Stability conditions and hypermultiplet space

许多现代物理理论，如弦论，本质上都是几何的，粒子和力的性质取决于理论中额外维度的卷曲方式。不幸的是，弦理论目前还没有被很好地理解，到目前为止还不可能做出可以被实验验证的预言。这个项目适合于当前纯数学研究的一个大领域，其目的是更好地理解弦理论的几何结构。人们可以断言，这将使我们有一天能够对真实的世界的量进行计算，然后用实验进行检验。虽然现在还处于早期阶段，我们的研究主要集中在弦理论中出现的卷曲维度的性质上，在数学中被称为Calabi-Yau流形。这个特别的提议涉及到弦理论中出现的某些代数对象，物理学家称之为BPS膜范畴，数学家称之为Calabi-Yaucategories。我们将关注称为唐纳森-托马斯不变量的整数，它测量理论中出现的BPS膜的精确数量。最终目的是更好地理解一个叫做超多重态空间的对象，它是弦理论中出现的一个辅助空间，但目前还没有专业的定义。物理学表明，这个空间可以配备一个几何结构，以一种有趣的方式编码唐纳森-托马斯不变量。这种几何结构被称为超卡勒度规，这种结构的例子在数学和理论物理中都很有意义。

Many modern physical theories such as string theory are geometrical in nature,with the properties of particles and forces being determined by the way extradimensions in the theory are curled up on themselves. Unfortunately stringtheory is not at all well-understood at present, and it has not so far beenpossible to make predictions that can be experimentally verified. This projectfits into a large area of current research in pure mathematics which aims at abetter understanding the mathematcal structure of string theory. One could hopethat this will one day enable us to make calculations of real world quantitiesthat can then be checked against experiment. For now though it is early days,and our research focuses on properties of the curled up dimensions appearing instring theory, known in mathematics as Calabi-Yau manifolds.This particular proposal concerns certain algebraic objects appearing in string theorywhich physicists call categories of BPS branes, and mathematicians call Calabi-Yaucategories. We will be concerned with integers called Donaldson-Thomas invariantswhich measure the precise number of BPS branes appearing in the theory. Theultimate aim is to better understand an object called the hypermultiplet space,an auxilliary space appearing in string theory but which has no mathematicaldefinition at present. The physics suggests that this space can be equipped witha geometrical structure which encodes the Donaldson-Thomas invariants in aninteresting way. This geometrical structure is called a hyperkahler metric andexamples of such structures are of interest in both mathematics and theoretical physics.