Integrability for the spectrum of Jordanian AdS/CFT

TL;DR

This work addresses integrability in non-AdS holography by studying the Jordanian-deformed AdS5xS5 setup in the sl(2,R) sector, focusing on the XXX_{-1/2} spin chain with a non-abelian Drinfeld twist. The authors develop a Baxter/Separation of Variables framework in which the TQ relation retains its functional form but the Q-functions become nonpolynomial and are selected by regularity rather than polynomiality, enabling analytic results for arbitrary chain length J. They obtain the full J=2 spectrum and extend to arbitrary J via the TQ equation, with perturbative and numerical checks, and demonstrate a precise matching with the semiclassical string spectrum in the large-J limit, including BMN-like states, under a specific identification between twist charges. The results provide nontrivial tests of Jordanian AdS/CFT and lay the groundwork for future SoV implementations and a complete quantum spectral curve for non-abelian twists.

Abstract

Jordanian deformations offer rare integrable realisations of non-AdS holography, whose solvability methods differ from conventional AdS/CFT examples. Here we study the $\mathfrak{sl}(2,R)$ sector of the Jordanian deformed $AdS_5\times S^5$ string and its weak-coupling spin chain counterpart: the $\mathrm{XXX}_{-1/2}$ model with a non abelian Jordanian Drinfel'd twist. While the twist breaks the usual highest-weight structure that underlies conventional Bethe ansätze, we show that the complete spectrum remains solvable within the Baxter framework. We argue that the functional form of the $TQ$-relation is unchanged, yet the structure of the $Q$-functions is nontrivially modified. This allows us to obtain analytic expressions at arbitrary spin chain length $J$, which match the deformed string spectrum at the one-loop level and to subleading order in the large-$J$ expansion, despite the severely reduced symmetry. Our results provide nontrivial tests of the Jordanian AdS/CFT correspondence and lay the groundwork for implementing the Separation of Variables program in non-abelian Drinfel'd-twisted models.

Figures (2)

• Figure 1: The ground state ($S=0$) transfer matrix eigenvalue as a function of the deformation parameter $-i\phi$. The blue points are the numerical results. The purple curve at small $\phi$ is the 20-th order small-$\phi$ expansion obtained analytically from solving the $TQ$-relation. The green curve is a large $\phi$ asymptotic approximation, $\tau_0^\star=-\frac{1}{2} \cos(\phi)$, obtained from a preliminary WKB analysis, the details of which will be discussed elsewhere.
• Figure 2: The ground state ($S=0$), $J=2$ energy as a function of the deformation parameter $-i\phi$. The blue points are the numerical results. The purple curve is the 20-th order small-$\phi$ expansion obtained analytically from solving the $TQ$-relation and evaluating the energy formula \ref{['evq']}.