AdS3 Integrability, Tensionless Limits, and Deformations: A Review

TL;DR

This review surveys how integrability structures organize the AdS3 string spectrum across mixed NSNS/RR flux backgrounds, emphasizing tensionless limits and a rich set of deformations. It details the uniform lightcone gauge formulation, the centrally extended symmetry algebra, and the S-matrix bootstrap that constrain the spectrum, while connecting small-tension regimes to dual 2D CFTs such as the D1–D5 system and symmetric-product orbifolds. It further surveys integrable deformations (Yang–Baxter, TsT, bi-Yang–Baxter+WZ, elliptic) and their deformed backgrounds and S-matrices, highlighting how fluxes and quantum-group symmetries shape the theory. The Outlook identifies key open problems, including complete mirror-TBA/QSC formulations for mixed-flux AdS3 strings and holographic interpretations of deformed models.

Abstract

Motivated by the recent advances in the understanding of integrability for $AdS_3$ backgrounds, we present a lightning review of this approach, with particular attention to the "tensionless" limits (with zero and one unit of NSNS flux), and to the many integrable deformations of the supergravity backgrounds. Our aim is to concisely but comprehensively take stock of the state of the art in the field, in a way accessible to non-experts, and to highlight outstanding challenges. Along the way we reference where the various derivation of these results, which we mostly omit, can be found in full detail.

Figures (4)

• Figure 1: Figure taken from Brollo:2023rgp. We plot the energy of two-excitations states with mode numbers $\nu_1=-\nu_2>0$ for various values of the volume in the lightcone gauge $L$. The dashed line is the expectation for a free theory, i.e. $H_{(1)}=4\sin\tfrac{\pi\nu_1}{L}$. The cross is given by the solution of the asymptotic Bethe equations and the solid circle is the TBA value.
• Figure 2: Figure taken from Brollo:2023rgp. We plot the energy of four-excitations states with mode numbers $\nu_1=-\nu_2>0$ and $\nu_3=-\nu_4>0$ for various values of the volume in the lightcone gauge $L$. The dashed line is the expectation for a free theory, i.e. $H_{(1)}=4\sin\tfrac{\pi\nu_1}{L}+4\sin\tfrac{\pi\nu_3}{L}$. The cross is given by the solution of the asymptotic Bethe equations and the solid circle is the TBA value. Notice how the energy exact is not additive, indicating that the model is interacting.
• Figure 3: Figure taken from Brollo:2023rgp. We plot the difference between the asymptotic energy (as predicted from the Asymptotic Bethe Ansatz (Bethe-Yang) equations) and the exact energy (as predicted from the mirror TBA) for a class of two-excitation states with mode numbers $\nu_1=-\nu_2$ and length (volume) $L$. The differece, which corresponds to the size of the "wrapping" effects, decreses as $1/L$ and $L\to\infty$.
• Figure 4: The landscape of integrable deformations of $AdS_3 \times S^3 \times T^4$ strings. The (undeformed) theory preserves 16 SUSYs, the unilateral deformation preserves 8 SUSYs and the bi-YB deformation breaks all the supersymmetries. Dotted arrows represent Poisson-Lie dualities. Several hybrid YB-$\lambda$ models can be constructed.