PU(2) monopoles and gauge theoretic invariants

本文的目标是证明Witten关于光滑四流形的Donaldson和Seiberg-Witten不变量的猜想，理解Seiberg-Witten简单型和Kronheimer-Mrowka简单型的条件之间的关系，并寻找这些不变量可能的拓扑约束。本工作将通过探索包含反自对偶连接模空间的PU(2)单极子模空间和某些自旋C结构的Seiberg-Witten单极子模空间来进行。这意味着Donaldson不变量可以表示为在这些自旋结构上的一个表达式的和，该表达式是在PU(2)单极子的模空间的Uhlenbeck紧化中，将某些上同类与Seiberg-Witten单极子的模空间的连接配对所给出的。本工作的第一阶段是完成了这些上同类与Seiberg-Witten单极子模空间的链路的对偶可以仅依赖于Seiberg-Witten不变量和流形的同伦类型以一种泛形式表示的证明。这项工作也将得到关于Donaldson不变量的wall-crossing公式的Kotschick-Morgan猜想的证明。这项工作的第二阶段是充分详细地计算这个通用形式，以便计算Donaldson和Seiberg-Witten不变量之间的显式关系。我们打算通过使用已知的两个不变量的手术公式（例如膨胀公式），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来进行计算。Donaldson和Seiberg-Witteninvariants之间的这种关系可能是过度确定的，因此将揭示由四流形的拓扑类型给出的这些不变量的约束，正如Kronheimer和Mrowka在早期的工作中所做的那样。一个n维流形是一个局部看起来像n维欧几里德空间的拓扑空间。流形是重要的研究对象，因为它们无处不在：k个变量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不变量和Donaldson不变量。因此，理解这些不变性之间的关系对于理解四维拓扑结构至关重要。此外，有关这些不变量的猜想来自于Witten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谨，因此我们对威滕猜想的数学上严谨的证明可以被视为一种极其廉价的实验物理形式。

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.