Twisted Bethe equations from a twisted S-matrix

Abstract

All-loop asymptotic Bethe equations for a 3-parameter deformation of AdS5/CFT4 have been proposed by Beisert and Roiban. We propose a Drinfeld twist of the AdS5/CFT4 S-matrix, together with c-number diagonal twists of the boundary conditions, from which we derive these Bethe equations. Although the undeformed S-matrix factorizes into a product of two su(2|2) factors, the deformed S-matrix cannot be so factored. Diagonalization of the corresponding transfer matrix requires a generalization of the conventional algebraic Bethe ansatz approach, which we first illustrate for the simpler case of the twisted su(2) principal chiral model. We also demonstrate that the same twisted Bethe equations can alternatively be derived using instead untwisted S-matrices and boundary conditions with operatorial twists.Comment: 42 pages; v2: a new appendix on sl(2) grading, 2 additional references, and some minor changes; v3: improved Appendix D, additional references, and further minor changes, to appear in JHE

On classical Yang-Baxter based deformations of the AdS5 × S5 superstring

“…26 The thermodynamic Bethe ansatz can indeed be extended to this case [51][52][53], see also [8]. Interestingly, note also that there is a clear link between the R matrix (2.8) and the twist of the twisted exact S matrix picture of [54], which presumably extends beyond this case provided the form of the twist is compatible with the exact S matrix.…”

Lunin-Maldacena backgrounds from the classical Yang-Baxter equation — towards the gravity/CYBE correspondence

“…Now the relation between classical r-matrices and the γ-deformed geometries has been clarified. For the γ-deformed geometries, various things are understood such as the deformed potential in N =4 SYM [57][58][59], the twisted Bethe ansatz [64,65] and the worldsheet S-matrix [66]. The mirror TBA with twisted boundary conditions is also investigated in [67,68].…”

Integrability of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring and its deformations

“…We will now compare the Bethe equations following from this twist to those derived in [162] or equivalently to [171]. After a duality transformation to the sl(2) grading we find the following set of equations (see also the appendix of [171] for the explicit sl(2) grading) 40) where K i are the excitation labels from the Bethe equations and K 0 ≡ L. The minus sign in the first term of the left hand side is due to the fact that P = −(AK) 0 in the notation of [162].…”

“…After a duality transformation to the sl(2) grading we find the following set of equations (see also the appendix of [171] for the explicit sl(2) grading) 40) where K i are the excitation labels from the Bethe equations and K 0 ≡ L. The minus sign in the first term of the left hand side is due to the fact that P = −(AK) 0 in the notation of [162]. This over-determined system of equations indeed has a unique solution.…”

“…would only give us a total of six arbitrary parameters, whereas the most general TsT transformation involves fifteen parameters. However, the idea of [171] was not just to consider twisting the psu(2|2) c.e. invariant factors of the S-matrix individually, but rather consider twisting a hypothetical psu(2, 2|4) invariant S-matrix by a general twist element corresponding to the Cartan subalgebra of psu(2, 2|4), and infer its consequences on our S-matrix with reduced symmetry.…”