Gauge theory, gluing theorems, and their applications

规范理论、粘合定理及其应用

• 批准号: 0905786
• 负责人: Thomas Leness
• 金额: $ 10.26万
• 依托单位: Florida International University
• 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: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905786&HistoricalAwards=false
• 关键词: Gauge; theory; gluing; theorems; their

The goal of this project is to prove some conjectured relations among gauge theoretic invariants of smooth four-manifolds and to investigate whether it is possible to define a new family of invariants for four-manifolds with b^+ even. The first step is to prove some analytic results on the Taubesian gluing maps which model the ends of the moduli space of SO(3) monopoles. These results will complete the proof of the SO(3) monopole cobordism formula, giving a relation between the Seiberg-Witten and the Donaldson invariants and a relation between the Seiberg-Witten and the spin invariants. These relations resemble Witten's conjecture relating the Seiberg-Witten and the Donaldson invariants except that they contain some unknown coefficients depending only on the topological type of the underlying four-manifold. The second step is an extension of Taubes' work on the gluing maps for the moduli space of anti-self-dual connections to prove that the bubbletree compactification of the moduli space of anti-self-dual connections is a manifold with boundary. This result will provide a framework to prove a relation between the Donaldson and spin invariants. It is hoped that combining the three relations will yield new topological constraints on these gauge theoretic invariants. Finally, the analytic work on the SO(3) monopole invariants should answer the question of whether it is possible to define gauge-theoretic invariants for four-manifolds with b^+ even by using the moduli space of SO(3) monopoles.Manifolds are shapes which locally resemble the Euclidean space of our everyday experience. The solution set of n equations in (n+d) variables will usually be a d-dimensional manifold: thus manifolds appear throughout mathematics and its applications, from the knot theory used to describe DNA to the manifolds appearing in string theory. Two manifolds are diffeomorphic if one can be stretched, without wrinkling, into the other. Deciding when four-dimensional manifolds are diffeomorphic has proven to be a particularly difficult problem. An invariant is a rule assigning an algebraic object such as a number or polynomial to a topological space in such a manner that the algebraic object does not change when the space is stretched. Hence if I is an invariant and X and Y are topological spaces with I(X) not equal to I(Y) then X cannot be stretched into Y so X and Y are not diffeomorphic. The goal of this project is to look for relations between different invariants and to find new invariants.

该项目的目的是证明光滑四个manifolds的仪表理论不变性之间的某些猜想关系，并调查是否有可能为具有B^+偶数的四个manifolds定义一个新的不变性家族。 第一步是证明对陶布斯胶水图的一些分析结果，该结果模拟了SO（3）单极的模量空间的末端。 这些结果将完成SO（3）单极核心主义公式的证明，从而在Seiberg-Witten和Donaldson不变性之间建立了关系，以及Seiberg-Witten和Seiberg-Witten与Spin不变性之间的关系。 这些关系类似于Witten的猜想，该猜想与Seiberg-Witten和Donaldson的不变式相关，只是它们仅根据基础四个manifold的拓扑类型包含一些未知系数。 第二步是陶布斯在粘合图上的作品的延伸，以模量空间，以证明反对偶发连接的模量空间的泡泡性紧凑型是边界的歧管。 该结果将提供一个框架，以证明唐纳森与自旋不变性之间的关系。 希望将这三个关系结合起来将对这些规格理论不变性产生新的拓扑约束。 最后，关于SO（3）单极不变的分析工作应该回答一个问题，即是否可以通过使用SO（3）Monopoles的模量空间来定义具有B^+的四个manifolds的规格理论不变，而Manifolds则是局部与我们日常体验相似的局部形状。 （n+d）变量中n个方程的解集通常是d维流形的：因此，在整个数学及其应用中都会出现歧管，从用于描述DNA到弦理论中出现的歧管。如果可以将一个歧管伸展而不皱纹，则两个歧管是不同的。 事实证明，确定四维流形的何时差异是一个特别困难的问题。 不变的是一个规则，将代数对象（例如数字或多项式）分配给拓扑空间，以使代数对象在伸展空间时不会改变。 因此，如果我是不变的，而x和y是i（x）不等于i（y）的拓扑空间，那么x不能伸展到y（y）中，因此x和y不是差异的。 该项目的目的是寻找不同不变的人之间的关系并找到新的不变式。