On spectrally flowed local vertex operators in AdS$_3$

TL;DR

This paper introduces a new local definition for spectrally flowed vertex operators in the SL(2,ℝ)-WZW model, connecting correlator computations with recent methods and proving conjectures about three-point function structure constants.

Contribution

It generalizes the spectral flow operator definition to all spectral flow charges and links different computational approaches in the SL(2,ℝ)-WZW model.

Findings

Established the connection between spectral flow correlator computations and local Ward identities.

Derived recursion relations as null-state conditions for generalized spectral flowed operators.

Proved the conjecture on y-space structure constants for three-point functions with arbitrary spectral flow.

Abstract

We provide a novel local definition for spectrally flowed vertex operators in the SL(2,$\mathbb{R}$)-WZW model, generalising the proposal of Maldacena and Ooguri in [arXiv:hep-th/0111180] for the singly-flowed case to all $\omega > 1$. This allows us to establish the precise connection between the computation of correlators using the so-called spectral flow operator, and the methods introduced recently by Dei and Eberhardt in [arXiv:2105.12130] based on local Ward identities. We show that the auxiliary variable $y$ used in the latter paper arises naturally from a point-splitting procedure in the space-time coordinate. The recursion relations satisfied by spectrally flowed correlators, which take the form of partial differential equations in $y$-space, then correspond to null-state conditions for generalised spectral flowed operators. We highlight the role of the SL(2,$\mathbb{R}$) series identifications in this context, and prove the validity of the conjecture put forward in [arXiv:2105.12130] for $y$-space structure constants of three-point functions with arbitrary spectral flow charges.