Cutting the cylinder into squares: The square form factor

In this article we present a method for constructing two-point functions in the spirit of the hexagon proposal, which leads us to propose a "square form factor". Since cutting the square gives us two squares, we can write a consistency condition that heavily constrains such form factors. In particular, we are able to use this constraint to reconstruct the Gaudin through the forest expansion of the determinant appearing in its definition. We also use this procedure to compute the norm of off-shell Bethe states for some simple cases.Comment: 34 pages, 10 figure

Spinning strings in the η\eta-deformed Neumann-Rosochatius system

The sigma-model of closed strings spinning in the $\eta$-deformation of $AdS_{5} \times S^{5}$ leads to an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical system. In this article we construct general solutions to this system that can be written in terms of elliptic functions. The solutions correspond to closed strings with non-constant radii rotating with two different angular momenta in an $\eta$-deformed three-sphere. We analyse the reduction of the elliptic solutions for some limiting values of the deformation parameter. For the case of solutions with constant radii we find the dependence of the classical energy of the string on the angular momenta as an expansion in the 't Hooft coupling.Comment: 17 pages. Latex. v2: Additional references. v3: Minor changes and updated reference

Holographic correlation functions of hexagon Wilson loops with one local operator

We consider the ratio of the correlation function of an hexagon light-like Wilson loop with one local operator over the expectation value of the Wilson loop within the strong-coupling regime of the AdS/CFT correspondence. We choose the hexagon Wilson loop within a class of minimal solutions obtained by cutting and gluing light-like quadrangle loops. These surfaces do not have an interpretation in terms of dual scattering amplitudes but they still exhibit general features of the mixed correlation function. In the case of a regular null hexagon conformal symmetry constrains the space-time dependence of the correlator up to a function of three conformal cross-ratios. We obtain the leading-order contribution to the correlation function in the semiclassical approximation of large string tension, and express the result in terms of three conformal ratios in the case where the local operator is taken to be the dilaton. We include the analysis of an irregular Wilson loop obtained after a boost of the regular hexagon.Comment: 12 pages. Latex. v2: Reference added. v3: Added clarifications, published versio

Correlation functions and the algebraic Bethe ansatz in the AdS/CFT correspondence

Inverse scattering and the algebraic Bethe ansatz can be used to reduce the evaluation of form factors and correlation functions to the calculation of a product of Bethe states. In this article we develop a method to compute correlation functions of spin operators located at arbitrary sites of the spin chain. We will focus our analysis on the SU(2) sector of N=4 supersymmetric Yang-Mills at weak-coupling. At one-loop we provide a systematic treatment of the apparent divergences arising from the algebra of the elements of the monodromy matrix of an homogeneous spin chain. Beyond one-loop the analysis can be extended through the map of the long-range Bethe ansatz to an inhomogeneous spin chain. We also show that a careful normalization of states in the spin chain requires choosing them as Zamolodchikov-Faddeev states.Comment: 41 pages. Late

Elliptic solutions in the Neumann-Rosochatius system with mixed flux

Closed strings spinning in AdS_3 x S^3 x T^4 with mixed R-R and NS-NS three-form fluxes are described by a deformation of the one-dimensional Neumann-Rosochatius integrable system. In this article we find general solutions to this system that can be expressed in terms of elliptic functions. We consider closed strings rotating either in S^3 with two different angular momenta or in AdS_3 with one spin. In order to find the solutions we will need to extend the Uhlenbeck integrals of motion of the Neumann-Rosochatius system to include the contribution from the flux. In the limit of pure NS-NS flux, where the problem can be described by a supersymmetric WZW model, we find exact expressions for the classical energy in terms of the spin and the angular momenta of the spinning string.Comment: 18 pages. Latex. v2: Extended discussions, corrected misprints and added reference. Published versio

Spinning strings in AdS_3 x S^3 with NS-NS flux

The sigma model describing closed strings rotating in AdS_3 x S^3 is known to reduce to the one-dimensional Neumann-Rosochatius integrable system. In this article we show that closed spinning strings in AdS_3 x S^3 x T^4 in the presence of NS-NS three-form flux can be described by an extension of the Neumann-Rosochatius system. We consider closed strings rotating with one spin in AdS_3 and two different angular momenta in S^3. For a class of solutions with constant radii we find the dependence of the classical energy on the spin and the angular momenta as an expansion in the square of the 't Hooft coupling of the theory.Comment: 14 pages. Latex. v2: Equations (3.19) and (3.28) corrected and reference added. v3: Expanded discussion on the WZW limit and additional references. Published version. v4: Misprints correcte

One-loop quantization of rigid spinning strings in AdS3×S3×T4AdS_3 \times S^3 \times T^4 with mixed flux

We compute the one-loop correction to the classical dispersion relation of rigid closed spinning strings with two equal angular momenta in the $AdS_3 \times S^3 \times T^4$ background supported with a mixture of R-R and NS-NS three-form fluxes. This analysis is extended to the case of two arbitrary angular momenta in the pure NS-NS limit. We perform this computation by means of two different methods. The first method relies on the Euler-Lagrange equations for the quadratic fluctuations around the classical solution, while the second one exploits the underlying integrability of the problem through the finite-gap equations. We find that the one-loop correction vanishes in the pure NS-NS limit.Comment: 35 pages. v2: Minor changes and references updated. v3: Published versio

Bayesian non-linear matching of pairwise microarray gene expressions

In this paper, we present a Bayesian non-linear model to analyze matching pairs of microarray expression data. This model generalizes, in terms of neural networks, standard linear matching models. As a practical application, we analyze data of patients with Acute Lymphoblastic Leukemia and we find out the best neural net model that relates the expression levels of two types of cytogenetically different samples from them

Spatial matching of M configurations of points with a bioinformatics application

In this paper, we present a model to deal with the problem of matching M objects or configurations of points. This is a generalization of the model proposed by Green and Mardia (2006). We consider, as a direct and simple application, the case of three configurations with labelled and with unlabelled points. In both cases, we consider data from a microarray experiment of gorilla, bonobo and human cultured fibroblasts published by Karaman et al. (2003). We find out the matchings and the best affine transformation between the projections of genes in a two dimensional space, obtained by a Multidimensional Scaling technique