Towards integrability for AdS3/CFT2

TL;DR

This work surveys the development of worldsheet integrability for AdS3/CFT2, concentrating on the massive RR sector of AdS3×S3×T4 and outlining extensions to massless modes and mixed-flux backgrounds. It constructs the all-loop worldsheet S matrix from off-shell su(1|1)^2 central extensions, identifies two scalar dressing factors governed by crossing symmetry, and derives the associated Bethe-Yang equations while addressing finite-size corrections conceptually via TBA/Y-system. The analysis connects the worldsheet theory, its coset/sigma-model description, and a psu(1|1)^2-based spin-chain picture, highlighting the non-relativistic dispersion and the LR symmetry that structure the massive spectrum and bound-state spectrum. The results provide a coherent framework for studying AdS3/CFT2 integrability and set the stage for incorporating massless modes and broader backgrounds, with perturbative and semiclassical checks supporting the proposed S-matrix and crossing structure.

Abstract

We review the recent progress towards applying worldsheet integrability techniques to the $AdS_3/CFT_2$ correspondence to find its all-loop S matrix and Bethe-Yang equations. We study in full detail the massive sector of $AdS_3\times S^3\times T^4$ superstrings supported by pure Ramond-Ramond (RR) fluxes. The extension of this machinery to accommodate massless modes, to the $AdS_3\times S^3\times S^3\times S^1$ pure-RR background and to backgrounds supported by mixed background fluxes is also reviewed. While the results discussed here were found elsewhere, our presentation sometimes deviates from the one found in the original literature in an effort to be pedagogical and self-contained.

Figures (29)

• Figure 1: Two processes involving exchange of real or virtual particles that wrap around the worldsheet cylinder, and therefore are not captured by the Bethe-Yang equations.
• Figure 2: The quiver diagram of ABJM theory, where we have drawn an arrow from the fundamental to the anti-fundamental representation of each gauge group. Note that the scalars $Y^{A}$ and the fermions $\psi_{A}$, as well as their conjugates, carry an index of the R-symmetry group $SU(4)$, $A=1,\dots 4$.
• Figure 3: The elements of the $\mathfrak{psu}(1,1|2)^2$ matrix $\mathcal{M}$ of \ref{['eq:chargemap']}, distinguished by the dependence on $x_{\pm}$ in the resulting charge $\mathbf{Q}_{\mathcal{M}}$. Elements on a white background yield an $x_{+}$-dependent charge (that does not commute with $\mathbf{H}$), while the one highlighted in yellow $\colorbox{yellow}{$\mathtt{K}$},\colorbox{yellow}{$\mathtt{D}$}$ yield conserved charges. We further distinguish between kinematical ($\mathtt{K}$), i.e.$x_{-}$-independent charges, and dynamical ones ($\mathtt{D}$).
• Figure 4: The action of the supercharges on a zero-momentum (on-shell) worldsheet excitation \ref{['eq:wsexcitation']}. In this case an L-excitation is charged only under $\mathbf{Q}^{j\hbox{\tiny L}}$ and $\overline{\mathbf{Q}}{}^{j\hbox{\tiny L}}$, and similarly for R-excitations. As explained in the text, the raising operators are $\overline{\mathbf{Q}}{}^{j\hbox{\tiny L}}$ and ${\mathbf{Q}}^{j\hbox{\tiny R}}$.
• Figure 5: The action of the supercharges $\mathbf{q}^{\hbox{\tiny L},\hbox{\tiny R}}$ and $\bar{\mathbf{q}}^{\hbox{\tiny L},\hbox{\tiny R}}$ of $\mathfrak{psu}(1|1)_{\hbox{\tiny L}}\oplus\mathfrak{psu}(1|1)_{\hbox{\tiny R}}$ centrally extended on an arbitrary momentum (off-shell) excitation \ref{['eq:smallexcitation']}.