The Geometry of Moduli Spaces of Sheaves

滑轮模空间的几何

• 批准号: 0948296
• 负责人: Alina Marian
• 金额: $ 5.49万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-04-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0948296&HistoricalAwards=false
• 关键词: Geometry; Moduli; Spaces; Sheaves

The PI plans to explore the geometry of parameter spaces of sheaves on algebraic curves and surfaces. She wants to calculate some of the intersection theory, in particular Verlinde numbers, on Gieseker moduli spaces of semistable sheaves on a smooth surface. These intersection-theoretic calculations will help the PI probe whether the level-rank duality which is a feature of moduli spaces of bundles on a curve, can be associated with moduli spaces of sheaves on a surface as well. The author also wishes to explore some of the representation theory of classical and affine Lie algebras on the cohomology of Grothendieck quot schemes on curves, as well as on the cohomology of Gieseker moduli spaces of sheaves on a surface, which would result in a better understanding of the geometric structure of these moduli spaces.The study of vector bundles in algebraic geometry parallels that of gauge field theories in physics. The most familiar example of a gauge field is the electromagnetic field. Higher-rank vector bundles correspond to nonabelian gauge fields, which appear in high-energy physics, and mathematically are organized as the entries of a matrix. Studying the geometry and intersection theory on the moduli space of bundles, which is the goal of this project, leads in particular to understanding the structure of various integrals on the space of all possible gauge fields. These integrals are mathematical expressions for scattering amplitudes of various physical processes in high-energy physics.

PI 计划探索代数曲线和曲面上滑轮参数空间的几何形状。她想要计算光滑表面上半稳定滑轮的 Gieseker 模空间的一些交集理论，特别是 Verlinde 数。这些交集理论计算将帮助 PI 探测水平-等级对偶性（曲线上束的模空间的特征）是否也可以与表面上滑轮的模空间相关联。作者还希望探讨经典和仿射李代数关于曲线上格罗腾迪克曲线上同调的一些表示理论，以及表面上滑轮的 Gieseker 模空间上同调，这将导致更好的理解这些模空间的几何结构。代数几何中矢量丛的研究与物理学中规范场论的研究相似。规范场最熟悉的例子是电磁场。高阶向量丛对应于高能物理学中出现的非阿贝尔规范场，并且在数学上被组织为矩阵的条目。 研究束模空间的几何和相交理论是该项目的目标，尤其有助于理解所有可能规范场空间上的各种积分的结构。 这些积分是高能物理中各种物理过程的散射幅度的数学表达式。