Abstract
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such as the Grassmannian. We perform a detailed investigation of the combinatorial and geometric objects associated to these graphs. We mainly focus on their relation to polytopes and toric geometry, the Grassmannian and its stratification. Our work extends the current understanding of these connections along several important fronts, most notably eliminating restrictions imposed by planarity, positivity, reducibility and edge removability. We illustrate our ideas with several explicit examples and introduce concrete methods that considerably simplify computations. We consider it highly likely that the structures unveiled in this article will arise in the on-shell study of scattering amplitudes beyond the planar limit. Our results can be conversely regarded as an expansion in the understanding of the Grassmannian in terms of bipartite graphs.
Highlights
We are in the midst of what might become a profound reformulation of quantum field theory, one which privileges hidden infinite dimensional symmetries over manifest locality and unitarity [1,2,3,4,5]
The main laboratory for the new ideas is planar N = 4 SYM. This approach has led to a focus on on-shell diagrams, equivalently bipartite graphs, which determine well-defined physical quantities exhibiting all the symmetries of the quantum field theory [6]
Given a differential form related to an on-shell diagram, the singularity structure contains the information of the residues at the poles of the differential form, which are generically located at some βi = 0.16 These singularities correspond to elements in the Grassmannian where the number of degrees of freedom in the matrix C has been reduced, by turning off some βi
Summary
We are in the midst of what might become a profound reformulation of quantum field theory, one which privileges hidden infinite dimensional symmetries over manifest locality and unitarity [1,2,3,4,5]. It is possible to obtain the perfect matching from a perfect orientation by selecting the incoming arrow for white nodes and the outgoing arrow for black nodes. These paths necessarily live in the perfect orientation associated to pref, because all arrows point out of white nodes and into black nodes except for the ones belonging to pref, which have opposite orientation These paths are precisely the flows in the perfect orientation defined by pref, i.e. we can think about them as pμ = pμ − pref. The number of flows in each perfect orientation is equal to the number of perfect matchings, and they are found by subtracting the reference perfect matching from the corresponding perfect matchings
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