## Integrability from Chern-Simons theories

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 |