# 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

This paper computes the exact spectrum of superstrings on a specific AdS3 background using integrability techniques, showing the S matrix is simple and the TBA equations are solvable, indicating an integrable structure.

## Contribution

It conjectures the exact worldsheet S matrix for the AdS3 WZW model and derives solvable mirror TBA equations, connecting to integrable spin-chain models.

## Key findings

- Tree-level bosonic S matrix is proportional to the identity.
- Mirror TBA equations can be solved in closed form.
- Mirror-TBA spectrum matches worldsheet CFT calculations.

## 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 Ka\v{c}-Moody algebras, and show that the mirror-TBA prediction matches it on the nose.

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Source: https://tomesphere.com/paper/1812.08195