On correlation functions of operators dual to classical spinning string states

We explore how to compute, classically at strong coupling, correlation functions of local operators corresponding to classical spinning string states. The picture we obtain is of `fattened' Witten diagrams, the evaluation of which turns out to be surprisingly subtle and requires a modification of the naive classical action due to a necessary projection onto appropriate wave functions. We examine string solutions which compute the simplest case of a two-point function and reproduce the right scaling with the anomalous dimensions corresponding to the energies of the associated spinning string solutions. We also describe, under some simplifying assumptions, how the spacetime dependence of a conformal three-point correlation function arises in this setup.Comment: 27 pages, 3 figures; v2: references and comments added

Wave functions and correlation functions for GKP strings from integrability

We develop a general method of computing the contribution of the vertex operators to the semi-classical correlation functions of heavy string states, based on the state-operator correspondence and the integrable structure of the system. Our method requires only the knowledge of the local behavior of the saddle point configuration around each vertex insertion point and can be applied to cases where the precise forms of the vertex operators are not known. As an important application, we compute the contributions of the vertex operators to the three-point functions of the large spin limit of the Gubser-Klebanov-Polyakov (GKP) strings in $AdS_3$ spacetime, left unevaluated in our previous work [arXiv:1110.3949] which initiated such a study. Combining with the finite part of the action already computed previously and with the newly evaluated divergent part of the action, we obtain finite three-point functions with the expected dependence of the target space boundary coordinates on the dilatation charge and the spin.Comment: 80 pages, 7 figures, v2: typos and minor errors corrected, a reference added, v3: typos and a reference corrected, published versio

Boundary operators in minimal Liouville gravity and matrix models

We interpret the matrix boundaries of the one matrix model (1MM) recently constructed by two of the authors as an outcome of a relation among FZZT branes. In the double scaling limit, the 1MM is described by the (2,2p+1) minimal Liouville gravity. These matrix operators are shown to create a boundary with matter boundary conditions given by the Cardy states. We also demonstrate a recursion relation among the matrix disc correlator with two different boundaries. This construction is then extended to the two matrix model and the disc correlator with two boundaries is compared with the Liouville boundary two point functions. In addition, the realization within the matrix model of several symmetries among FZZT branes is discussed.Comment: 26 page

Holographic three-point functions of semiclassical states

We calculate the holographic three-point functions in N = 4 super-Yang-Mills theory in the case when two of the operators are semiclassical and one is dual to a supergravity mode. We further discuss the transition to the regime when all three operators are semiclassical.Comment: 17 pages, 3 figures; v2: refs. added, discussion in sec. 2.1 expanded; v3: misprint in (2.28) corrected, published versio

Correlation functions of three heavy operators - the AdS contribution

We consider operators in N=4 SYM theory which are dual, at strong coupling, to classical strings rotating in S^5. Three point correlation functions of such operators factorize into a universal contribution coming from the AdS part of the string sigma model and a state-dependent S^5 contribution. Consequently a similar factorization arises for the OPE coefficients. In this paper we evaluate the AdS universal factor of the OPE coefficients which is explicitly expressed just in terms of the anomalous dimensions of the three operators.Comment: 49 pages, 3 figures; v.2 references corrected; v3: corrected discussion in section 5, results unchange

Superconformal Yang-Mills quantum mechanics and Calogero model with OSp(N|2,R) symmetry

In spacetime dimension two, pure Yang-Mills possesses no physical degrees of freedom, and consequently it admits a supersymmetric extension to couple to an arbitrary number, N say, of Majorana-Weyl gauginos. This results in (N,0) super Yang-Mills. Further, its dimensional reduction to mechanics doubles the number of supersymmetries, from N to N+N, to include conformal supercharges, and leads to a superconformal Yang-Mills quantum mechanics with symmetry group OSp(N|2,R). We comment on its connection to AdS_2 \times S^{N-1} and reduction to a supersymmetric Calogero model.Comment: 1+28 pages, no figure; Refs added. To appear in JHE

Spinning strings and integrable spin chains in the AdS/CFT correspondence

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the limit of large angular momenta on the S^5. The energies of the folded and circular spinning string solutions rotating on a S^3 within the S^5 are derived, which yield all loop predictions for the dual gauge theory scaling dimensions. These follow from the eigenvalues of the dilatation operator of N=4 super Yang-Mills in a minimal SU(2) subsector and we display its reformulation in terms of a Heisenberg s=1/2 spin chain along with the coordinate Bethe ansatz for its explicit diagonalization. In order to make contact to the spinning string energies we then study the thermodynamic limit of the one-loop gauge theory Bethe equations and demonstrate the matching with the folded and closed string result at this loop order. Finally the known gauge theory results at higher-loop orders are reviewed and the associated long-range spin chain Bethe ansatz is introduced, leading to an asymptotic all-loop conjecture for the gauge theory Bethe equations. This uncovers discrepancies at the three-loop order between gauge theory scaling dimensions and string theory energies and the implications of this are discussed. Along the way we comment on further developments and generalizations of the subject and point to the relevant literature.Comment: 40 pages, invited contribution to Living Reviews in Relativity. v2: improvements in the text and references adde