A Combinatorial Tale of Two Scattering Amplitudes: See Two Bijections
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Date
2022-01-07
Authors
Hu, Simeng Simone
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
In this thesis, we take a journey through two different but not dissimilar stories with an underlying
theme of combinatorics emerging from scattering amplitudes in quantum field theories.
The first part tells the tale of the c2-invariant, an arithmetic invariant related to the Feynman integral in
𝜙4-theory, which studies the zeros of the Kirchoff polynomial and related graph polynomials. Through
reformulating the c2-invariant as a purely combinatorial problem, we show how enumerating certain
edge bipartitions through fixed-point free involutions can complete a special case of the long sought
after c2 completion conjecture.
The second part tells the tale of the positive Grassmannian and a combinatorial T-duality map on its
cells, as related to scattering amplitudes in planar N = 4 SYM theory. In particular, T-duality is a bridge
between triangulations of the hypersimplex and triangulations of the amplituhedron, two objects that
appear as images of the positive Grassmannian. We give an algorithm for viewing T-duality as a map
on Le diagrams and characterize a nice structure to the Le diagrams (which can then be used in lieu
of the algorithm). Through this Le diagram perspective on T-duality, we show how the dimensional
relationship between the positroid cells on either side of the map can be directly explained.
Description
Keywords
c2 invariant, le diagrams, combinatorics, quantum field theory, scattering amplitudes