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

首席研究人员将进一步发展和使用他们在奥比霍尔德上同调、奥比霍尔德K-理论和奥比霍尔德切恩特征性方面的最新结果。他们将把他们在阻塞丛上的工作扩展到包括奥比霍尔德·格罗莫夫-威腾理论，这不仅对计算有意义，而且有重要的理论意义。其次，他们将把弦K-理论和上同调的构造从有限群推广到具有无限稳定子的非交换无限群的情形。第三，他们将研究由其串和或折字符产生的不变量。特别是，他们将研究弦和奥比福尔德环境中的陈类和类似的结构。最后，他们将研究这些不变量与它们在各种超Kaehler空间上的对应关系以及基本奇异空间的明确分辨率。空间的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具有重要意义，因为它为识别和描述几何和拓扑空间的本质性质提供了新的工具。例如，不变量也出现在理论物理中，作为拓扑量子场论中的可观测变量。在许多物理和数学环境中，最重要的空间也有对称性，了解这些对称性如何与空间的几何和拓扑属性相互作用是很重要的。最近，PI发展了具有对称性的空间的新的不变量(串K-理论)，并且在描述它们的新的不变量与已知的不变量之间的联系方面也取得了重要进展。他们还使用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。在这笔赠款的支持下，PI们将使用他们的弦K-理论以及他们对Orbiold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，以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，这在数学和物理中是常见的。他们还将开发这类空间的新不变量，包括对著名的经典不变量的增强，如陈类，但考虑到对称性。这种不变量是由拓扑弦理论提出的，它将为理解这些空间提供强有力的工具。

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.