  Recently, string theory on $\text{AdS}_3 \times \text{S}^3 \times
\mathbb{T}^4$ with one unit of NS-NS flux ($k=1$) was argued to be exactly dual
to the symmetric orbifold of $ \mathbb{T}^4$, and in particular, the full
(unprotected) spectrum was matched between the two descriptions. This duality
was later extended to the case with higher NS-NS background flux for which the
long string sector was argued to be described by the symmetric product orbifold
of $(\mathcal{N}=4$ $\text{Liouville}) \times \mathbb{T}^4$. In this paper we
study correlation functions for the bosonic analogue of this duality, relating
bosonic string theory on $\text{AdS}_3 \times X$ to the symmetric orbifold of
$\text{Liouville} \times X$. More specifically, we show that the low-lying null
vectors of Liouville theory correspond to BRST exact states from the worldsheet
perspective, and we demonstrate that they give rise to the expected BPZ
differential equations for the dual CFT correlators. Since the structure
constants of Liouville theory are uniquely fixed by these constraints, this
shows that the seed theory of the dual CFT contains indeed the Liouville
factor.
