# Three-point functions in AdS$_3$/CFT$_2$ holography

## TL;DR

This paper explores the duality between bosonic string theory on AdS3 and symmetric orbifolds involving Liouville theory, demonstrating how correlation functions and structure constants support the duality's consistency.

## Contribution

It establishes the correspondence between Liouville theory null vectors and BRST states, deriving BPZ equations that confirm Liouville's role in the dual CFT.

## Key findings

- Null vectors correspond to BRST exact states.
- BPZ equations are derived for dual correlators.
- Liouville structure constants are uniquely determined.

## Abstract

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.