Symmetric Instantons on 6-Sphere

Lead Research Organisation: University of Leeds
Department Name: Pure Mathematics


One of the great breakthroughs in mathematics during the 1980s was the discovery of Yang-Mills instantons. These are special solutions of a set of partial differential equations. These partial differential equations in question are a generalisation of Maxwell's equations of electromagnetism, which are familiar to students of physics and mathematics.
The instantons that have received the most attention until now are inherently four-dimensional. They play a fundamental role in quantum field theory and the standard model of particle physics. At the same time, they have been used by geometers extensively as a tool to learn about four-dimensional manifolds, exemplified in the Fields Medal awarded to Sir Simon Donaldson in 1986. More recently, instantons have been identified as an important part of supersymmetric theories such as string theory, and have had a profound impact on the subject of integrable systems.
This project will focus on instantons not in four but rather in eight dimensions. Dimension eight is of particular interest at the moment because it is one of the least-understood cases in Marcel Berger's classification of holonomy groups of Riemannian manifolds. In 1955 Berger produced a list of all possible holonomy groups, and at the time all but two (of dimensions seven and eight) had been seen before in concrete examples. It was only much later in 1989 that Robert Bryant and Simon Salamon showed that the remaining two cases are genuine, producing explicit examples of manifolds with the appropriate holonomy group. These two cases remain relatively poorly-understood compared with the other cases in Berger's list.
One way for the mathematical community to enhance its understanding these remaining cases is to study instantons on them. Doing so will create a pool of researchers who are familiar with these geometries. In the longer term, it is hoped that studying instantons will lead to a breakthrough comparable to Donaldson's work on four-manifolds. These instantons are also of interest to string theorists, as solving the instanton equations is one step in the process of constructing a string theory background.
This particular project will seek to construct explicit instantons on some specific eight-dimensional manifolds discovered within the last twenty years. These explicit solutions will provide a testing-ground for theoretical ideas about the general properties of instantons. They could in the future be used for gluing constructions of instantons on other manifolds. The project will furthermore construct supersymmetric string theory backgrounds utilising these instantons, deepening understanding of the backgrounds allowed by string theory. The project will also analyse whether the instanton equations on these particular manifolds are integrable, with the hope of improving the community's understanding of the links between this type of instanton and integrable systems


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509681/1 01/10/2016 30/09/2021
1801251 Studentship EP/N509681/1 01/10/2016 01/06/2018 Ben Daniel-Thorpe