A Yangian double for the AdS/CFT classical r-matrix

Abstract

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.

The Classical r-Matrix of AdS/CFT and its Lie Bialgebra Structure

“…The argument of the authors of [29] not to use the standard evaluation representation J n = x n J was due to the fact that that this would lead to a classical r-matrix with poles only in x 1 = x 2 and none at x 1 = 1/x 2 , in agreement with Table 1. Furthermore, the proposed Lie brackets look quite complicated; they are not of a standard loop algebra form, and we shall not reproduce them here.…”

“…Furthermore, the proposed Lie brackets look quite complicated; they are not of a standard loop algebra form, and we shall not reproduce them here. We do not know whether the brackets obey the Jacobi identities and whether the r-matrix satisfies the CYBE using these brackets; no statement was made in [29]. In contrast, it was shown in [17] that the psu(2|2) R 3 fundamental R-matrix is invariant under Yangian generators for an ordinary evaluation representation with evaluation parameter iu with u = x + 1/x.…”

“…We should note that the procedure applied in [29] is not necessarily unique. There may be several representations leading to the same fundamental r-matrix in Table 1 upon inserting into the above classical r-matrix because often several terms of the classical r-matrix contribute to a single matrix element of the fundamental r-matrix.…”

An exceptional algebraic origin of the AdS/CFT Yangian symmetry

“…We shall reproduce the coproducts of the generators in psu(2|2) ⋉ R 3 from those in d(2, 1; ε). In the derivation, we will see the first sign that the novel generator I [8,9,10] can be interpreted as the ε-correction of the last su(2) generators C a b . A model is solvable if it has an equal number of conserved charges and degrees of freedom.…”

“…The second one [7] is to couple the central charges to the su(2) outer automorphism, which has a similar origin as the above case. In [8], to find an expression for the classical r-matrix similar to (1.1), another regularization was adopted by introducing a new generator I which complements the original superalgebra su(2|2) into u(2|2) and couples to the center of su(2|2). Subsequently, the corresponding coproduct and antipode of this new generator I at level-1 were proposed [9] …”