Seiberg-Witten and Gromov invariants of symplectic 4-orbifolds and some applications

陈伟民首席研究员计划继续他在规范理论和辛、低维拓扑方面的工作。更具体地说，他计划研究以下问题：(1)辛4流形的辛形态有限群的一般性质；(2)如果一个辛圆的不动点集是非空有限的，它是否作用于一个6维流形的哈密顿量；(3)代数曲面上商奇点出现的一般、全局限制。这些研究的一个统一主题是一个理论，但在这个项目中还有待发展，关于辛4-轨道的Seiberg-Witten和gromovov不变量，它将建立在Taubes关于辛4流形的相关基础工作的基础上并实质性地扩展。四维轨道是一个局部看起来像我们所有人生活的时空的空间，除了局部发生的有限对称造成一定程度的模糊性。在数学（如拓扑学和几何学）和理论物理模型（如轨道弦理论）中，可以很容易地找到许多这种空间的例子。所提出的研究旨在理解在四维轨道中分布在整个空间中的某种类型的场与在二维子空间中被压缩的空间中的一组弦所消灭的某种类型的世界表之间的基本对偶性。当有限对称不存在这种模糊性时，这种对偶性就很好理解了，并且是数学的一个基本部分。

DMS-0304956Weimin ChenThe Principal Investigator plans to continue his work in gaugetheory and symplectic, low-dimensional topology. More concretely,he plans to study the following problems: (1) general properties offinite groups of symplectomorphisms of a symplectic 4-manifold,(2) whether a symplectic circle action on a 6-dimensional manifoldis Hamiltonian if its fixed-point set is non-empty and finite, and(3) general, global restrictions on the occurrence of quotient singularitiesin an algebraic surface. A unifying theme in these studies is a theory,yet to be developed in this project, on the Seiberg-Witten and Gromovinvariants of a symplectic 4-orbifold, which will be built on andsubstantially extend the related fundamental work of Taubes onsymplectic 4-manifolds.A 4-dimensional orbifold is a space which locally looks like the space-time we all live in, except that there is a certain degree of ambiguity caused bya finite symmetry occured locally. Many examples of this kind of space can befound easily and naturally in Mathematics (e.g. topology and geometry) as wellas models in Theoretical Physics (e.g. orbifold string theory). The proposedresearch seeks to understand a fundamental duality in a 4-dimensional orbifoldbetween a certain type of fields that are distributed all over the space and a certaintype of worldsheet that is wiped out by a collection of strings in the space whichis condensed in a 2-dimensional subspace. When there is no such ambiguity of finite symmetries, such a duality is well-understood, and is a fundamental piece ofmathematics.