Proof of the conjectured four-point correlator formula for AdS3 strings

Prove that the conjectured formulas (equations (4ptf-even) and (4ptf-odd)) for spectrally flowed four-point worldsheet correlation functions in the SL(2,R) WZW model on AdS3 are correct for generic spins, spectral flow charges, and insertion points, including establishing the required factorization properties and crossing symmetry for k>3 to complete the solution of four-point functions.

Background

The paper presents a conjectural closed-form expression for four-point functions with arbitrary spectral flow charges, written in terms of generalized cross-ratios built from holomorphic covering maps and unflowed conformal blocks. While this framework reproduces various non-trivial factorization limits and special cases (e.g., tensionless k=3), a general proof is lacking. Establishing factorization and crossing symmetry for generic k>3 would validate the conjecture and finalize the solution of four-point dynamics in the AdS3 string worldsheet theory.

References

At the moment of writing these notes, the conjecture put forward in remains unproven, although a number of important aspects were considered in from the holographic perspective.

— Lecture notes on strings in AdS$_3$ from the worldsheet and the AdS$_3$/CFT$_2$ duality  (2601.06697 - Kovensky, 10 Jan 2026) in Section "The conjecture for four-point functions" (eqs. (4ptf-even), (4ptf-odd))