# 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.

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.

### 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 | In physics there is a natural Energy, called the Yang-Mills energy, associated to a mathematical object called a connnection (these are objects that generalise the notion of a magentic field). It is natural to look for optimal connections- such object are called "instanton connections", they actually minimise the Yang-Mills energy and have important applications in physics. Instantons have been studied in depth on 4 dimensional spaces (this is the critical dimension for physicists as space time is four dimensional), but more recently mathematicians have been interested in studying these connections on more general spaces, particularly in higher dimensions. In dimensions 7 there are spaces called G2 manifolds that have a very special geometric structure only occurring in this dimension and are a suitable background for studying instantons. Furthermore G2 manifolds appear in physics when one studies a branch of string theory called "M-Theory" and again the instanton equation has physical relevance. The simplest G2 manifolds is flat Euclidean space in dimension 7 (where one has 7 Cartesian coordinates and straight lines minimise the distance between points). There is an example of a G2 instanton connection on this space, which is in some sense the most basic example and was found by two physicists Gunaydin and Nicolai in 1995. It is natural to ask if this is the only example in this setting, or if there are in fact other solutions to the equation that we do not know about. The main result of this award is that this basic G2 instanton is in fact unique- it is the only solution to the G2 instanton equation in this setting. |

Exploitation Route | I have shown that a recently found (2018) example of a G2 instanton admits at least a 1 dimensional space of perturbations, in the paper where this instantons was constructed the authors have a candidate for understanding this perturbation, although they do not believe (via numerical investigation) that this candidate extands to a global solution. It would be interesting to compare the two results and identify if this is indeed the deformation in question. Recently (2017) new examples of spaces suitable for the application of my work were found. It is ongoing research to find G2 instantons on this spaces. When examples have been found, one will be able to use the theory I have developed to try to understand if these examples are unique and if not what the dimension of the space of solutions is. The project relies heavily on Harland and Charbonneau's work on nearly Kahler Instantons. It would be interesting to extend their results to the bundles relevant to asymptotically conical G2 instantons- then one could compare the moduli spaces of the G2 instanton and its asymptotic nearly kahler instanton- one can ask when this map is surjective, if the moduli spaces are always invariant under a group action. The work of Lotay-Oliveria would then be relevant for proving more general uniqueness theorems. |

Sectors | Other |

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 |