Tensionless strings on ${\rm AdS}_3$ orbifolds

The bound state of one NS5 brane (wrapped on a $\mathbb{T}^4$) and $N$ NS1-branes has two dual descriptions: its low-energy dynamics is described by the symmetric orbifold of $\mathbb{T}^4$, while the near horizon geometry is captured by string theory on ${\rm AdS}_3 \times {\rm S}^3\times \mathbb{T}^4$ with one unit of NS flux. The latter theory is exactly solvable in the hybrid formalism, and this allows one to prove the equivalence of the two descriptions. In this paper we extend this duality to $\mathbb{Z}_k$ orbifolds of this ${\rm AdS}_3 \times {\rm S}^3$ background. In particular, we show that the corresponding worldsheet spectrum reproduces exactly the perturbative excitations on top of a certain non-perturbative state in the dual symmetric orbifold theory. Since the ${\rm AdS}/{\rm CFT}$ duality map is exact for these models, we obtain an interesting picture of how the duality relates boundary and bulk descriptions.