dc.contributor.author | Moerman, Robert William | |
dc.date.accessioned | 2023-08-31T09:33:14Z | |
dc.date.available | 2023-08-31T09:33:14Z | |
dc.date.issued | 2023-05-25 | |
dc.identifier.uri | http://hdl.handle.net/2299/26621 | |
dc.description.abstract | Positive geometries provide a purely geometric point of departure for studying scattering
amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic
set equipped with a unique rational top form—the canonical form. There are known
examples where the canonical form of some positive geometry, defined in some kinematic
space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of
the positive geometry are in bijection with the physical singularities of the scattering
amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical
positive geometry. It lives in momentum twistor space and describes tree-level (and
the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric
Yang-Mills theory.
In this dissertation, we study three positive geometries defined in on-shell momentum
space: the Arkani-Hamed–Bai–He–Yan (ABHY) associahedron, the Momentum
Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level
scattering amplitudes for different theories in different spacetime dimensions. The three
positive geometries share a series of interrelations in terms of their boundary posets
and canonical forms. We review these relationships in detail, highlighting the author’s
contributions. We study their boundary posets, classifying all boundaries and hence all
physical singularities at the tree level. We develop new combinatorial results to derive
rank-generating functions which enumerate boundaries according to their dimension.
These generating functions allow us to prove that the Euler characteristics of the three
positive geometries are one. In addition, we discuss methods for manipulating canonical
forms using ideas from computational algebraic geometry. | en_US |
dc.language.iso | en | en_US |
dc.rights | info:eu-repo/semantics/openAccess | en_US |
dc.rights | Attribution 3.0 United States | * |
dc.rights.uri | http://creativecommons.org/licenses/by/3.0/us/ | * |
dc.subject | Scattering Amplitudes | en_US |
dc.subject | Supersymmetric Gauge Theory | en_US |
dc.subject | Positive Geometries | en_US |
dc.subject | Enumerative Combinatorics | en_US |
dc.subject | Differential and Algebraic Geometry | en_US |
dc.subject | Algebraic Geometry | en_US |
dc.subject | Differential | en_US |
dc.subject | Differential Geometry | en_US |
dc.title | Positive Geometries for Scattering Amplitudes in Momentum Space | en_US |
dc.type | info:eu-repo/semantics/doctoralThesis | en_US |
dc.identifier.doi | doi:10.18745/th.26621 | * |
dc.identifier.doi | 10.18745/th.26621 | |
dc.type.qualificationlevel | Doctoral | en_US |
dc.type.qualificationname | PhD | en_US |
dcterms.dateAccepted | 2023-05-25 | |
rioxxterms.funder | Default funder | en_US |
rioxxterms.identifier.project | Default project | en_US |
rioxxterms.version | NA | en_US |
rioxxterms.licenseref.uri | https://creativecommons.org/licenses/by/4.0/ | en_US |
rioxxterms.licenseref.startdate | 2023-08-31 | |
herts.preservation.rarelyaccessed | true | |
rioxxterms.funder.project | ba3b3abd-b137-4d1d-949a-23012ce7d7b9 | en_US |