On Drinfeld’s second realization of the AdS/CFT su(2|2) Yangian

Abstract

We construct Drinfeld's second realization of the Yangian based on psu(2|2) ⋉ R 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 realizations 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.

On Yangian and long representations of the centrally extended $ \mathfrak{s}\mathfrak{u}\left( {\left. 2 \right|2} \right) $ superalgebra

“…Analogously, V 16d (P, q) is a long 16-dimensional representation described by the momentum P , the coupling constant g and the parameter q. We find an explicit relation between the pairs (p 1 , p 2 ) and (P, q) at fixed g, in 1 Given the existence of an invertible map between the generators of Drinfeld's second [43] and first [34] realization of the Yangian, we will use either realizations according to the needs considering them as completely equivalent. A rigorous proof of this equivalence has however never been derived.…”

“…It naturally does that in both branches of the double-covering. Importantly, this (double-branched) tensor product representation of the Yangian is not isomorphic to the long evaluation representation we were discussing before, even though the two short representations composing it are the short evaluation representations of the Yangian described in [34,43]. This is also clear from the fact that the long evaluation representation discussed before does not admit an S-matrix.…”

“…We report the details in appendix B for convenience of the reader. Drinfeld's second realization is also obtained by applying a similar (Drinfeld's) map as in [43]. 12 This ensures the fulfilment of the Serre relations (see also [55]).…”

“…The origin of this problem can be clearly seen in Drinfeld's second realization [43], 1 where it can be traced back to non-co-commutativity of the higher Yangian central charges C n with n ≥ 2. Co-commutativity means that the coproduct map coincides with its opposite, namely ∆ = ∆ op .…”

The AdS3× S3× S3× S1worldsheetSmatrix

“…Together with the presence of massless modes this provides a rich setting for investigating more fully the integrability structures present here. It would be very interesting to understand, for example, the asymptotic Bethe ansatz [84], finite-gap equations [20,32,43,46], the thermodynamic Bethe ansatz [63,[85][86][87][88][89][90], Yangian symmetries [91][92][93][94][95], and quantum spectral curve [96][97][98] for these models. The study of boundary integrable boundary conditions for these models is another interesting direction, which has recently been investigated in [99].…”