The complete worldsheet S matrix of superstrings on AdS_3 x S^3 x T^4 with mixed three-form flux

TL;DR

This work extends integrability techniques to Type IIB strings on AdS3×S3×T4 with mixed RR and NSNS flux, deriving the complete off-shell symmetry algebra and the non-perturbative worldsheet S matrix for all excitations, including massless modes. By deforming the pure-RR representations and carefully building tensor-product S-matrices, the authors obtain exact all-loop representation parameters, a momentum-dependent mass structure, and a crossing framework that constrains four dressing factors across massive, massless, and mixed sectors. The results unify massive and massless scattering in a single integrable framework and reveal a rich parameter space controlled by the flux q and the WZW level k, with clear paths toward a full spectral problem solution via Thermodynamic Bethe Ansatz or Quantum Spectral Curve. The methodology and findings pave the way for exploring broader AdS3/CFT2 dualities and potential connections to hybrid formalisms and NS-NS-dominated limits.

Abstract

We determine the off-shell symmetry algebra and representations of Type IIB superstring theory on $AdS_3\times S^3 \times T^4$ with mixed R-R and NS-NS three-form flux. We use these to derive the non-perturbative worldsheet S matrix of fundamental excitations of the superstring theory. Our analysis includes both massive and massless modes and shows how turning on mixed three-form flux results in an integrable deformation of the S matrix of the pure R-R theory.

Figures (2)

• Figure 1: The massive excitations and their transformation properties under $\mathcal{A}$. The left and right panel depict the left and right representations respectively. The bosons $Z^{\hbox{\tiny L},\hbox{\tiny R}}$ are excitations on $\text{AdS}_3$ while $Y^{\hbox{\tiny L},\hbox{\tiny R}}$ are excitations on $\mathrm{S}^3$. Note that the massive fermions $\eta^{\hbox{\tiny L} \dot{a}},\eta^{\hbox{\tiny R} \dot{a}}$ are charged under $\mathfrak{su}(2)_{\bullet}$. The tensor products below each diagram indicate how each representation can be obtained from the short fundamental representations of $\mathfrak{su}(1|1)^2_{\text{c.e.}}$ introduced in section \ref{['sec:repr-smallalgebra']}.
• Figure 2: The massless excitations transform in two irreducible representations of $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$, which form a doublet under $\mathfrak{su}(2)_{\circ}$. Each of these representation can equivalently be taken to be the massless limit of a left or a right representation with a fermionic highest-weight state. For definiteness, here we take both of them to be given by left representations. Below each $\mathfrak{psu}(1|1)^4_{\text{c.e.}}$ diagram we indicate how each representation can be obtained from one of two (left or right) isomorphic tensor products of fundamental $\mathfrak{su}(1|1)^2_{\text{c.e.}}$ representations, see section \ref{['sec:repr-smallalgebra']}.