dc.contributor.author Bittleston, Roland dc.date.accessioned 2022-01-12T06:10:15Z dc.date.available 2022-01-12T06:10:15Z dc.date.submitted 2022-08-02 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/332645 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.rights All Rights Reserved dc.rights.uri https://www.rioxx.net/licenses/all-rights-reserved/ dc.subject Chern-Simons dc.subject Integrability dc.subject Twistor dc.subject Integrable dc.title Integrability from Chern-Simons theories dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.date.updated 2022-01-10T14:34:41Z dc.identifier.doi 10.17863/CAM.80090 rioxxterms.licenseref.uri https://www.rioxx.net/licenses/all-rights-reserved/ rioxxterms.type Thesis dc.publisher.college Trinity pubs.funder-project-id EPSRC (1936254) pubs.funder-project-id Engineering and Physical Sciences Research Council (1936254) cam.supervisor Skinner, David cam.supervisor.orcid Skinner, David [0000-0002-3014-9127] cam.depositDate 2022-01-10 pubs.licence-identifier apollo-deposit-licence-2-1 pubs.licence-display-name Apollo Repository Deposit Licence Agreement
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