We construct Drinfeld's second realization of the Yangian based on psu(2|2)xR^3 symmetry. The second realization is traditionally more suitable for deriving the quantum double and the universal R matrix with respect to the first realization, originally obtained by Beisert, and it is generically more useful in order to study finite dimensional representations. We show that the two realization are isomorphic, where the isomorphism is almost the standard one given by Drinfeld for simple Lie algebras, but needs some crucial corrections to account for the central charges. We also evaluate the generators of the second realization on the fundamental representation, finding the interesting result that the rapidity variable for some generators gets boosted by the energy eigenvalue.
We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.
We construct highest-weight modules and a Yangian extension of the centrally extended sl(1|1)^2 superalgebra, that is a symmetry of the worldsheet scattering associated with the AdS3/CFT3 duality. We demonstrate that the R-matrix intertwining atypical modules is of a trigonometric type. We also consider a quantum deformation of this superalgebra, its modules, and obtain a quantum affine extension of the Drinfeld-Jimbo type that describes a deformed worldsheet scattering.
We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.
The reflection matrices of multi magnon bound states are obtained explicitely by exploiting the Yangian symmetry of boundary scattering on the Y=0 maximal giant graviton brane.
We review the study of Hopf algebras, classical and quantum R-matrices, infinite-dimensional Yangian symmetries and their representations in the context of integrability for the N=4 vs AdS5xS5 correspondence.
We consider the problem of boundary scattering for Y=0 maximal giant graviton branes. We show that the boundary S-matrix for the fundamental excitations has a Yangian symmetry. We then exploit this symmetry to determine the boundary S-matrix for two-particle bound states. We verify that this boundary S-matrix satisfies the boundary Yang-Baxter equations.
We review the algebraic construction of the S-matrix of AdS/CFT. We also present its symmetry algebra which turns out to be a Yangian of the centrally extended su(2|2) superalgebra.
We study the Yangian of the sl(2|1) Lie superalgebra in a multi-parametric four-dimensional representation. We use Drinfeld's second realization to independently rederive the R-matrix, and to obtain the antiparticle representation, the crossing and the unitarity condition. We consistently apply the Yangian antipode and its inverse to the individual particles involved in the scattering. We explicitly find a scalar factor solving the crossing and unitarity conditions, and study the analytic structure of the resulting dressed R-matrix. The formulas we obtain bear some similarities with those fami…
We express the classical r-matrix of AdS/CFT in terms of tensor products involving an infinite family of generators, which takes a form suggestive of the universal expression obtained from a Yangian double. This should provide an insight into the structure of the infinite dimensional symmetry algebra underlying the integrability of the model, and give a clue to the construction of its universal R-matrix. We derive the commutation relations under which the algebra of these new generators close.
In the su(2|2) spin chain motivated by the AdS/CFT correspondence, a novel symmetry extending the superalgebra su(2|2) into u(2|2) was found. We pursue the origin of this symmetry in the exceptional superalgebra d(2,1;epsilon), which recovers su(2|2) when the parameter epsilon is taken to zero. Especially, we rederive the Yangian symmetries of the AdS/CFT spin chain using the exceptional superalgebra and find that the epsilon-correction corresponds to the novel symmetry. Also, we reproduce the non-canonical classical r-matrix of the AdS/CFT spin chain expressed with this symmetry from the cano…
We consider the worldsheet boundary scattering and the corresponding boundary algebras for the Z=0 giant graviton and the Z=0 D7-brane in the AdS/CFT correspondence. We consider two approaches to the boundary scattering, the usual one governed by the (generalized) twisted Yangians and the q-deformed model of these boundaries governed by the quantum affine coideal subalgebras. We show that the q-deformed approach leads to boundary algebras that are of a more compact form than the corresponding twisted Yangians, and thus are favourable to use for explicit calculations. We obtain the q-deformed r…
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.
This paper is devoted to study of integrable structures in superconformal field theory and more general coset CFT's related to the affine Yangian $\textrm{Y}\big(\widehat{\mathfrak{gl}}(2)\big)$. We derive the relation between the RLL and current realizations and prove Bethe anzatz equations for the spectrum of Integrals of Motion.
It was shown that the spin chain model coming from AdS/CFT correspondence satisfies the Yangian symmetry if we assume evaluation representation, though so far there is no explicit proof that the evaluation representation satisfies the Serre relation, which is one of the defining equations of the Yangian algebra imposing constraints on the whole algebraic structure. We prove completely that the evaluation representation adopted in the model satisfies the Serre relation by introducing a three-dimensional gamma matrix. After studying the Serre relation, we proceed to the whole Yangian algebraic s…
We find new twisted Yangian symmetries of the AdS/CFT reflection matrices for the Y=0 maximal giant graviton and D5-brane. These new symmetries originate from the known secret symmetries of the Yangian symmetry of the AdS/CFT S-matrix.
In this paper we work out the RTT-realization for the Yangian algebra of the Hubbard model and AdS/CFT correspondence. We find that this Yangian algebra is of a non-standard type in which the levels of the Yangian mix. The crucial feature that allows this is a braiding factor that deforms the coproduct and generates the central extensions of the underlying sl(2|2) Lie algebra. In our RTT-realization we have also been able to incorporate the so-called secret symmetry and we were able to extend it to higher Yangian levels. Finally, we discuss the center of the Yangian and the automorphisms relat…
We study quantized enveloping algebras called twisted Yangians. They are analogues of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in Y(gl(N)) and coideals with respect to the coproduct in Y(gl(N)). We give a complete description of their finite-dimensional irreducible representations. Every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite-dimensional. The result is analogous to Drinfeld's theorem for the ordinary Yangians. Its de…
We identify certain blocks in the S-matrix describing the scattering of bound states of the AdS5 x S5 superstring that allow for a representation in terms of universal R-matrices of Yangian doubles. For these cases, we use the formulas for Drinfeld's second realization of the Yangian in arbitrary bound-state representations to obtain the explicit expressions for the corresponding R-matrices. We then show that these expressions perfectly match with the previously obtained S-matrix blocks.
We find a new quantum Yangian symmetry of the AdS/CFT S-matrix, which complements the original su(2|2) symmetry to gl(2|2) and does not have a Lie algebra analog. Our finding is motivated by the Yangian double structure discovered at the classical level.
We review some aspects of the free field approach to two-dimensional conformal field theories. Specifically, we discuss the construction of free field resolutions for the integrable highest weight modules of untwisted affine Kac-Moody algebras, and extend the construction to a certain class of admissible highest weight modules. Using these, we construct resolutions of the completely degenerate highest weight modules of W-algebras by means of the quantum Drinfeld-Sokolov reduction. As a corollary we derive character formulae for these degenerate highest weight modules.
We study the remnants of Yangian symmetry of AdS/CFT magnons reflecting from boundaries with no degrees of freedom. We present the generalized twisted boundary Yangian of open strings ending on boundaries which preserve only a subalgebra h of the bulk algebra g, where (g,h) is a symmetric pair. This is realized by open strings ending on the D3 brane known as the Y=0 maximal giant graviton in AdS_5 x S^5. We also consider the Yangian symmetry of the boundary which preserves an su(1|2) subalgebra only.
We give a general account of nonlocal symmetries in symmetric space models and their relation to the AdS/CFT correspondence. In particular, we study a master symmetry which generates the spectral parameter and acts as a level-raising operator on the classical Yangian generators. The master symmetry extends to an infinite tower of symmetries with nonlocal Casimir elements as associated conserved charges. We discuss the algebraic properties of these symmetries and establish their role in explaining the recently observed one-parameter deformation of holographic Wilson loops. Finally, we provide a…
Using considerations based on the thermodynamical Bethe ansatz as well representation theory of twisted Yangians we derive an exact expression for the overlaps between the Bethe eigenstates of the $SO(6)$ spin chain and matrix product states built from matrices whose commutators generate an irreducible representation of $\mathfrak{so}(5)$. The latter play the role of boundary states in a domain wall version of ${\cal N}=4$ SYM theory which has non-vanishing, $SO(5)$ symmetric vacuum expectation values on one side of a co-dimension one wall. This theory, which constitutes a defect CFT, is known…
We introduce a series of articles reviewing various aspects of integrable models relevant to the AdS/CFT correspondence. Topics covered in these reviews are: classical integrability, Yangian symmetry, factorized scattering, the Bethe ansatz, the thermodynamic Bethe ansatz, and integrable structures in (conformal) quantum field theory. In the present article we highlight how these concepts have found application in AdS/CFT, and provide a brief overview of the material contained in this series.
Scattering amplitudes in planar N=4 super Yang-Mills theory reveal a remarkable symmetry structure. In addition to the superconformal symmetry of the Lagrangian of the theory, the planar amplitudes exhibit a dual superconformal symmetry. The presence of this additional symmetry imposes strong restrictions on the amplitudes and is connected to a duality relating scattering amplitudes to Wilson loops defined on polygonal light-like contours. The combination of the superconformal and dual superconformal symmetries gives rise to a Yangian, an algebraic structure which is known to be related to the…
It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus …
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 le…
We give an AdS/CFT interpretation to homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring as noncommutative deformations of the dual gauge theory, going well beyond the canonical noncommutative case. These homogeneous Yang-Baxter deformations can be of so-called abelian or jordanian type. While abelian deformations have a clear interpretation in string theory and many already had well understood gauge theory duals, jordanian deformations appear novel on both counts. We discuss the symmetry structure of the deformed string from the uniformizing perspective of Drinfeld twists and …
We consider the exact S-matrix governing the planar spectral problem for strings on $AdS_5\times S^5$ and $\mathcal N=4$ super Yang-Mills, and we show that it is invariant under a novel "boost" symmetry, which acts as a differentiation with respect to the particle momentum. This generator leads us also to reinterpret the usual centrally extended $\mathfrak{psu}(2|2)$ symmetry, and to conclude that the S-matrix is invariant under a $q$-Poincaré supersymmetry algebra, where the deformation parameter is related to the 't Hooft coupling. We determine the two-particle action (coproduct) that turns …
