Spacetime dilaton in ${\rm AdS}_3\times X$ holography

We study the spectrum generating operators (DDF operators) for bosonic strings and superstrings in ${\rm AdS}_3\times X$ with pure, $\textit{generic}$ NSNS flux. The dual CFT is conjectured to be a deformed symmetric orbifold of $\mathbb{R}_{\widetilde{Q}}\times X$, where $\mathbb{R}_{\widetilde{Q}}$ denotes a linear dilaton theory with slope $\widetilde{Q}$. In this paper, we construct DDF operators corresponding to the spacetime linear dilaton field in the limit where the worldsheet theory becomes free. The DDF operator gives another evidence supporting the proposed dual CFT. Furthermore, the DDF operator is constructed purely from the $\rm AdS_3$ fields and thus, strongly implies the existence of $\mathbb{R}_{\widetilde{Q}}$ for generic $X$ that defines a consistent background.