Rafael I. Nepomechie

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010.

Gauge invariance versus masslessness in de Sitter spaces

Abstract The connection between gauge invariance, masslessness and null cone propagation is a flat space property which does not persist even in constant curvature geometries. In particular, we show that both the gauge invariant spin 3 2 and 2 fields in anti-de Sitter space have support inside the cone, whereas where are conformally invariant, but gauge variant, models which do propagate on the light cone. The Maxwell field in constant curvature spaces of dimension other than four also does not have null cone propagation; again there is a conformally invariant model which does.

q-Deformations of the O(3) symmetric spin-1 Heisenberg chain

The authors present the general expression for the spin-1 Heisenberg chain invariant under the Uq(SO(3)) quantum algebra. Several physical and mathematical implications are discussed.

N=6 super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations

We propose the exact S-matrix for the planar limit of the = 6 super Chern-Simons theory recently proposed by Aharony, Bergman, Jafferis, and Maldacena for the AdS4/CFT3 correspondence. Assuming SU(2|2) symmetry, factorizability and certain crossing-unitarity relations, we find the S-matrix including the dressing phase. We use this S-matrix to formulate the asymptotic Bethe ansatz. Our result for the Bethe-Yang equations and corresponding Bethe ansatz equations confirms the all-loop Bethe ansatz equations recently conjectured by Gromov and Vieira.

Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms

We consider the integrable open XX quantum spin chain with non-diagonal boundary terms. We derive an exact inversion identity, by which we obtain the eigenvalues of the transfer matrix and the Bethe ansatz equations. For generic values of the boundary parameters, the Bethe ansatz solution is formulated in terms of the Jacobian elliptic functions.

Functional Relations and Bethe Ansatz for the XXZ Chain

There is an approach due to Bazhanov and Reshetikhin for solving integrable RSOS models which consists of solving the functional relations which result from the truncation of the fusion hierarchy. We demonstrate that this is also an effective means of solving integrable vertex models. Indeed, we use this method to recover the known Bethe Ansatz solutions of both the closed and open XXZ quantum spin chains with U(1) symmetry. Moreover, since this method does not rely on the existence of a pseudovacuum state, we also use this method to solve a special case of the open XXZ chain with nondiagonal boundary terms.

Anomalous Propagation of Gauge Fields in Conformally Flat Spaces

Abstract Gauge invariant spin 3 2 and 2 fields in (anti-) de Sitter space propagate inside as well as on the null cone. This is also true for antisymmetric tensor gauge fields and ( D ≠ 4) phitins. In each case, there exist gauge variant but Wey invariant models with purely null cone support.

Fusion procedure for open chains

The authors have generalized Sklyanins approach of constructing open integrable quantum spin chains to the case of PT-invariant R matrices. They formulate a fusion procedure for such chains. In particular, they show that the fused transfer matrix can be expressed in terms of products of the original transfer matrix and products of certain quantum determinants which can be explicitly evaluated. Applications of these results include constructing open integrable higher-spin chains, as well as obtaining functional equations for transfer-matrix eigenvalues, which may be solved by an analytical Bethe ansatz.

Bethe Ansatz Solution of the Fateev-zamolodchikov Quantum Spin Chain With Boundary Terms

Abstract We solve the Fateev-Zamolodchikov quantum spin chain (i.e., the spin-1 XXZ quantum Heisenberg chain) with a class of boundary terms by the quantum inverse scattering method. For a particular choice of boundary terms, the model has the quantum symmetry U q [SU (2)].

Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

A Bethe Ansatz solution of the open spin- XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.