Description
Electromagnetism is the simplest example of a gauge theory, where gauge transformations involve real-valued functions. Non-abelian gauge theories are a generalization of electromagnetism in which gauge transformations are matrix-valued functions. Theories of this type form the basis of the standard model of high energy particle physics.
Some non-abelian gauge theories possess solutions that represent magnetic monopoles, that is, sources of magnetic charge. There are a number of sophisticated mathematical approaches to studying the equations for magnetic monopoles, but it is difficult to find exact solutions. Rather remarkably, it turns out that if simple flat space is replaced by a more complicated curved space (called hyperbolic space) then the equations can be easier to solve. The image shows pictures of the energy density for three different magnetic monopole solutions in hyperbolic space.
The project will begin with an introduction to non-abelian gauge theories. Next, we will consider the equations for magnetic monopoles and see how to obtain the simplest solution, together with some properties of more complicated solutions. Finally, we will study hyperbolic space and see how the equations for magnetic monopoles can be simplified by choosing to curve space by just the right amount.
Prerequisites
Special Relativity and Electromagnetism II and Mathematical Physics II are prerequisites.Differential Geometry III and/or General Relativity IV would be useful but are not essential.
Resources
Books:Topological solitons, by Manton and Sutcliffe, Chapters 2 and 8.
Magnetic monopoles, by Shnir.
Research papers:
Platonic hyperbolic monopoles
Hyperbolic monopoles, JNR data and spectral curves
Spectral curves of hyperbolic monopoles from ADHM