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Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

Published online by Cambridge University Press:  26 February 2021

Susama Agarwala*
Johns Hopkins Applied Physics Lab, Laurel, MD, USA
Siân Fryer
UC Santa Barbara, Santa Barbara, CA93106, USA e-mail:
Karen Yeats
Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada e-mail:


Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

MSC classification

© Canadian Mathematical Society 2021

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SA was partially supported by an Office of Naval Research grant. KY is supported by an NSERC Discovery grant, by the Canada Research Chair program, and also, over some of the time this work was developed, by a Humboldt Fellowship from the Alexander von Humboldt foundation.


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