标题： PU(2)单极子和四流形不变量之间的关系 显示英文标题

标题： PU(2) monopoles and relations between four-manifold invariants

摘要： 使用量子场论的论点，维滕已经建立了光滑四流形的唐纳森不变量和塞伯格-威滕不变量之间的关系。 在这篇综述文章中，我们描述了利用PU(2) = SO(3)单极子的模空间作为唐纳森模空间的反自对偶SO(3)联络和U(1)单极子模空间之间的cobordism来证明这一关系的方案。 我们概述了关于PU(2)单极子的一些横截性和乌伦贝克紧性结果，以及一些将唐纳森不变量用塞伯格-威滕不变量表示的计算。 我们简要概述了粘合理论方面的问题，重点讨论了一些与PU(2)单极子相关的分析困难，以及其在PU(2)单极子计划中的应用，以证明唐纳森和塞伯格-威滕不变量之间的关系。 显示英文摘要

摘要： Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants. We give a brief outline of issues concerning the gluing theory, focussing on some of the analytical difficulties that are particular to PU(2) monopoles, and its application to the PU(2) monopole program to prove the relation between Donaldson and Seiberg-Witten invariants.