One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

The high-energy asymptotic form of the scattering amplitude of colorless particles in quantum chromodynamics is obtained in the leading logarithmic approximation. It is argued that such a calculation is justified for the amplitudes of scattering in which charmed quarks participate. The cross section for formation of two pairs of charmed quarks in ..gamma gamma.. collisions is found in explicit form.

On the Pomeranchuk singularity in the quantum chromodynamics

We compute super Yang–Mills planar amplitudes at strong coupling by considering minimal surfaces in AdS5 space. The surfaces end on a null polygonal contour at the boundary of AdS. We show how to compute the area of the surfaces as a function of the conformal cross ratios characterizing the polygon at the boundary. We reduce the problem to a simple set of functional equations for the cross ratios as functions of the spectral parameter. These equations have the form of thermodynamic Bethe ansatz (TBA) equations. The area is the free energy of the TBA system. We consider any number of gluons and in any kinematic configuration.

Y-system for scattering amplitudes

We argue that every CFT contains light-ray operators labeled by a continuous spin J. When J is a positive integer, light-ray operators become integrals of local operators over a null line. However for non-integer J , light-ray operators are genuinely nonlocal and give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in our construction is played by a novel set of intrinsically Lorentzian integral transforms that generalize the shadow transform. Matrix elements of light-ray operators can be computed via the integral of a double-commutator against a conformal block. This gives a simple derivation of Caron-Huot’s Lorentzian OPE inversion formula and lets us generalize it to arbitrary four-point functions. Furthermore, we show that light-ray operators enter the Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-point functions. The average null energy operator is an important example of a light-ray operator. Using our construction, we find a new proof of the average null energy condition (ANEC), and furthermore generalize the ANEC to continuous spin.

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Light-ray operators in conformal field theory

In this thesis we consider certain supersymmetric string theories on curved backgrounds containing the three-dimensional anti-de Sitter space, AdS3. On the one hand, these are gravity theories in three dimensions, which makes them simpler than real-world gravity but still rich in interesting phenomena. On the other hand, according to the holographic duality, each of these gravitational theories should be equivalent to a suitable quantum field theory with conformal symmetry in two dimensions, a CFT2. In that low dimension, such conformal field theories enjoy an infinite-dimensional Virasoro symmetry which strongly constrains them, but remain non-trivial. Our study is confined to a particular regime, the 't Hooft limit, in which the strings propagate freely. Free string theory can be described as a non-linear sigma model from the string worldsheet (geometrically, a two-dimensional cylinder) into a target space of the form AdS3xM, where M is a manifold without boundary. For us, M is a product of circles and three-dimensional spheres, namely M=S3xT4 or M=S3xS3xS1, both of which enjoy 16 supersymmetries. For the sake of brevity, our presentation is mostly focused on the massive sector of the former geometry, while more general cases are discussed in the concluding chapter. Our main result is to describe how these theories are integrable at the quantum level, i.e. how their worldsheet scattering matrix factorises and their spectrum can be efficiently computed by Bethe ansatz techniques. This is done both from the point of view of the worldsheet theory and of a dual spin chain picture. We also discuss how this procedure is supported by overwhelming evidence from independent (perturbative or semiclassical) calculations, up to two loops in the worldsheet theory expansion.

/pdf/towards-integrability-for-ads3-cft2-nz9zwp7qsu.pdf

Towards integrability for AdS3/CFT2

We obtain an analytical expression for the next-to-next-to-leading order of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron eigenvalue in planar N=4 SYM using quantum spectral curve (QSC) integrability-based method. The result is verified with more than 60-digit precision using the numerical method developed by us in a previous paper [N. Gromov, F. Levkovich-Maslyuk, and G. Sizov, arXiv:1504.06640]. As a by-product, we developed a general analytic method of solving the QSC perturbatively.

/pdf/pomeron-eigenvalue-at-three-loops-in-n-4-supersymmetric-yang-2wx46rmwzk.pdf

Pomeron Eigenvalue at Three Loops in N = 4 Supersymmetric Yang-Mills Theory

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS
 3 × S
 3 × T
 4 supported by RR flux. We find that the phase should be different from the phase appearing in the AdS
 5 × S
 5 case. We use the simplest example of a rigid circular string with two equal spins in S
 3 and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS
 3 × S
 3 × S
 3 × S
 1 theory and find the dependence of the 1-loop correction to the effective string tension function h(λ) (expected to enter the magnon dispersion relation) on the parameter α related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS
 3 × S
 3 × T
 4 case.

Quantum corrections to spinning superstrings in AdS 3 × S 3 × M 4 : determining the dressing phase

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B
 good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.

/pdf/new-construction-of-eigenstates-and-separation-of-variables-4aqcnv90lz.pdf

New construction of eigenstates and separation of variables for SU(N) quantum spin chains

We developed an efficient numerical algorithm for computing the spectrum of anomalous dimensions of the planar ${\cal N}=4$ Super-Yang--Mills at finite coupling. The method is based on the Quantum Spectral Curve formalism. In contrast to Thermodynamic Bethe Ansatz, worked out only for some very special operators, this method is applicable for generic states/operators and is much faster and precise due to its Q-quadratic convergence rate. 
To demonstrate the method we evaluate the dimensions $\Delta$ of twist operators in $sl(2)$ sector directly for any value of the spin $S$ including non-integer values. In particular, we compute the BFKL pomeron intercept in a wide range of the 't Hooft coupling constant with up to $20$ significant figures precision, confirming two previously known from the perturbation theory orders and giving prediction for several new coefficients. Furthermore, we explore numerically a rich branch cut structure for complexified spin $S$.

/pdf/quantum-spectral-curve-and-the-numerical-solution-of-the-1u7d77o1vs.pdf

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4

We show that the Quantum Spectral Curve (QSC) formalism, initially formulated for the spectrum of anomalous dimensions of all local single trace operators in $$ \mathcal{N}=4 $$
 SYM, can be extended to the generalized cusp anomalous dimension for all values of the parameters. We find that the large spectral parameter asymptotics and some analyticity properties have to be modified, but the functional relations are unchanged. As a demonstration, we find an all-loop analytic expression for the first two nontrivial terms in the small |ϕ ± θ| expansion. We also present nonperturbative numerical results at generic angles which match perfectly 4-loop perturbation theory and the classical string prediction. The reformulation of the problem in terms of the QSC opens the possibility to explore many open questions. We attach to this paper several Mathematica notebooks which should facilitate future studies.

/pdf/quantum-spectral-curve-for-a-cusped-wilson-line-in-n-4-sym-29abf5zih1.pdf

Fedor Levkovich-Maslyuk

Papers

Pomeron Eigenvalue at Three Loops in N = 4 Supersymmetric Yang-Mills Theory

Quantum corrections to spinning superstrings in AdS 3 × S 3 × M 4 : determining the dressing phase

New construction of eigenstates and separation of variables for SU(N) quantum spin chains

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4

Quantum Spectral Curve for a cusped Wilson line in N=4 SYM