Proof of KLT relations beyond tree level

Prove Kawai–Lewellen–Tye relations beyond tree level, establishing a rigorous formulation that holds at loop order or higher genus, and clarifying their connection to intersection theory.

Background

KLT relations connect closed-string amplitudes to products of open-string amplitudes and motivate the color–kinematics duality in field theory. While tree-level formulations are well established, extending them beyond tree level remains challenging.

The authors emphasize that despite recent advances using intersection-theoretic ideas, a complete proof beyond tree level has not yet been achieved.