Integrable S matrix, mirror TBA and spectrum for the stringy AdS 3 × S 3 × S 3 × S 1 WZW model

We compute the tree-level bosonic S matrix in light-cone gauge for superstrings on pure-NSNS AdS 3 ×S 3 ×S 3 ×S 1 . We show that it is proportional to the identity and that it takes the same form as for AdS 3 × S 3 × 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 AdS 3 × S 3 × 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.