Non-planar on-shell diagrams

Authors

Sebastían Franco, CUNY City CollegeFollow
Daniele Galloni, Durham University
Brenda Penante, Queen Mary University of London
Congkao Wen, University of Rome Tor Vergata

Document Type

Article

Publication Date

2015

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 d log 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 Plücker coordinates beyond Plücker 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 Plücker 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.

Comments

This article was originally published in Journal of High Energy Physics, available at http://dx.doi.org/10.1007/JHEP06(2015)199

This work is distributed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).