Formal proof of the Dei–Iguri–Kovensky–Toro integral relation for flowed four-point functions

Prove the conjectured integral relation that expresses spectrally flowed SL(2,R) WZW four-point functions in terms of unflowed four-point functions, as proposed in Dei et al. (2021). Specifically, establish rigorously that the y-basis correlators with total spectral flow charge either even or odd are given by the integral transforms in equations (even4pt) and (odd4pt), and that the corresponding x-basis correlators follow from the Mellin-type transform in equation (xtoybasis), for arbitrary external spectral flow charges and spins. The proof should derive the formula from first principles (e.g., local Ward identities and the Knizhnik–Zamolodchikov equations) and hold under general conditions on the kinematics and insertion points.

Background

The paper builds on a conjectural formula due to Dei et al. (2021) that relates spectrally flowed bosonic SL(2,R) four-point functions to unflowed correlators via explicit integral expressions. These expressions, presented in equations (even4pt) and (odd4pt), are key to extending factorization analyses and matching spacetime OPEs, including long- and short-string exchanges.

The authors provide substantial evidence for the conjecture by demonstrating consistent spacetime factorization across channels and exact matching with known holographic data, both in the bosonic and supersymmetric settings. Despite this evidence, a formal derivation establishing the integral relation remains unavailable, and the authors emphasize the need for a rigorous proof.

A formal proof would cement the foundation for computing spectrally flowed four-point functions and strengthen the bridge between worldsheet and spacetime correlators beyond the tensionless point.