Spectral flow and localisation in $\text{AdS}_3$ string theory

We study string theory in three-dimensional Anti-de Sitter spacetime in the path integral formalism. We derive expressions for generic spectrally-flowed near-boundary vertex operators in the Wakimoto representation, and relate their correlation functions to covering maps from the worldsheet to the target space boundary. We show that the path integral structurally reproduces correlation functions of the dual symmetric orbifold theory. By rephrasing spectral flow as the introduction of a background gauge field, we provide a path integral derivation of the localisation property of the near boundary theory. We then focus on the case of IIB string theory on $\text{AdS}_3\times\text{S}^3\times\mathbb{T}^4$ with $k=1$ units of NS-NS flux, where the relationship between correlation functions and covering maps can be made sharp. We also comment on the relation of the $k=1$ theory and twistor theory.