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.

本项目的目标是证明光滑四维流形的规范理论不变量之间的某些特殊关系，并研究是否有可能定义一类新的具有B^+偶的四维流形的不变量。 第一步是证明关于模拟SO（3）单极模空间端点的Taubesian胶合映射的一些解析结果。 这些结果将完成SO（3）协边公式的证明，给出Seiberg-Witten和唐纳森不变量之间的关系以及Seiberg-Witten和自旋不变量之间的关系。 这些关系类似于维滕的猜想有关的Seiberg-维滕和唐纳森不变量，除了他们包含一些未知的系数，只依赖于拓扑类型的基础四维流形。 第二步是Taubes关于反自对偶联络模空间的胶合映射的工作的推广，证明了反自对偶联络模空间的Bubbletree紧化是一个有边界流形。 这一结果将提供一个框架来证明唐纳森和自旋不变量之间的关系。 人们希望结合这三个关系将产生新的拓扑约束这些规范理论的不变量。 最后，对SO（3）单极子不变量的分析工作应该回答这样一个问题：即使使用SO（3）单极子的模空间，是否有可能定义具有B^+的四流形的规范论不变量。在（n+d）个变量中n个方程的解集通常是一个d维流形：因此流形出现在数学及其应用中，从用来描述DNA的纽结理论到出现在弦理论中的流形。两个流形是同构的，如果一个可以拉伸到另一个中，而不需要重叠。 判定四维流形何时是复纯的已被证明是一个特别困难的问题。 不变量是一种规则，它将一个代数对象（如一个数或多项式）分配给一个拓扑空间，使得当空间被拉伸时，代数对象不会改变。 因此，如果I是一个不变量，X和Y是拓扑空间，且I（X）不等于I（Y），则X不能拉伸到Y中，因此X和Y不是同构的。 这个项目的目标是寻找不同的不变量之间的关系，并找到新的不变量。