Collaborative Proposal: Stringy Invariants, Orbicurves, and Topological Field Theory

主要研究人员将进一步发展和利用他们在轨道上同调、轨道k理论和轨道陈氏性质方面的最新成果。他们将扩展他们关于阻塞束的工作，包括轨道Gromov-Witten理论，这不仅对计算有影响，而且具有重要的理论意义。其次，他们将弦和轨道k理论和上同调的构造从有限群推广到具有可能无限稳定子的非阿贝尔无限群的情况。第三，他们将研究由它们的弦和轨道陈氏特征引起的不变量。特别是，他们将研究陈类和类似的结构在弦和轨道设置。最后，他们将研究这些不变量与它们的对应物在潜在奇异空间的各种超kaehler和crepetresolution上的关系。空间不变量是拓扑学和几何学中的基本工具。新不变量的发展对这些领域非常重要，因为它为识别和描述几何和拓扑空间的基本性质提供了新的工具。不变量也出现在理论物理中，例如拓扑量子场论中的可观测值。在许多物理和数学环境中，最重要的空间也具有对称性，了解这些对称性如何与空间的几何和拓扑特性相互作用是很重要的。最近，pi开发了具有对称性的空间的新不变量（stringyK-theory），并在描述新不变量与先前已知不变量之间的联系（如轨道上同调）方面取得了重要进展。他们还使用他们新开发的工具来改进和简化那些以前已知的不变量的许多方面。在这笔资金的支持下，pi将利用他们的弦K理论以及他们对轨道上同调的改进来研究具有对称性的空间。他们还将进一步开发这些工具，将它们的适用性扩展到更多类型的空间，包括具有连续对称性的空间，这在数学和物理中很常见。他们还将开发这种空间的新不变量，包括对著名的经典不变量（如陈氏类）的增强，但要考虑对称性。这种不变量是由拓扑弦理论提出的，将为理解这些空间提供强大的工具。

The Principal Investigators will further develop and use their recent results in orbifold cohomology, orbifold K-theory and the orbifold Cherncharacter. They will expand their work on the obstruction bundle to include orbifold Gromov-Witten theory which will have implications not only forcalculations, but also important theoretical significance. Secondly, they will generalize their constructions of stringy and orbifold K-theory andcohomology from the case of a finite group to the case of a non-Abelian,infinite group with possibly infinite stabilizers. Third, they will study invariants arising from their stringy and orbifold Chern characters. Inparticular, they will investigate the Chern classes and similar structures in the stringy and orbifold settings. Finally, they will examine the relation of these invariants to their counterparts on various hyper-Kaehler andcrepant resolutions of the underlying singular spaces.Invariants of spaces are fundamental tools in topology and geometry. The development of new invariants is of great importance to these fields, as it provides new tools to identify and describe essential properties of geometric and topological spaces. Invariants also appear in theoretical physicsas observables in topological quantum field theories, for example. In many physical and mathematical settings, the spaces of greatest importance also havesymmetries, and it is important to understand how those symmetries interact with the geometric and topological properties of the space. Recently, the PIs have developed new invariants of spaces with symmetries (stringyK-theory) and have also made important progress in describing connections between their new invariants and previously known invariants, such asorbifold cohomology. They have also used their newly developed tools torefine and simplify many aspects of those previously known invariants. With the support of this grant, the PIs will use their theory of stringy K- theory as well as their improvements on orbifold cohomology to study spaces with symmetries. They will also further develop these tools to extend theirapplicability to more types of spaces, including spaces with continuoussymmetries, which are common throughout mathematics and physics.They will also develop new invariants of such spaces, includingenhancements of well-known classical invariants such as Chern classes,but accounting for symmetries. Such invariants are suggested bytopological string theory and will provide powerful tools for understanding these spaces.