Integrable S matrix, mirror TBA and spectrum for the stringy $\text{AdS}_{3}\times\text{S}^3\times\text{S}^3\times\text{S}^1$ WZW model

TL;DR

The paper analyzes superstrings on pure-NSNS AdS$_3\times$S$^3\times$S$^3\times$S$^1$ and shows the tree-level bosonic S matrix is effectively diagonal, matching the simple form found in flat space and AdS$_3\times$S$^3\times$T$^4$. Building on this, the authors propose an all-loop diagonal worldsheet S matrix given by $\mathbf{S}(p_i,p_j)=e^{i\Phi(p_i,p_j)}\mathbf{1}$, and derive the corresponding mirror TBA that describes the finite-volume spectrum. Remarkably, the ground-state and excited-state TBA equations admit closed-form solutions that reproduce the known worldsheet WZW spectrum, including the highest-weight representations and, by extension, spectrally flowed sectors, indicating an underlying integrable spin-chain structure with a pseudo-vacuum in non-supersymmetric settings. The results provide a coherent framework linking perturbative worldsheet calculations, exact S-matrix proposals, and CFT spectra, and point toward a tractable, integrable description of pure-NSNS AdS$_3$ backgrounds via a spin-chain picture.

Abstract

We compute the tree-level bosonic S matrix in light-cone gauge for superstrings on pure-NSNS $\text{AdS}_{3}\times\text{S}^3\times\text{S}^3\times\text{S}^1$. We show that it is proportional to the identity and that it takes the same form as for $\text{AdS}_{3} \times \text{S}^3\times\text{T}^4$ and for flat space. Based on this, we make a conjecture for the exact worldsheet S matrix and derive the mirror thermodynamic Bethe ansatz (TBA) equations describing the spectrum. Despite a non-trivial vacuum energy, they can be solved in closed form and coincide with a simple set of Bethe ansatz equations - again much like $\text{AdS}_{3}\times\text{S}^3\times\text{T}^4$ and flat space. This suggests that the model may have an integrable spin-chain interpretation. Finally, as a check of our proposal, we compute the spectrum from the worldsheet CFT in the case of highest-weight representations of the underlying Kač-Moody algebras, and show that the mirror-TBA prediction matches it on the nose.