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A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator

Chih-Hao Fu, Pierre Vanhove, Yihong Wang

TL;DR

This work develops a vertex-operator algebra framework to derive BCJ kinematic numerators from open-string monodromy, formulating the numerators as $α'$-weighted commutators of vertex operators and off-shell currents. It identifies a diffeomorphism subalgebra in the vector sector and expresses higher-point numerators as nested commutators, linking disc-integrals to Berends–Giele currents and hypergeometric functions. At 3 and 4 points, the construction reproduces YM amplitudes and satisfies Jacobi-type relations, providing a string-inspired perspective on color–kinematics duality and the double-copy principle. The results suggest a string-origin generalization of BCJ numerators and motivate further exploration of loop-level structures and alternative representations within a vertex-operator framework.

Abstract

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an $α'$-weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the $α'$ deformed colour structure observed in open string and semi-abelian $Z$-theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra.

A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator

TL;DR

This work develops a vertex-operator algebra framework to derive BCJ kinematic numerators from open-string monodromy, formulating the numerators as -weighted commutators of vertex operators and off-shell currents. It identifies a diffeomorphism subalgebra in the vector sector and expresses higher-point numerators as nested commutators, linking disc-integrals to Berends–Giele currents and hypergeometric functions. At 3 and 4 points, the construction reproduces YM amplitudes and satisfies Jacobi-type relations, providing a string-inspired perspective on color–kinematics duality and the double-copy principle. The results suggest a string-origin generalization of BCJ numerators and motivate further exploration of loop-level structures and alternative representations within a vertex-operator framework.

Abstract

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an -weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the deformed colour structure observed in open string and semi-abelian -theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra.

Paper Structure

This paper contains 13 sections, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An off-shell continuation of the open string amplitude
  4. Half-ladder basis numerators and the vertex operator algebra
  5. The three point numerator
  6. Four points and higher
  7. Explicit generators
  8. Structure constants of the vertex operator algebra
  9. Reproducing the diffeomorphism algebra
  10. Tensors and scalars contributions
  11. Structure constants and disc integrals
  12. Jacobi identities and the four-point numerators
  13. Conclusion

Figures (4)

  • Figure 1: Integration contours associated with world-sheet variables $v_{i}^{-}$ in the KLT monodromy relations.
  • Figure 2: Integration contours present in the evaluation of three point kinematic numerator
  • Figure 3: The contours corresponding to $V_{1}V_{2}$ and $V_{2}V_{1}$ respectively.
  • Figure 4: The contours present in the $\alpha'$-weighted commutator calculation