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Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators

Published online by Cambridge University Press:  23 July 2021

Susama Agarwala
Affiliation:
Applied Physics Laboratory, Johns Hopkins University, Laurel, MD, USA e-mail: susama@alum.mit.edu
Siân Fryer
Affiliation:
UC Santa Barbara, Department of Mathematics Yeats, Santa Barbara, CA, USA e-mail: fryer@math.ucsb.edu
Karen Yeats*
Affiliation:
University of Waterloo, Department of Combinatorics and Optimization, Waterloo, ON, Canada

Abstract

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. In this paper, we study the structure of the associated positroids, as well as the structure of the denominator of the integrand defined by each diagram. We give an algorithm to derive the Grassmann necklace of the associated positroid directly from the Wilson loop diagram, and a recursive proof that the dimension of these cells is thrice the number of propagators in the diagram. We also show that the ideal generated by the denominator in the integrand is the radical of the ideal generated by the product of Grassmann necklace minors.

MSC classification

Type
Article
Copyright
© Canadian Mathematical Society 2021

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Footnotes

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, through some the time this work was developed, by a Humboldt Fellowship from the Alexander von Humboldt Foundation.

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