String theory on $\mathrm{<msub><mtext>AdS</mtext><mn\; mathvariant="bold">3</mn></msub>}$ and the symmetric orbifold of Liouville theory

For string theory on AdS${}_3$ with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows us to show that the CFT dual of superstring theory on ${\rm AdS}_3 \times {\rm S}^3 \times $\mathbb{T}$^4$ for generic NS-NS flux is the symmetric orbifold of $(\mathcal{N}=4$ Liouville theory$)\times {\mathbb{T}}^4$. For the case of minimal flux ($k=1$), the Liouville factor disappears, and we just obtain the symmetric orbifold of ${\mathbb{T}}^4$, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times X$.