PU(2) monopoles and gauge theoretic invariants

PU(2) 单极子和规范理论不变量

• 批准号: 0103677
• 负责人: Thomas Leness
• 金额: $ 6.57万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103677&HistoricalAwards=false
• 关键词: PU; monopoles; gauge; theoretic; invariants

Thomas G. LenessThe goals of this proposal, to be done in collaboration with P. Feehan, are to prove Witten's conjecture relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds, to understand the relation betweenthe conditions of Seiberg-Witten simple type and Kronheimer-Mrowka simple type, and to search for possible topological constraints on these invariants. This work will be carried out by exploring the moduli space of PU(2) monopoles which contains a moduli space of anti-self-dual connections and the moduli spaces of Seiberg-Witten monopoles for certain Spin C structures. This implies that the Donaldson invariant can be expressed as a sum, over these spinc structures, of an expression given by pairing certain cohomology classes with the link of the moduli space of Seiberg-Witten monopoles in the Uhlenbeck compactification of the moduli space of PU(2) monopoles.The first phase of this work is to complete the proof that the pairing of these cohomology classes with the link of the moduli space of Seiberg-Witten monopoles can be expressed in a universal form depending only on the Seiberg-Witten invariant and the homotopy type of the manifold. This work will also yield a proof of the Kotschick-Morgan conjecture on wall-crossing formulas for Donaldson invariants. The second phase of this work is to calculate this universal form in sufficient detail to allow the computation of the explicit relation between the Donaldson and Seiberg-Witten invariants. We intend to do this calculation by using known surgery formulas forboth invariants (e.g. blow-up formulas), examples where both invariants are known, and some internal symmetries of the sum mentioned above. It is possible that this relation between the Donaldson and Seiberg-Witteninvariants is over-determined and thus will reveal constraints on these invariants given by the topological type of the four-manifold, as was done in earlier work with Kronheimer and Mrowka.An n-dimensional manifold is a topological space that locally looks like n-dimensional Euclidean space. Manifolds are important objects to study because they are ubiquitous: the solution set of k equations inn variables will usually be an (n-k)-dimensional manifold. The main tools for distinguishing between four-dimensional manifolds are the Seiberg-Witten and Donaldson invariants. Thus, understanding the relation between these invariantsis crucial to an understanding of four-dimensionaltopology. In addition, the conjectures relating these invariants arise from Witten's work using quantum field theory. These methods of quantum field theory are not mathematically rigorous, so our mathematically rigorous proof of Witten's conjecture can be viewed as an extremelyinexpensive form of experimental physics.

托马斯G. Leness这个建议的目标是与P. Feehan合作完成，证明维滕关于光滑四维流形的唐纳森和Seiberg-维滕不变量的猜想，理解Seiberg-维滕简单型和Kronheimer-Mrowka简单型的条件之间的关系，并寻找这些不变量上可能的拓扑约束。 本文的工作将通过探索PU（2）单极子的模空间（其中包含反自对偶联络的模空间）和Seiberg-Witten单极子的模空间（对于某些SpinC结构）来进行。 这意味着唐纳森不变量可以表示为这些自旋结构上的和，在PU（2）单极子模空间的Uhlenbeck紧化中，对Seiberg-Witten单极子模空间的链与某些上同调类的配对给出了一个表达式，本文的第一阶段工作是完成对Seiberg-Witten单极子模空间的链与这些上同调类的配对的证明。维滕单极子可以表示为一种普遍形式，这只依赖于流形的Seiberg-维滕不变量和同伦类型。 这项工作也将产生一个证明的Kotschick-Morgan猜想的跨壁公式的唐纳森不变量。 这项工作的第二阶段是计算这个普遍的形式，在足够的细节，使计算之间的显式关系的唐纳森和塞伯格-威滕不变量。 我们打算通过使用已知的外科手术公式来计算这两个不变量（例如，爆破公式），两个不变量都已知的例子，以及上面提到的和的一些内部对称性。 唐纳森和塞伯格-维滕不变量之间的这种关系可能是超定的，因此将揭示由四流形的拓扑类型给出的对这些不变量的约束，就像Kronheimer和Mrowka在早期的工作中所做的那样。n维流形是局部看起来像n维欧氏空间的拓扑空间。 流形是重要的研究对象，因为它们无处不在：k个方程的解集通常是一个（n-k）维流形。 区分四维流形的主要工具是Seiberg-Witten和唐纳森不变量。 因此，理解这些不变量之间的关系对于理解四维拓扑至关重要。 此外，与这些不变量相关的结构来自于维滕使用量子场论的工作。 量子场论的这些方法在数学上并不严格，所以我们对维滕猜想的数学严格证明可以被看作是实验物理学的一种极其廉价的形式。