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.