The University of Southampton
University of Southampton Institutional Repository

On the singularities of scattering amplitudes: from cluster adjacency tropical geometry

On the singularities of scattering amplitudes: from cluster adjacency tropical geometry
On the singularities of scattering amplitudes: from cluster adjacency tropical geometry
Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a unique quantum field theory. Its interesting properties include conformal symmetry at the quantum level, expected integrability, and being a prime example of the AdS/CFT correspondence. Witten’s twistor string theory [1] reignited an interest in this theory, leading to enormous progress in utilising its special features to uncover new mathematical structures, such as the Grassmannian [2], and to better understand gauge theory in general. One such feature is the hidden dual conformal symmetry in SYM [3, 4] which led to the introduction of momentum twistors [5] as useful variables when describing scattering amplitudes in SYM. It was then shown in [6] that the kinematic space parametrised by these momentum twistors, itself a Grassmannian, comes equipped with a particular mathematical structure called a cluster algebra. It was shown that scattering amplitudes in SYM depend on cluster coordinates and, at least for six and seven-points, the cluster algebra provided all the singularities, or alphabet, for all known amplitudes in SYM. The ultimate hope is that by understanding all the analytic structure of the functions scattering amplitudes consist of, one could simply write down the form of an amplitude without the need for an explicit calculation. The overarching theme of this thesis is to better understand the analytic properties of scattering amplitudes, mainly (but not exclusively) in SYM. Although this thesis has two parts, the main focus is on cluster algebras; extracting the information they contain about amplitudes and exploring their relationship with other areas of mathematics, as well as what those relationships imply for the singularity properties of amplitudes. In the first part, we introduce and develop the notion of cluster adjacency and how it controls the poles and branch cuts an amplitude is allowed to have. We also provide an example of how cluster adjacency can aid in computations in calculating the seven-point, four-loop, NMHV scattering amplitude. The second part expands on the discussion started in [7] linking tropical geometry to scattering amplitudes in the biadjoint φ 3 theory. We demonstrate how the connection between tropical Grassmannians and cluster algebras allows for straightforward calculation of amplitudes in this theory. We go on to utilise this connection to generalise the notion of cluster adjacency and conjecture how different helicity configurations (MHV, NMHV, etc.) may each have their own cluster adjacency rules. Finally, tropical geometry allows us to approach certain issues with eight-point scattering in SYM. Namely, the infinite set of cluster coordinates provided by cluster algebra associated with eight-point scattering, as well as generating the algebraic singularities known to appear even at one-loop for N2MHV amplitudes. This thesis is based on [8–13] with considerable overlap with these papers.
University of Southampton
Foster, Jack Alexander
375960ff-578f-4309-93ef-4c55cecf76d2
Foster, Jack Alexander
375960ff-578f-4309-93ef-4c55cecf76d2
Drummond, James
3ea15544-457f-4e72-8ad0-60f3136841db

Foster, Jack Alexander (2020) On the singularities of scattering amplitudes: from cluster adjacency tropical geometry. Doctoral Thesis, 181pp.

Record type: Thesis (Doctoral)

Abstract

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a unique quantum field theory. Its interesting properties include conformal symmetry at the quantum level, expected integrability, and being a prime example of the AdS/CFT correspondence. Witten’s twistor string theory [1] reignited an interest in this theory, leading to enormous progress in utilising its special features to uncover new mathematical structures, such as the Grassmannian [2], and to better understand gauge theory in general. One such feature is the hidden dual conformal symmetry in SYM [3, 4] which led to the introduction of momentum twistors [5] as useful variables when describing scattering amplitudes in SYM. It was then shown in [6] that the kinematic space parametrised by these momentum twistors, itself a Grassmannian, comes equipped with a particular mathematical structure called a cluster algebra. It was shown that scattering amplitudes in SYM depend on cluster coordinates and, at least for six and seven-points, the cluster algebra provided all the singularities, or alphabet, for all known amplitudes in SYM. The ultimate hope is that by understanding all the analytic structure of the functions scattering amplitudes consist of, one could simply write down the form of an amplitude without the need for an explicit calculation. The overarching theme of this thesis is to better understand the analytic properties of scattering amplitudes, mainly (but not exclusively) in SYM. Although this thesis has two parts, the main focus is on cluster algebras; extracting the information they contain about amplitudes and exploring their relationship with other areas of mathematics, as well as what those relationships imply for the singularity properties of amplitudes. In the first part, we introduce and develop the notion of cluster adjacency and how it controls the poles and branch cuts an amplitude is allowed to have. We also provide an example of how cluster adjacency can aid in computations in calculating the seven-point, four-loop, NMHV scattering amplitude. The second part expands on the discussion started in [7] linking tropical geometry to scattering amplitudes in the biadjoint φ 3 theory. We demonstrate how the connection between tropical Grassmannians and cluster algebras allows for straightforward calculation of amplitudes in this theory. We go on to utilise this connection to generalise the notion of cluster adjacency and conjecture how different helicity configurations (MHV, NMHV, etc.) may each have their own cluster adjacency rules. Finally, tropical geometry allows us to approach certain issues with eight-point scattering in SYM. Namely, the infinite set of cluster coordinates provided by cluster algebra associated with eight-point scattering, as well as generating the algebraic singularities known to appear even at one-loop for N2MHV amplitudes. This thesis is based on [8–13] with considerable overlap with these papers.

Text
Jack_Foster_Thesis_Final
Available under License University of Southampton Thesis Licence.
Download (1MB)
Text
PTDThesis_Foster-SIGNED
Restricted to Repository staff only

More information

Published date: October 2020

Identifiers

Local EPrints ID: 447601
URI: http://eprints.soton.ac.uk/id/eprint/447601
PURE UUID: 8dab5146-dfae-45c1-82eb-ba1ee73049a1

Catalogue record

Date deposited: 16 Mar 2021 17:46
Last modified: 16 Mar 2021 17:46

Export record

Contributors

Author: Jack Alexander Foster
Thesis advisor: James Drummond

University divisions

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×