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

Part of: Quantum field theory; related classical field theories

Published online by Cambridge University Press:  23 July 2021

Susama Agarwala ,
Siân Fryer  and
Karen Yeats
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
*
e-mail: kayeats@uwaterloo.ca
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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.

Keywords

Wilson loop diagramspositroidsGrassmann necklaceLe diagram

MSC classification

Primary: 81T60: Supersymmetric field theories
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1625 - 1672
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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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