Solutions to the Haydys-Witten equations in 5 dimensions

This project falls within the Geometry and Topology, and Mathematical Analysis EPSRC research areas.

Towards the end of the 20th century, the study of anti-self-dual instantons and Seiberg-Witten theory, and their corresponding moduli spaces, laid the foundations for low dimensional gauge theory. In recent years, attempts have been made to use analogous equations to develop gauge theories on higher dimensional manifolds. One such set of equations is the Haydys-Witten equations. These are gauge theoretic equations on a 5-manifold that arise as a reduction of the Spin(7)-instanton equation in 8 dimensions. The equations were first studied by Haydys and Witten independently about 10 years ago.

The Haydys-Witten equations share many of the same properties as well studied equations on manifolds of various dimensions, which have been used to develop gauge theories in other contexts, but, even for simple manifolds, not much is known about their space of solutions. For example, it is not known whether the space of solutions of the equations over a compact manifold is itself compact. If this is the case, counting the solutions to the Haydys-Witten equations over a compact manifold would give a new integer invariant.

The proposed research is to study the space of solutions to the Haydys-Witten equations. Initially there will be a focus on studying the solutions in simple cases, for example on a 5-sphere. This can be achieved by using imposed symmetry to reduce the dimension of the problem, as the equations then reduce to well known problems on lower dimensional manifolds. Once the space of solutions has been described in some simple cases, this should indicate directions of study to start formulating a more general theory of solutions to the Haydys-Witten equations in 5 dimensions, and hence the formulation of a five dimensional gauge theory.

As shown by Haydys, by studying dimensional reductions of the Haydys-Witten equations one can obtain invariants for three and four dimensional manifolds. While there is interest in the Haydys-Witten equations in their own right, as described in Witten's original paper where equations were introduced to study the Jones polynomial of knots, which has links to string theory, the research has the potential to improve the understanding of 5-manifolds through a five dimensional gauge theory, and also the understanding of three and four dimensional manifolds through dimensional reduction. The development of this gauge theory also has the potential so shed light on how approaches to gauge theories may be generalised to manifolds of arbitrary dimension.