Strong-coupling duals of integrable sigma models

Ben Hoare

This is a project outline for the 4th year of the Mathematics degree at Durham University.

Please send comments and corrections to Ben Hoare at

Durham, 5 October 2020

last updated 2 February 2021

Description. Integrable field theories play a key role in a number of different areas of theoretical physics, including string theory and statistical mechanics, as well as being interesting physical models in their own right. The hallmark of integrability is the presence of a large, but hidden, symmetry that opens up new avenues to exploring the physics of these models.

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.

Integrable sigma models are typically difficult to quantize. One approach is to construct a dual massive integrable model that can be studied using standard methods. To see how this works let us consider a relativistic string moving on a two-sphere. We can continuously deform the two-sphere into a sausage while preserving integrability. Taking the deformation to its limit we find the cigar sigma model.

Reversing this logic, we may think of the sausage sigma model as a perturbation of the cigar sigma model. This new perspective allows us to construct a "dual" perturbation, which defines a massive integrable model describing the same physics as the deformed integrable sigma model. This phenomenon is known as duality.

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 based on the two-sphere and three-sphere. From this point you will be able to explore many different directions, including higher-dimensional spheres, more general symmetric spaces, supersymmetric models and the interplay between different integrable deformations.

Prerequisties and corequisites. Quantum Mechanics III and Advanced Quantum Theory IV, or the equivalent courses in Physics, are essential. Other courses that may be useful, but not essential, include Solitons III/IV, Statistical Mechanics III/IV and General Relativity IV.

Resources. The following pedagogical reviews introduce the two types of integrable model and will form a basis for the first part of the project:

The following two papers introducing the duality are from the 1990s:

They are significantly more advanced and the aim is that you will understand them over the course of the project. A more modern perspective can be found in:

These are the primary references for the initial stages of the project. Additional references will be based on the directions you choose to explore.