Solutions to the Haydys-Witten equations in 5 dimensions

这个项目属于几何与拓扑学和数学分析EPSRC的研究领域。直到20世纪末，反自对偶瞬子和Seiberg-Witten理论以及它们对应的模空间的研究，为低维规范理论奠定了基础。近年来，人们试图利用相似方程来发展高维流形上的规范理论。其中一组方程就是Haydys-Witten方程。这些是5维流形上的规范理论方程，它是8维自旋(7)-瞬子方程的约化。这些方程最初是由Haydys和Witten在大约10年前独立研究的。Haydys-Witten方程有许多相同的性质，就像在不同维度流形上研究的方程一样，这些方程已经被用来在其他背景下发展规范理论，但是，即使对于简单的流形，关于它们的解空间也知之甚少。例如，紧流形上的方程的解的空间本身是否是紧的就不得而知。如果是这样的话，计算紧致流形上的Haydys-Witten方程的解将得到一个新的整数不变量。拟研究的是Haydys-Witten方程的解空间。最初的重点是研究简单情况下的解，例如5-球面上的解。这可以通过使用强加的对称性来降低问题的维度来实现，因为方程随后简化为低维流形上的众所周知的问题。一旦在一些简单的情况下描述了解的空间，这应该指示开始建立更一般的5维Haydys-Witten方程的解的理论的研究方向，从而形成5维规范理论。如Haydys所示，通过研究Haydys-Witten方程的降维，我们可以获得三维和四维流形的不变量。虽然人们对Haydys-Witten方程本身很感兴趣，正如Witten的原始论文中描述的那样，引入方程来研究纽结的琼斯多项式，这与弦理论有联系，但这项研究有可能通过五维规范理论提高对5-流形的理解，并通过降维提高对三维和四维流形的理解。这种规范理论的发展也具有潜力，从而阐明了如何将规范理论的方法推广到任意维度的流形上。

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.