Perturbations of G2 Instantons

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

Abstract

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 seven dimensions. Dimension seven 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 study instantons on a particular class of seven-dimensional manifolds which includes for example the seven-dimensional sphere. The main goal is develop general tools to identify whether or not instantons can be deformed - in other words, to determine whether it is possible to construct a new instanton by taking an existing instanton and making a small change to it. The motivation comes from one dimension higher: to apply Donaldson's ideas to instantons on eight-dimensional manifolds one must consider the various ways that instantons can collapse, and one known route of collapse is provided by instantons on seven-dimensional spheres. Thus the outcome of the project will either be the identification of new routes for instanton collapse, or the ruling out a whole class of routes for instanton collapse, either of which will be valuable information for the research community.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509681/1 01/10/2016 30/09/2021
1801930 Studentship EP/N509681/1 01/10/2016 31/03/2020 Joseph John Driscoll
 
Description Sixth form Conferece, Leeds 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Schools
Results and Impact Talk for sixth form students from across the country. I explained to them the ideas involved in my research.
Year(s) Of Engagement Activity 2018