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Integrable field theories in two dimensions have the special property that they can be solved exactly. While they are rare, it is remarkable that they can be constructed systematically from four-dimensional Chern-Simons theory with defects. In this project you will investigate this construction, and explore its implications for the landscape and physics of integrable field theories.
Due to the presence of an infinite number of conserved charges, integrable models can be solved exactly. In this project, you will investigate a class of such models called quantum spin chains. These have applications to a wide range of topics, from toy models of magnetism to anomalous dimensions in the 4-d gauge theory N=4 Super Yang-Mills. The main goal of the project is to understand which boundaries preserve the integrability of quantum spin chains and to investigate the properties of these models. There are several different possible models to focus on, including an su(2)⊕su(2) model describing electrons that only move when in pairs, a deformation of the Hubbard model and a perturbative long-range deformation of the Heisenberg spin chain.
In this project you will investigate two types of integrable model in two dimensions and explore the relation between them. The first are integrable sigma models, which describe the classical motion of relativistic strings on a special class of curved space-times. The second are massive integrable models, which have the remarkable property that the scattering of excitations in these theories can be described exactly. In the first part of the project, you will familiarise yourself with the two types of integrable model and some of the methods that are used to study them. The aim of the second part is then to use this knowledge to investigate the simplest examples of the duality.