gauge theory in the SU(2) sector. We also discuss some other slow-string solutions with one spin component in AdS_5 and one in S^5.Comment: 32 pages, Latex v2: one footnote and references adde

Double-helix Wilson loops: case of two angular momenta

Recently, Wilson loops with the shape of a double helix have played an important role in studying large spin operators in gauge theories. They correspond to a quark and an anti-quark moving in circles on an S3 (and therefore each of them describes a helix in RxS3). In this paper we consider the case where the particles have two angular momenta on the S3. The string solution corresponding to such Wilson loop can be found using the relation to the Neumann-Rosochatius system allowing the computation of the energy and angular momenta of the configuration. The particular case of only one angular momentum is also considered. It can be thought as an analytic continuation of the rotating strings which are dual to operators in the SL(2) sector of N=4 SYM.Comment: 30 pages, 2 figures, LaTeX. v2: Small corrections, reference adde

Energy-Spin Trajectories in AdS_5 x S^5 from Semiclassical Vertex Operators

We study the relation between vertex operators in AdS_5 x S^5 and classical spinning string solutions. In the limit of large quantum numbers the treatment of vertex operators becomes semiclassical. In this regime, a given vertex operator carrying a certain set of quantum numbers defines a singular solution. We show in a number of examples that this solution coincides with the classical string solution with the same quantum numbers but written in a different two-dimensional coordinate system. The marginality condition imposed on an operator yields a relation between the energy and the other quantum numbers which is shown to coincide with that of the corresponding classical string solution. We also argue that in some cases vertex operators in AdS_5 x S^5 cannot be given by expressions similar to the ones in flat space and a more involved consideration is required.Comment: 23 pages, 1 Figur

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

Strong coupling from the Hubbard model

It was recently observed that the one dimensional half-filled Hubbard model reproduces the known part of the perturbative spectrum of planar N=4 super Yang-Mills in the SU(2) sector. Assuming that this identification is valid beyond perturbation theory, we investigate the behavior of this spectrum as the 't Hooft parameter \lambda becomes large. We show that the full dimension \Delta of the Konishi superpartner is the solution of a sixth order polynomial while \Delta for a bare dimension 5 operator is the solution of a cubic. In both cases the equations can be solved easily as a series expansion for both small and large \lambda and the equations can be inverted to express \lambda as an explicit function of \Delta. We then consider more general operators and show how \Delta depends on \lambda in the strong coupling limit. We are also able to distinguish those states in the Hubbard model which correspond to the gauge invariant operators for all values of \lambda. Finally, we compare our results with known results for strings on AdS_5\times S^5, where we find agreement for a range of R-charges.Comment: 14 pages; v2: 17 pages, 2 figures, appendix and references added; typos fixed, minor changes; v3 fixed figures; v4 more references added, minor correctio

On correlation functions of Wilson loops, local and non-local operators

We discuss and extend recent conjectures relating partial null limits of correlation functions of local gauge invariant operators and the expectation value of null polygonal Wilson loops and local gauge invariant operators. We point out that a particular partial null limit provides a strategy for the calculation of the anomalous dimension of short twist-two operators at weak and strong coupling.Comment: 29 pages, 8 figure

Exact computation of one-loop correction to energy of pulsating strings in AdS_5 x S^5

In the present paper, which is a sequel to arXiv:1001:4018, we compute the one-loop correction to the energy of pulsating string solutions in AdS_5 x S^5. We show that, as for rigid spinning string elliptic solutions, the fluctuation operators for pulsating solutions can be also put into the single-gap Lame' form. A novel aspect of pulsating solutions is that the one-loop correction to their energy is expressed in terms of the stability angles of the quadratic fluctuation operators. We explicitly study the "short string" limit of the corresponding one-loop energies, demonstrating a certain universality of the form of the energy of "small" semiclassical strings. Our results may help to shed light on the structure of strong-coupling expansion of anomalous dimensions of dual gauge theory operators.Comment: 49 pages; v2: appendix F and note about antiperiodic fermions added, typos corrected, references adde

Two-loop world-sheet corrections in AdS_5 x S^5 superstring

We initiate the computation of the 2-loop quantum AdS_5 x S^5 string corrections on the example of a certain string configuration in S^5 related by an analytic continuation to a folded rotating string in AdS_5 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 \kappa-symmetry gauge. While we are unable to verify the cancellation of 2d UV divergences we compute the bosonic contribution to the effective action and also determine the non-trivial finite part of the fermionic contribution. Both the bosonic and the fermionic contributions to the string energy happen to be proportional to the Catalan's constant. The resulting value for 2-loop superstring prediction for the subleading coefficient a_2 in the scaling function matches the numerical value found in hep-th/0611135 from the BES equation.Comment: 48 pages, 1 Figure. v3: several mistakes corrected, the finite result for the 2-loop coefficient is found to agree with the numerical value found by Benna et al in hep-th/061113

Bethe Ansatz in Stringy Sigma Models

We compute the exact S-matrix and give the Bethe ansatz solution for three sigma-models which arise as subsectors of string theory in AdS(5)xS(5): Landau-Lifshitz model (non-relativistic sigma-model on S(2)), Alday-Arutyunov-Frolov model (fermionic sigma-model with su(1|1) symmetry), and Faddeev-Reshetikhin model (string sigma-model on S(3)xR).Comment: 37 pages, 11 figure