On spectrally flowed local vertex operators in AdS$_3$

Abstract

We provide a novel local definition for spectrally flowed vertex operators in the SL(2,\mathbb{R})SL(2,ℝ)-WZW model, generalising the proposal of Maldacena and Ooguri in [arXiv:hep-th/0111180] for the singly-flowed case to all \omega>1ω>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 yy 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 yy-space, then correspond to null-state conditions for generalised spectral flowed operators. We highlight the role of certain SL(2,\mathbb{R})SL(2,ℝ) discrete module isomorphisms in this context, and prove the validity of the conjecture put forward in [arXiv:2105.12130] for yy-space structure constants of three-point functions with arbitrary spectral flow charges.