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Peer-reviewedCC BY 4.0Journal of High Energy Physics

Grassmannian geometries for non-planar on-shell diagrams

  • Artyom Lisitsyn1
  • Umut Oktem1
  • Melissa Sherman-Bennett1
  • Jaroslav Trnka1
  1. 1University of California
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Abstract

A bstract : On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the 𝒩 N = 4 Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian G ( k , n ). Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large N limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian G β‚Š ( k , n ) which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite N . In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian G (2, n ). These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are internally planar there always exists a strongly connected geometry, and for those that are irreducible , there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

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Cite

BibTeXRIS
BibTeX
@article{lisitsyn2026grassmannian,
  title = {{Grassmannian geometries for non-planar on-shell diagrams}},
  author = {Artyom Lisitsyn and Umut Oktem and Melissa Sherman-Bennett and Jaroslav Trnka},
  journal = {Journal of High Energy Physics},
  year = {2026},
  month = nov,
  doi = {10.1007/jhep11(2026)017},
  url = {https://link.springer.com/article/10.1007/JHEP11(2026)017},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Grassmannian geometries for non-planar on-shell diagrams
AU  - Artyom Lisitsyn
AU  - Umut Oktem
AU  - Melissa Sherman-Bennett
AU  - Jaroslav Trnka
PY  - 2026
DA  - 2026/11/04
JO  - Journal of High Energy Physics
DO  - 10.1007/jhep11(2026)017
UR  - https://link.springer.com/article/10.1007/JHEP11(2026)017
AB  - A bstract : On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the 𝒩 N = 4 Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian G ( k , n ). Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large N limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian G β‚Š ( k , n ) which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite N . In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian G (2, n ). These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are internally planar there always exists a strongly connected geometry, and for those that are irreducible , there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.
SN  - 1029-8479
ER  -