A proof for string three-point functions in AdS$_3$

TL;DR

This paper proves a conjectured integral expression for three-point functions involving spectrally flowed vertex operators in the SL(2,R)-WZW model, advancing understanding of AdS3/CFT2 holography.

Contribution

It provides a rigorous proof of a recent conjecture for three-point functions, extending previous methods and emphasizing the role of holomorphic covering maps.

Findings

Confirmed the integral expression for three-point functions.

Extended methods based on SL(2,R) series identifications.

Highlighted the importance of holomorphic covering maps.

Abstract

Correlation functions of the $\text{SL}(2,\mathbb{R})$-WZW model involving spectrally flowed vertex operators are notoriously difficult to compute. An explicit integral expression for the corresponding three-point functions was recently conjectured in [arXiv:2105.12130v2]. In this paper, we provide a proof for this conjecture. For this, we extend the methods of [arXiv:2208.00978] based on the so-called $\text{SL}(2,\mathbb{R})$ series identifications, which relate vertex operators belonging to different spectral flow sectors. We also highlight the role of holomorphic covering maps in this context. Our results constitute an important milestone for proving this instance of the AdS$_3$/CFT$_2$ holographic duality at finite 't Hooft coupling.