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Non-Planar On-Shell Diagrams

Sebastian Franco, Daniele Galloni, Brenda Penante, Congkao Wen

TL;DR

This work extends the on-shell diagram program beyond planarity in $ ${N}=4$ SYM by introducing generalized face variables, a boundary measurement to the Grassmannian, and a combinatorial framework built on generalized matching and matroid polytopes. It develops a consistent $d\log$ formulation for non-planar diagrams, proposes a boundary map valid for arbitrary genus, and reveals novel phenomena such as non-Plücker constraints and new pole structures that do not arise in planar cases. The authors provide a comprehensive combinatorial method to compute non-planar on-shell forms directly in terms of minors, illustrate with genus-one and genus-two examples, and analyze reducibility with explicit non-planar counterexamples. The work lays groundwork for a systematic classification of non-planar leading singularities and offers tools potentially relevant to extending positivity and amplituhedron concepts beyond the planar limit.

Abstract

We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in terms of generalized matroid polytopes. Interestingly, non-planar diagrams exhibit novel phenomena, such as the emergence of constraints on Plucker coordinates beyond Plucker relations when deleting edges, which are neatly captured by the generalized matching and matroid polytopes. This behavior is tied to the existence of a new type of poles in the on-shell form at which combinations of Plucker coordinates vanish. Finally, we introduce a prescription, applicable beyond the MHV case, for writing the on-shell form as a function of minors directly from the graph.

Non-Planar On-Shell Diagrams

TL;DR

This work extends the on-shell diagram program beyond planarity in {N}=4d\log$ formulation for non-planar diagrams, proposes a boundary map valid for arbitrary genus, and reveals novel phenomena such as non-Plücker constraints and new pole structures that do not arise in planar cases. The authors provide a comprehensive combinatorial method to compute non-planar on-shell forms directly in terms of minors, illustrate with genus-one and genus-two examples, and analyze reducibility with explicit non-planar counterexamples. The work lays groundwork for a systematic classification of non-planar leading singularities and offers tools potentially relevant to extending positivity and amplituhedron concepts beyond the planar limit.

Abstract

We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in terms of generalized matroid polytopes. Interestingly, non-planar diagrams exhibit novel phenomena, such as the emergence of constraints on Plucker coordinates beyond Plucker relations when deleting edges, which are neatly captured by the generalized matching and matroid polytopes. This behavior is tied to the existence of a new type of poles in the on-shell form at which combinations of Plucker coordinates vanish. Finally, we introduce a prescription, applicable beyond the MHV case, for writing the on-shell form as a function of minors directly from the graph.

Paper Structure

This paper contains 52 sections, 121 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Planar On-Shell Diagrams and Bipartite Technology
  3. On-Shell Diagrams and On-Shell Forms
  4. Equivalence Moves and Reductions
  5. Bipartite Graph Technology and the Boundary Measurement
  6. Non-Planar On-Shell Diagrams and Non-Adjacent BCFW Shifts
  7. Generalized Face Variables
  8. Embedding Into a Riemann Surface
  9. Canonical Variables for Non-Planar Diagrams: Generalized Faces
  10. The $\boldsymbol{d}$log Form
  11. A Genus-One, $\boldsymbol{B=2}$ Example
  12. Combinatorial Characterization of Non-Planar Diagrams: Generalized Matching and Matroid Polytopes
  13. Constructing the Polytopes
  14. $\bullet$ Method 1.
  15. $\bullet$ Method 2.
  16. ...and 37 more sections

Figures (34)

  • Figure 1: In a merger move, two connected internal nodes of the same color are condensed. When two internal nodes of the same color are connected by an edge, we can also introduce a 2-valent node of the opposite color between them.
  • Figure 2: Square move.
  • Figure 3: Bubble reduction.
  • Figure 4: (a) The graph for $\mathcal{A}^{\text{MHV}}_4$, (b) a choice of a possible perfect matching is shown in red and (c) the perfect orientation associated to it. Here 3 and 4 are the sources while 1 and 2 are the sinks.
  • Figure 5: On-shell diagram for the tree-level four-point MHV amplitude $\mathcal{A}^{\text{MHV}}_4$. The number of degrees of freedom is $d=4$. Faces are labeled in green, external nodes in black and edges in red.
  • ...and 29 more figures