Monopole moduli spaces and manifolds with corners

Lead Research Organisation: University College London
Department Name: Mathematics

Abstract

Non-abelian magnetic monopoles are particle-like solutions of a differential equation that live on ordinary three-dimensional space. It is known that some of the solutions correspond to widely separated particles, but that as the separations decrease, the particles lose their individual identity and can no longer be distinguished. These monopoles are part of a physical theory (Yang--Mills--Higgs theory) which cannot be solved exactly but which predicts that monopoles will interact non-trivially with each other when they move. When they move at low speeds, their motion is well approximated by geodesics on a certain curved space, much as particles in Einstein's theory of general relativity move under the influence of gravity. This project will study the distance function (metric) underlying this curved space M, and will study in particular how it looks at large distances.

By undertaking a detailed study of the metric, we shall be also be able to study natural differential equations which are defined on the space M. This is of interest in the study of differential equations generally, since the large-scale structure of M is rather complicated, and can only be built up recursively. The behaviour of differential equations on M is important since they are necessary for the understanding of the quantum theory of the original Yang--Mills--Higgs theory. The important ingredient here again is the large-scale structure of M and its metric.

This research will be important not only for understanding monopoles and features of the Yang--Mills--Higgs theory of which they are a part, but also for the development of a suite of sophisticated tools for understanding differential equations in this type of setting. These tools will be useful in similar problems and can be expected to have an importance beyond their applications to the theory of monopoles.

Planned Impact

This project will contribute knowledge impacts in terms of key scientific advances and new techniques. The advances will include a complete understanding of the asymptotic properties of the monopole metrics and a solution of the Sen conjecture. The new techniques will include a very sophisticated set of analytic tools tailored to the geometry of the monopole moduli spaces, but of much wider applicability.

The project will also include training of highly skilled PDRA and, through the proposed workshop, a larger body of early-career researchers skilled in the techniques being developed in this project.

As part of this proposal, we shall widen the impact by communicating and engaging a wider community with the main research themes of this project. This will contribute in a small way to raising public awareness of key research themes in the UK today.

Publications

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Description The goal of the grant was to understand the metric on monopole moduli spaces in the asymptotic regions. At the conclusion of the grant, we have succeeded in this goal, though the details require to be written up carefully. This is ongoing.
Exploitation Route The metric calculation feeds into other research on analytical tools on non-compact spaces, such as Hodge Theory and Index Theory. The monopole moduli spaces provide key examples in this area. The issue of understanding the Hodge theory completely is a future project.
Sectors Creative Economy

URL https://arxiv.org/abs/1512.02979
 
Description The research is largely complete and the project was successful. The team is engaged in writing up the results. The research has been presented at numerous workshops, colloquia and seminars.
First Year Of Impact 2016
Sector Creative Economy