Spectral flow and the conformal block expansion for strings in AdS$_3$

TL;DR

This paper analyzes spectrally flowed four-point functions in the SL(2,R) WZW model, providing a detailed conformal block decomposition and supporting a conjecture relating flowed and unflowed correlators.

Contribution

It introduces a method to describe s-channel exchanges involving spectral flow, linking flowed and unflowed conformal blocks, and computes their normalization factors.

Findings

Supports Dei and Eberhardt's conjecture on spectral flow relations.

Provides explicit normalization formulas for flowed conformal blocks.

Validates the computational framework for spectral flow in string theory.

Abstract

We present a detailed study of spectrally flowed four-point functions in the SL(2,$\mathbb{R}$) WZW model, focusing on their conformal block decomposition. Dei and Eberhardt conjectured a general formula relating these observables to their unflowed counterparts. Although the latter are not known in closed form, their conformal block expansion has been formally established. By combining this information with the integral transform that encodes the effect of spectral flow, we show how to describe a considerable number of $s$-channel exchanges, including cases with both flowed and unflowed intermediate states. For all such processes, we compute the normalization of the corresponding conformal blocks in terms of products of the recently derived flowed three-point functions with arbitrary spectral flow charges. Our results constitute a highly non-trivial consistency check, thus strongly supporting the aforementioned conjecture, and establishing its computational power.