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Trivalent Feynman Diagrams as a Flag

Lili Yang

TL;DR

This work constructs a geometric bridge between flag theory on the moduli space $\mathcal{M}_{0,n}$ and scattering amplitudes by identifying each standard flag with a trivalent Feynman diagram, allowing propagators to be read from flag data via $S^{(a)}(F_m,F_m)$. It recovers the bi-adjoint scalar amplitude as an intersection pairing of twisted cocycles, derives the $Z$-theory amplitude through flag simplices in the $\alpha'\to0$ limit, and connects to the CHY representation by projecting onto special flags associated with the scattering-equation critical points. The framework unifies BCJ/duality, KLT, and CHY within a single flag-theoretic structure built from the Orlik–Solomon algebra and twisted (co)homology, with the residue theorem furnishing the Jacobi relations. Collectively, it provides a concrete, geometric route to reproduce known field- and string-theory amplitudes and suggests extensions to gauge-theory color factors and broader algebraic structures.

Abstract

By identifying each standard flag with a trivalent Feynman diagram, the corresponding propagators can be read directly from the flag itself. Within the flag representation, the kinematic Jacobi identity (equivalently, the residue theorem on moduli spaces) admits a natural interpretation as the equivalence between a complete flag and its gapped counterpart. Using flags together with Orlik-Solomon algebras, we reconstruct the intersection numbers of twisted cocycles, thereby obtaining the bi-adjoint amplitude. Moreover, employing flag simplices enables the construction of the Z-amplitude in the alpha' -> 0 limit. By further examining pairings of specific flags, we also recover the Cachazo-He-Yuan (CHY) representation of the bi-adjoint amplitude.

Trivalent Feynman Diagrams as a Flag

TL;DR

This work constructs a geometric bridge between flag theory on the moduli space and scattering amplitudes by identifying each standard flag with a trivalent Feynman diagram, allowing propagators to be read from flag data via . It recovers the bi-adjoint scalar amplitude as an intersection pairing of twisted cocycles, derives the -theory amplitude through flag simplices in the limit, and connects to the CHY representation by projecting onto special flags associated with the scattering-equation critical points. The framework unifies BCJ/duality, KLT, and CHY within a single flag-theoretic structure built from the Orlik–Solomon algebra and twisted (co)homology, with the residue theorem furnishing the Jacobi relations. Collectively, it provides a concrete, geometric route to reproduce known field- and string-theory amplitudes and suggests extensions to gauge-theory color factors and broader algebraic structures.

Abstract

By identifying each standard flag with a trivalent Feynman diagram, the corresponding propagators can be read directly from the flag itself. Within the flag representation, the kinematic Jacobi identity (equivalently, the residue theorem on moduli spaces) admits a natural interpretation as the equivalence between a complete flag and its gapped counterpart. Using flags together with Orlik-Solomon algebras, we reconstruct the intersection numbers of twisted cocycles, thereby obtaining the bi-adjoint amplitude. Moreover, employing flag simplices enables the construction of the Z-amplitude in the alpha' -> 0 limit. By further examining pairings of specific flags, we also recover the Cachazo-He-Yuan (CHY) representation of the bi-adjoint amplitude.
Paper Structure (11 sections, 138 equations, 3 figures)

This paper contains 11 sections, 138 equations, 3 figures.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Scattering Amplitudes
  4. Hyperplanes Arrangements and Flag
  5. Residue Theorem as Flag with a Gap
  6. Flag Representation of the Scalar Amplitude
  7. Intersection Number of Cocycles Flags and Scalar Amplitudes
  8. Flag Simplex and Z-Amplitude in Filed Theory Limit
  9. CHY Formula and Special Flag
  10. Conclusion
  11. The Detail of Equation (\ref{['A']})

Figures (3)

  • Figure 1: The 5-point example illustrating the relation between trivalent tree diagrams and flags generated by the hyperplanes $H_{12},H_{13},H_{23}$.
  • Figure 2: The 6-point example illustrating the relation between trivalent tree diagrams and flags generated by the hyperplanes $H_{12},H_{14},H_{24}$.
  • Figure 3: The left figure, taken from Mizera_2017, shows all twisted cycles $\hat{C}(\alpha)$ of the Riemann space $\mathcal{M}_{0,5}$ after the blow‑up. The right figure decomposes the integral region $\hat{C}(12345)$ into 10 simplices according to equation (\ref{['s']}).