The su(2|2) Dynamic S-Matrix

Abstract

We derive and investigate the S-matrix for the su(2|3) dynamic spin chain and for planar N = 4 super Yang-Mills. Due to the large amount of residual symmetry in the excitation picture, the S-matrix turns out to be fully constrained up to an overall phase. We carry on by diagonalizing it and obtain Bethe equations for periodic states. This proves an earlier proposal for the asymptotic Bethe equations for the su(2|3) dynamic spin chain and for N = 4 SYM.

An exceptional algebraic origin of the AdS/CFT Yangian symmetry

“…Actually this idea is realized in some sense 4 . Speaking in a more sophisticated way, we will diagonalize the action of (1/ε)T C 12 by Fourier-transforming the above coordinate (site) picture into the momentum picture as in the asymptotic state (2.10).…”

“…In this section we would like to review the exceptional Lie superalgebra d(2, 1; ε), the limit which recovers the symmetry psu(2|2) ⋉ R 3 and the AdS/CFT spin chain with this symmetry [4].…”

“…In the above expression D comes from the classical limit of the centers P and K and relates to C by D = 2u −1 C. The Casimir-like operator T psu takes the opposite sign on bosons and fermions 4) and serves the same role as I. Interestingly, it was further noted that the eigenvalue of the new generator I and the composite operator T psu C −1 coincides and that both the composite operator T psu C −1 and the Cartan generator of the su(2) outer automorphism 2B = B 1 1 − B 2 2 satisfy similar commutation relations.…”

“…To cure the degeneracy of the Killing form, several different regularizations were proposed. The first one [4] is to consider the exceptional superalgebra d(2, 1; ε) which recovers psu(2|2)⋉R 3 when the parameter ε is taken to zero. In the exceptional algebra d(2, 1; ε), the generators C, P and K are not central but form the usual su(2) algebra.…”

“…The N = 4 super Yang-Mills theory has a global symmetry of the Lie superalgebra psu(2, 2|4), which is broken down to two copies of su(2|2) algebra once we fix a vacuum. The spin chain model with this su(2|2) symmetry, which interpolates the two coupling regions, was constructed in [4]. To take care of the off-shell trace operators, the centrally extended su(2|2) superalgebra was considered, which contains not only the center of su(2|2), C, whose eigenvalue is the energy of the spin chain, but also two additional centers, P and K, whose eigenvalues correspond to the momentum and should vanish on the on-shell trace operators where the total momentum vanishes.…”