PU(2) monopoles and gauge theoretic invariants

本文的目的是证明Witten关于光滑四维流形的Donaldson和Seiberg-Witten不变量的猜想，理解Seiberg-Witten简单类型和Kronheimer-Mrowka简单类型条件之间的关系，并寻找这些不变量的可能的拓扑约束。这项工作将通过探索PU(2)单极的模空间和Seiberg-Witten单极对某些自旋C结构的模空间来进行。PU(2)单极的模空间包含一个反自对偶连接的模空间。这意味着Donaldson不变量可以表示为通过将某些上同调类与PU(2)单极的模空间的Uhlenbeck紧化中的Seiberg-Witten单极子的模空间的链接配对而给出的表达式的和。这项工作的第一阶段是完成证明这些上同调类与Seiberg-Witten单极子的模空间的链接的配对可以以普适的形式表示，仅依赖于流形的Seiberg-Witten不变量和同伦类型。这项工作还将证明Kotschick-Morgan关于Donaldson不变量的跨墙公式的猜想。这项工作的第二阶段是足够详细地计算这种普遍形式，以允许计算Donaldson和Seiberg-Witten不变量之间的显式关系。我们打算通过使用两个不变量的已知运算公式(例如，Blow-up公式)、两个不变量都已知的例子以及上述和的一些内部对称性来进行计算。Donaldson和Seiberg-Wittenant不变量之间的这种关系可能是过度确定的，从而揭示了由四维流形的拓扑型给出的对这些不变量的约束，就像以前Kronheimer和Mrowk.流形是重要的研究对象，因为它们是无处不在的：变量中的k个方程的解集通常是(n-k)维流形。区分四维流形的主要工具是Seiberg-Witten和Donaldson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是至关重要的。此外，与这些不变量相关的猜想源于Witten使用量子场论的工作。这些量子场论的方法在数学上并不严格，所以我们对Witten猜想的数学严格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.