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We study in detail the analytic properties of the Thermodynamic Bethe Ansatz (TBA) equations for the anomalous dimensions of composite operators in the planar limit of the 3D N=6 superconformal Chern-Simons gauge theory and derive functional relations for the jump discontinuities across the branch cuts in the complex rapidity plane. These relations encode the analytic structure of the Y functions and are extremely similar to the ones obtained for the previously-studied AdS(5)/CFT(4) case. Together with the Y-system and more basic analyticity conditions, they are completely equivalent to the TBA equations. We expect these results to be useful to derive alternative nonlinear integral equations for the AdS(4)/CFT(3) spectrum., Comment: 33 pages, 9 figures

In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction - a quartic polynomial in the field derivatives - is exactly the composite field $T\bar{T}$, built with the chiral components, $T$ and $\bar{T}$, of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA),the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected., Comment: 25 pages, 2 PDF figures, PDFLatex. v2: typos corrected, two references added, new comments in sections 6 and 7. v3: minor amendments, to be published in JHEP

We study the analytic properties of the $AdS_5/CFT_4$ Y functions. It is shown that the TBA equations, including the dressing factor, can be obtained from the Y-system with some additional information on the square-root discontinuities across semi-infinite segments in the complex plane. The Y-system extended by the discontinuity relations constitutes a fundamental set of local functional constraints that can be easily transformed into integral form through Cauchy's theorem., Comment: LaTeX2e, 42 pages, 7 figures. v2: 45 pages, references and a new appendix included, typos corrected. v3: minor typos corrected, references added

We consider high spin, $s$, long twist, $L$, planar operators (asymptotic Bethe Ansatz) of strong ${\cal N}=4$ SYM. Precisely, we compute the minimal anomalous dimensions for large 't Hooft coupling $\lambda$ to the lowest order of the (string) scaling variable $\ell \sim L/ (\ln \mathcal{S} \sqrt{\lambda})$ with GKP string size $\sim\ln \mathcal{S}\equiv 2 \ln (s/\sqrt{\lambda})$. At the leading order $(\ln \mathcal{S}) \cdot \ell ^2 $, we can confirm the O(6) non-linear sigma model description for this bulk term, without boundary term $(\ln \mathcal{S})^0$. Going further, we derive, extending the O(6) regime, the exact effect of the size finiteness. In particular, we compute, at all loops, the first Casimir correction $\ell ^0/\ln \mathcal{S}$ (in terms of the infinite size O(6) NLSM), which reveals only one massless mode (out of five), as predictable once the O(6) description has been extended. Consequently, upon comparing with string theory expansion, at one loop our findings agree for large twist, while reveal for negligible twist, already at this order, the appearance of wrapping. At two loops, as well as for next loops and orders, we can produce predictions, which may guide future string computations., Comment: Version 2 with: new exact expression for the Casimir energy derived (beyond the first two loops of the previous version); UV theory formulated and analysed extensively in the Appendix C; origin of the O(6) NLSM scattering clarified; typos correct and references added

We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. This term is still governed by a linear integral equation which we study in the weak and strong coupling regimes. In the strong coupling regime we find agreement with the string theory computations, Comment: 5 pages, contribution to the proceedings of the workshop Diffraction 2010, Otranto, 10th-15th September, talk given by M.Rossi; v2: references added

Moving from the mirror theory Bethe-Yang equations proposed by Arutyunov and Frolov, we derive the thermodynamic Bethe Ansatz equations which should control the spectrum of the planar $\text{AdS}_5/\text{CFT}_4$ correspondence. The associated set of universal functional relations (Y-system) satisfied by the exponentials of the TBA pseudoenergies is deduced, confirming the structure inferred by Gromov, Kazakov and Vieira., Comment: Main typos corrected, notations fixed, references added

We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward., Comment: Latex version

A method for determining the generalised scaling function(s) arising in the high spin behaviour of long operator anomalous dimensions in the planar $sl(2)$ sector of ${\cal N}=4$ SYM is proposed. The all-order perturbative expansion around the strong coupling is detailed for the prototypical third and fourth scaling functions, showing the emergence of the O(6) Non-Linear Sigma Model mass-gap from different SYM 'mass' functions. Remarkably, only the fourth one gains contribution from the non-BES reducible densities and also shows up, as first, NLSM interaction and specific model dependence. Finally, the computation of the $n$-th generalised function is sketched and might be easily finalised for checks versus the computations in the sigma model or the complete string theory., Latex version, typos corrected, references and clarifications added

The generalized (1+1)-D non-linear Schrodinger (NLS) theory with particular integrable boundary conditions is considered. More precisely, two distinct types of boundary conditions, known as soliton preserving (SP) and soliton non-preserving (SNP), are implemented into the classical $gl_N$ NLS model. Based on this choice of boundaries the relevant conserved quantities are computed and the corresponding equations of motion are derived. A suitable quantum lattice version of the boundary generalized NLS model is also investigated. The first non-trivial local integral of motion is explicitly computed, and the spectrum and Bethe Ansatz equations are derived for the soliton non-preserving boundary conditions., Comment: 33 pages, Latex. Minor misprints corrected

We describe a procedure for determining the generalised scaling functions $f_n(g)$ at all the values of the coupling constant. These functions describe the high spin contribution to the anomalous dimension of large twist operators (in the $sl(2)$ sector) of ${\cal N}=4$ SYM. At fixed $n$, $f_n(g)$ can be obtained by solving a linear integral equation (or, equivalently, a linear system with an infinite number of equations), whose inhomogeneous term only depends on the solutions at smaller $n$. In other words, the solution can be written in a recursive form and then explicitly worked out in the strong coupling regime. In this regime, we also emphasise the peculiar convergence of different quantities ('masses', related to the $f_n(g)$) to the unique mass gap of the $O(6)$ nonlinear sigma model and analyse the first next-to-leading order corrections., Comment: Latex version, journal version (with explanatory appendices and more references)

As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU(2) sector of the planar $ {\cal N}=4$ SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, $L$, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, $\lambda$. For instance, we will show that the large $L$ (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on $L$ and $\lambda$ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easier-looking theories., Comment: On the 75-th Anniversary of Bethe Ansatz, 37 Pages, Latex file

Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable limit and therefore the scattering factors between them are explicitly computed by inspecting the corresponding Non-Linear Integral Equations. As a consequence, abstracting from the physical model the Zamolodchikov-Faddeev algebra of two $n$-th elliptic breathers defines a tower of $n$-order Deformed Virasoro Algebras, reproducing the $n=1$ case the usual well-known algebra of Shiraishi-Kubo-Awata-Odake \cite{SKAO}., Latex version, 13 pages

Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus paralleling similar results by Kl\"umper \cite{KLU}, achieved through a different technique in the {\it antiferroelectric regime}. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther \cite{LUT} and Johnson et al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe} and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description of unknown integrable field theories., Comment: Article, 41 pages, Latex

The g-function was introduced by Affleck and Ludwig in the context of critical quantum systems with boundaries. In the framework of the thermodynamic Bethe ansatz (TBA) method for relativistic scattering theories, all attempts to write an exact integral equation for the off-critical version of this quantity have, up to now, been unsuccesful. We tackle this problem by using an n-particle cluster expansion, close in spirit to form-factor calculations of correlators on the plane. The leading contribution already disagrees with all previous proposals, but a study of this and subsequent terms allows us to deduce an exact infrared expansion for g, written purely in terms of TBA pseudoenergies. Although we only treat the thermally-perturbed Ising and the scaling Lee-Yang models in detail, we propose a general formula for g which should be valid for any model with entirely diagonal scattering., 21 pages, 9 figures, Latex 2e. v2: typos fixed and comments added. v3: Published version: minor typos corrected, numerical results included and a note added

We will describe the appearance of specific algebraic KdV potentials as a consequence of a requirement on a integro-differential expression. This expression belongs to a class generated by means of Virasoro vector fields acting on the KdV field. The ``almost'' rational KdV fields are described in terms of a geometrical locus of complex points. A class of solutions of this locus has recently appeared as a description of any conformal Verma module without degeneration., Comment: LaTex, 9 pages

With the aim of exploring a massive model corresponding to the perturbation of the conformal model [hep-th/0211094] the nonlinear integral equation for a quantum system consisting of left and right KdV equations coupled on the cylinder is derived from an integrable lattice field theory. The eigenvalues of the energy and of the transfer matrix (and of all the other local integrals of motion) are expressed in terms of the corresponding solutions of the nonlinear integral equation. The analytic and asymptotic behaviours of the transfer matrix are studied and given., Comment: enlarged version before sending to jurnal, second part of hep-th/0211094

The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin $+1/2$ chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers \cite{BLZ, BLZ2} by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point., Comment: Journal version: references added and minor corrections performed

Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and {\it unbraided} (usual) Yang-Baxter algebras is derived and also analysed., 13 Latex pages

In the framework of the Sine-Gordon (SG) theory we will present the construction of a dynamicalVirasorosymmetry which has nothing to do with the space-time Virasorosymmetry of 2D CFT. Although, it is non-local in the SGeld theory, nevertheless it gives rise to a local action on specic N-soliton solutionvariables. Theseanalyticvariables possess a beautiful geometricalmeaning and enter the Form Factor expressions. At the end, we will also give some preliminary hints about the quantisation.

We generalize the dressing symmetry construction in mKdV hierarchy. This leads to non-local vector fields (expressed in terms of vertex operators) closing a Virasoro algebra. We argue that this algebra realization should play an important role in the study of 2D integrable field theories and in particular should be related to the Deformed Virasoro Algebra (DVA) when the construction is perturbed out of the critical theory., 11 pages, LaTex file

We propose an alternative description of 2 dimensional Conformal Field Theory in terms of Quantum Inverse Scattering. It is based on the generalized KdV systems attached to $A_2^{(2)}$, yielding the classical limit of Virasoro as Poisson bracket structure. The corresponding T-system is shown to coincide with the one recently proposed by Kuniba and Suzuki. We classify the primary operators of the minimal models that commute with all the Integrals of Motion, and that are therefore candidates to perturb the model by keeping the conservation laws. For our $A_2^{(2)}$ structure these happen to be $\phi_{1,2},\phi_{2,1},\phi_{1,5}$, in contrast to the $A_1^{(1)}$ case, studied by Bazhanov, Lukyanov and Zamolodchikov~\cite{BLZ}, related to $\phi_{1,3}$., Comment: 12 pages, latex. 1 reference added

We compute classical and first quantum finite-size corrections to the recently found giant magnon solutions in two different subspaces of 3. We use the L?scher approach on the recently proposed exact S-matrix for = 6 superconformal Chern-Simons theory. We compare our results with the string and algebraic curve computations and find agreement, thus providing a non-trivial test for the new AdS4/CFT3 correspondence within an integrability framework.

Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formul\ae and compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations., Comment: 38 pages, Latex revised version: draft formulae indicator deleted, one reference added, typos corrected, few minor text modifications