{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/332645"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/332645","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Integrability from Chern-Simons theories","abstract":"This thesis details my work exploring connections between integrable systems and Chern-Simons theories. It is divided into two parts. The first concerns the application of 4d Chern-Simons theory to describe integrable models with boundary, while the second concerns relations between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. Part one opens with a review of 4d Chern-Simons theory, including a discussion of its connections to both quantum and classical integrable systems. It then turns to the results of this thesis concerning the application of 4d Chern-Simons theory to generate solutions of the boundary Yang-Baxter equation. They include: defining the boundary analogue of a quasi-classical $R$-matrix and classical $r$-matrix; realising $K$-matrices as the vacuum expectation values of Wilson lines in 4d Chern-Simons theory on a $\\bbZ_2$ orbifold; deriving the order $\\hbar$ contribution to a $K$-matrix in the rational case and verifying that it obeys the boundary Yang-Baxter equation to second order in $\\hbar$; determining the OPE of bulk and boundary Wilson lines; demonstrating that boundary line operators are labelled by representations of twisted Yangians; giving the gauge theory realisation of boundary unitarity and the Sklyanin determinant; proving the uniqueness of the rational $K$-matrix; obtaining explicit formulae for the order $\\hbar$ contributions to trigonometric and elliptic $K$-matrices and matching them to examples in the literature. Part two begins with a review of twistor theory. This is followed by the results of this thesis concerning the connections between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. They include: showing that holomorphic Chern-Simons theory on twistor space for a meromorphic measure descends to an integrable theory on 4d spacetime; extending these results to indefinite signatures; identifying 4d Chern-Simons theory as the quotient of a 6d Chern-Simons theory on twistor correspondence space by an appropriate lift of the 2d translation group on spacetime; quotienting holomorphic Chern-Simons theory on twistor space by a 1 dimensional group of translations to obtain a 5d Chern-Simons theory on minitwistor correspondence space describing the Bogomolny equations.","abstract_html":"This thesis details my work exploring connections between integrable systems and Chern-Simons theories. It is divided into two parts. The first concerns the application of 4d Chern-Simons theory to describe integrable models with boundary, while the second concerns relations between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. Part one opens with a review of 4d Chern-Simons theory, including a discussion of its connections to both quantum and classical integrable systems. It then turns to the results of this thesis concerning the application of 4d Chern-Simons theory to generate solutions of the boundary Yang-Baxter equation. They include: defining the boundary analogue of a quasi-classical $R$-matrix and classical $r$-matrix; realising $K$-matrices as the vacuum expectation values of Wilson lines in 4d Chern-Simons theory on a <span class=\"etd-inline-math\">\\bbZ<sub>2</sub></span> orbifold; deriving the order $\\hbar$ contribution to a $K$-matrix in the rational case and verifying that it obeys the boundary Yang-Baxter equation to second order in $\\hbar$; determining the OPE of bulk and boundary Wilson lines; demonstrating that boundary line operators are labelled by representations of twisted Yangians; giving the gauge theory realisation of boundary unitarity and the Sklyanin determinant; proving the uniqueness of the rational $K$-matrix; obtaining explicit formulae for the order $\\hbar$ contributions to trigonometric and elliptic $K$-matrices and matching them to examples in the literature. Part two begins with a review of twistor theory. This is followed by the results of this thesis concerning the connections between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. They include: showing that holomorphic Chern-Simons theory on twistor space for a meromorphic measure descends to an integrable theory on 4d spacetime; extending these results to indefinite signatures; identifying 4d Chern-Simons theory as the quotient of a 6d Chern-Simons theory on twistor correspondence space by an appropriate lift of the 2d translation group on spacetime; quotienting holomorphic Chern-Simons theory on twistor space by a 1 dimensional group of translations to obtain a 5d Chern-Simons theory on minitwistor correspondence space describing the Bogomolny equations.","abstract_has_math":true,"creators":["Bittleston, Roland"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Skinner, David"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08-02","date_published":"2022-08-02","updated_at":"2026-07-22T22:24:01Z","subjects":["Chern-Simons","Integrability","Twistor","Integrable"],"languages":["eng"],"rights":[],"rights_urls":["http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000230149127"],"render_values":[{"text":"0000-0002-3014-9127","href":"https://orcid.org/0000-0002-3014-9127","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.80090","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Skinner, David"]},{"key":"dc:creator","label":"Author","values":["Bittleston, Roland"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000230149127"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2022-08-02"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/332645"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Chern-Simons","Integrability","Twistor","Integrable"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.80090"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/00f3b31b-be82-4953-8aab-ae8bb6bd1d1a/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis details my work exploring connections between integrable systems and Chern-Simons theories. It is divided into two parts. The first concerns the application of 4d Chern-Simons theory to describe integrable models with boundary, while the second concerns relations between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. Part one opens with a review of 4d Chern-Simons theory, including a discussion of its connections to both quantum and classical integrable systems. It then turns to the results of this thesis concerning the application of 4d Chern-Simons theory to generate solutions of the boundary Yang-Baxter equation. They include: defining the boundary analogue of a quasi-classical $R$-matrix and classical $r$-matrix; realising $K$-matrices as the vacuum expectation values of Wilson lines in 4d Chern-Simons theory on a $\\bbZ_2$ orbifold; deriving the order $\\hbar$ contribution to a $K$-matrix in the rational case and verifying that it obeys the boundary Yang-Baxter equation to second order in $\\hbar$; determining the OPE of bulk and boundary Wilson lines; demonstrating that boundary line operators are labelled by representations of twisted Yangians; giving the gauge theory realisation of boundary unitarity and the Sklyanin determinant; proving the uniqueness of the rational $K$-matrix; obtaining explicit formulae for the order $\\hbar$ contributions to trigonometric and elliptic $K$-matrices and matching them to examples in the literature. Part two begins with a review of twistor theory. This is followed by the results of this thesis concerning the connections between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. They include: showing that holomorphic Chern-Simons theory on twistor space for a meromorphic measure descends to an integrable theory on 4d spacetime; extending these results to indefinite signatures; identifying 4d Chern-Simons theory as the quotient of a 6d Chern-Simons theory on twistor correspondence space by an appropriate lift of the 2d translation group on spacetime; quotienting holomorphic Chern-Simons theory on twistor space by a 1 dimensional group of translations to obtain a 5d Chern-Simons theory on minitwistor correspondence space describing the Bogomolny equations."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["e0856e1ff3c1a2568f30cc4bce77ce9a"]},{"key":"dc:title","label":"Title","values":["Integrability from Chern-Simons theories"]}]}],"canonical_facts":{"dc:contributor.advisor":["Skinner, David"],"dc:creator":["Bittleston, Roland"],"dc:creator.authoridentifier":["0000000230149127"],"dc:date.issued":["2022-08-02"],"dc:description.abstract":["This thesis details my work exploring connections between integrable systems and Chern-Simons theories. It is divided into two parts. The first concerns the application of 4d Chern-Simons theory to describe integrable models with boundary, while the second concerns relations between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. Part one opens with a review of 4d Chern-Simons theory, including a discussion of its connections to both quantum and classical integrable systems. It then turns to the results of this thesis concerning the application of 4d Chern-Simons theory to generate solutions of the boundary Yang-Baxter equation. They include: defining the boundary analogue of a quasi-classical $R$-matrix and classical $r$-matrix; realising $K$-matrices as the vacuum expectation values of Wilson lines in 4d Chern-Simons theory on a $\\bbZ_2$ orbifold; deriving the order $\\hbar$ contribution to a $K$-matrix in the rational case and verifying that it obeys the boundary Yang-Baxter equation to second order in $\\hbar$; determining the OPE of bulk and boundary Wilson lines; demonstrating that boundary line operators are labelled by representations of twisted Yangians; giving the gauge theory realisation of boundary unitarity and the Sklyanin determinant; proving the uniqueness of the rational $K$-matrix; obtaining explicit formulae for the order $\\hbar$ contributions to trigonometric and elliptic $K$-matrices and matching them to examples in the literature. Part two begins with a review of twistor theory. This is followed by the results of this thesis concerning the connections between holomorphic Chern-Simons theory on twistor space, 4d Chern-Simons theory and the anti-self-dual Yang-Mills equations. They include: showing that holomorphic Chern-Simons theory on twistor space for a meromorphic measure descends to an integrable theory on 4d spacetime; extending these results to indefinite signatures; identifying 4d Chern-Simons theory as the quotient of a 6d Chern-Simons theory on twistor correspondence space by an appropriate lift of the 2d translation group on spacetime; quotienting holomorphic Chern-Simons theory on twistor space by a 1 dimensional group of translations to obtain a 5d Chern-Simons theory on minitwistor correspondence space describing the Bogomolny equations."],"dc:format.checksum.md5":["e0856e1ff3c1a2568f30cc4bce77ce9a"],"dc:identifier.doi":["10.17863/CAM.80090"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/00f3b31b-be82-4953-8aab-ae8bb6bd1d1a/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/332645"],"dc:rights":["http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["Chern-Simons","Integrability","Twistor","Integrable"],"dc:title":["Integrability from Chern-Simons theories"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:01Z"}