Yangians, S-matrices and AdS/CFT

Journal of Physics A Mathematical and Theoretical (Impact Factor: 1.69). 04/2011; 44(26). DOI: 10.1088/1751-8113/44/26/263001
Source: arXiv

ABSTRACT This review is meant to be an account of the properties of the
infinite-dimensional quantum group (specifically, Yangian) symmetry lying
behind the integrability of the AdS/CFT spectral problem. In passing, the
chance is taken to give a concise anthology of basic facts concerning Yangians
and integrable systems, and to store a series of remarks, observations and
proofs the author has collected in a five-year span of research on the subject.
We hope this exercise will be useful for future attempts to study Yangians in
field and string theories, with or without supersymmetry.

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    ABSTRACT: We establish features of so-called Yangian secret symmetries for AdS3 type IIB superstring backgrounds, thus verifying the persistence of such symmetries to this new instance of the AdS/CFT correspondence. Specifically, we find two a priori different classes of secret symmetry generators. One class of generators, anticipated from the previous literature, is more naturally embedded in the algebra governing the integrable scattering problem. The other class of generators is more elusive and somewhat closer in its form to its higher-dimensional AdS5 counterpart. All of these symmetries respect left-right crossing. In addition, by considering the interplay between left and right representations, we gain a new perspective on the AdS5 case. We also study the &$R\mathcal{T}\mathcal{T}$;-realisation of the Yangian in AdS3 backgrounds, thus establishing a new incarnation of the Beisert–de Leeuw construction.
    Journal of Physics A Mathematical and Theoretical 11/2014; 47(45). DOI:10.1088/1751-8113/47/45/455402 · 1.69 Impact Factor
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    ABSTRACT: Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author's original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain 2d lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions.
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    ABSTRACT: We propose Drinfeld's second realisation of the quantum group relevant to the Lax-operator approach developed in the work of Bazhanov, Frassek, Lukowski, Meneghelli and Staudacher.
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