Invariants for Hamiltonian Spaces

摘要奖：DMS-1104670主要研究人员：Eduardo Gonzalez主要研究人员将使用辛几何技术研究与哈密尔顿群作用有关的不变量。第一个项目将研究(与A.Ott、C.Woodward和F.Ziltener合作)固定完整穿孔表面上的gugedGromov-Witten不变量的构造。PI还将调查仿射复空间上的扭转作用的特殊情况。我们的第二个项目是研究仿射空间上环面作用量的规范Gromov-Witten势的不变性。我们将用已建立的公式来证明，对于Calabi-Yau穿墙，广义Gromov-Witten势是不变的。在这个项目完成后，我们将研究势在非圆周情况下的行为。这将与C.Woodward共同完成。我们与H.Iritani共同提出的第三个活动将解释Seidel元素与环面非奇异射影簇的镜像变换之间的关系。然后，我们将尝试将我们的方法应用于非nef情况。我们与A·科顿-克莱提出的最后一个项目将试图解决汤姆的一个猜想。撒迪厄斯用有限群联系了辛同构的Floer上同调和全局商的Or(Stacky)上同调。我们的研究将促进数学和物理中几个学科的相互作用，其中对称性起着相关的控制作用。这笔赠款中的第一个项目将促进辛几何和规范理论(基本粒子物理的现代语言)在称为“规范的西格玛模型”的物理理论中的理解和应用。我们的其他项目将使用辛几何不变量提供一种不同的方法来研究镜像对称现象。

AbstractAward: DMS-1104670Principal Investigator: Eduardo GonzalezThe principal investigator will investigate, using symplecticgeometry techniques, invariants associated to Hamiltonian groupactions. The first project will study (joint with A. Ott,C. Woodward and F. Ziltener) the construction of gaugedGromov-Witten invariants on punctured surfaces with fixedholonomies. The PI will also investigate the particular case oftoric actions on affine complex spaces. Our second project is onthe study of the invariance of gauged Gromov-Witten potentialsfor toric actions on affine spaces. We will use establishedformulae to show that for Calabi-Yau wall-crossing the gaugedGromov-Witten potentials do not change. After this project isfinished we will study the behaviour of the potential in thenon-toric case. This will be done jointly with C. Woodward. Ourthird proposed activity, joint with H. Iritani, will explain therelation between Seidel elements and Mirror Transformations fornef toric non-singular projective varieties. We will then try toapply our methods to the non-nef case. Our last proposed projectwith A. Cotton-Clay will attempt to resolve a conjecture due toM. Thaddeus relating Floer cohomology for symplectomorphisms andorbifold (stacky) cohomology for global quotients by finitegroups.Our research will advance the interplay of several subjects inmathematics and physics where symmetries play a relevantrole. The first project in this grant will advance theunderstanding and applications of symplectic geometry and gaugetheory (the modern language of elementary particle physics) intophysical theories called "gauged sigma models". Our otherprojects will provide a different approach to the Mirror Symmetryphenomena using symplectic geometry invariants.