All-loop worldsheet S matrix for AdS_3 x S^3 x T^4

TL;DR

This work derives the all-loop worldsheet S matrix for fundamental excitations on AdS3×S3×T4 by analyzing the off-shell symmetry algebra of the lightcone-gauge Green-Schwarz action, thereby including massless modes that were previously neglected. The S matrix is fixed by symmetry up to a few scalar phases and is organized into massive, mixed, and massless sectors, each with explicit block structures and non-relativistic massless kinematics. Crucially, the massless sector exhibits a non-analytic dispersion and a Heisenberg-type SU(2) invariant scattering matrix, while unitarity and crossing impose a set of equations that constrain the scalar factors. The results lay a foundation for complete AdS3/CFT2 integrability, with clear next steps toward Bethe-Yang equations, bound-state spectra, and extensions to deformations and mixed flux backgrounds.

Abstract

We obtain the all-loop worldsheet S matrix for fundamental excitations on AdS_3 x S^3 x T^4 by studying the off-shell symmetry algebra of the superspace action in lightcone gauge. The massless modes, unaccounted for in earlier works, are automatically included in our treatment. Their exact dispersion relation is found to be non-relativistic, of giant-magnon form and their scattering is naturally well-defined. This opens the way to a complete investigation of AdS_3/CFT_2 integrability.

Figures (2)

• Figure 1: Each of the two (left and right) massive $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ multiplets consists of two bosons $Y^{\hbox{\tiny L},\hbox{\tiny R}}$, $Z^{\hbox{\tiny L},\hbox{\tiny R}}$ and of two fermions $\eta_{\ a}^{\hbox{\tiny L},\hbox{\tiny R}}$, the latter carrying the fundamental $\mathfrak{su}(2)_{\bullet}$ index $a$. For clarity we only indicate the supercharges that do not vanish on shell.
• Figure 2: The two massless $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ multiplets, in the representation $(\varrho_{\hbox{\tiny L}}\otimes\widetilde{\varrho}_{\hbox{\tiny L}})^{\oplus2}$. Overall we have four bosons $T^{a\alpha}$ and four fermions $\chi^{\alpha}, \widetilde{\chi}^{\alpha}$, where $a$ and $\alpha$ are fundamental indices of $\mathfrak{su}(2)_{\bullet}$ and $\mathfrak{su}(2)_{\circ}$. Again we show only some supercharges. Note that $\mathfrak{su}(2)_{\circ}$ relates the two $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ modules.