Mathematical Sciences: Geometry of Yang-Mills Moduli Spaces

主要研究人员将为四维杨-米尔斯理论中的模空间创建一个正则度量。他将把Donaldson程序推广到四球面上的1瞬子空间和复射影2空间。将特别注意模空间边界的性质；它们的层的性质是什么，以及这些层是如何结合在一起的？用来解决这些问题的技巧将是那些用来理解更简单的正定流形的技巧的推广。几何拓扑学家直到最近才能理解四维超曲面。令人惊讶的是，唐纳森使用了几何偏微分方程组理论中的技术来证明四维时空的非唯一性。首席研究人员将以一种自然的方式将这一理论扩展到更奇异的超曲面。这些工作对规范场物理中的超弦理论有相当大的影响。

The principal investigator will create a canonical metric for moduli spaces arising in four-dimensional Yang-Mills theory. He will extend the Donaldson program to spaces of 1-instantons over the four sphere and complex projective two-space. Particular attention will be paid to the nature of the boundary of moduli spaces; what is the nature of their strata and how do these strata fit together? The techniques used to approach these questions will be extensions of those used to understand simpler positive-definite manifolds. Four dimensional hypersurfaces have only recently become understandable to geometric topologists. Surprisingly, Donaldson used techniques from the theory of geometric partial differential equations to demonstrate the nonuniqueness of four dimensional space time. The principal investigator will extend this theory to more exotic hypersurfaces in a natural way. Much of this work has considerable impact on the superstring theories of gauge field physics.