Cobordism of Configuration Spaces and Its Applications

9802616 Morava在以前的工作中，作者证明了Witten的二维拓扑引力的tau函数具有同伦论解释为一种拓扑场论，它取值于环谱上的模范畴中的值。这个环谱是稳定代数曲线的模空间的大亏格完备化，它是由稳定代数曲线上光滑点的构形空间用同伦论方法构造的。它的上同调有许多令人联想到顶点算子代数的特征，而顶点算子代数又与对称函数理论有关。协边理论的方法还没有被系统地应用于构形空间的研究，但许多领域的问题表明这是值得研究的。特别重要的是，将上述结果推广到与Sigma模型耦合的拓扑引力，以包含T.Eguchi和他的学派建议的量子上同调。物理学家对二维引力的兴趣揭示了黎曼曲面的模空间的拓扑结构，比这个非常经典的主题的所有先前研究的总和还多，但他们的工作回答了一些问题，乍一看，数学家可能不想问什么，而且从这个主题的传统数学方法来看，很难理解这些新的结果。越来越清楚的是，配置空间和协边技术是物理学家构造的几何核心，它们的交集位于代数拓扑学的丰富部分，而到目前为止，这些拓扑学还没有得到系统的研究。在物理学中，当“实验”应用提出新的“理论”问题进行研究时，总是被认为是一个令人鼓舞的迹象；这似乎就是数学中正在发生的事情。***

9802616 Morava In previous work, the author showed that Witten's tau-function for topological gravity in two dimensions has a homotopy-theoretic interpretation as a kind of topological field theory taking values in a category of modules over a ring-spectrum. This ring-spectrum is a large-genus completion of the moduli space of stable algebraic curves; it is constructed by homotopy-theoretic methods from spaces of configurations of smooth points on such curves. Its cohomology has many features reminiscent of a vertex operator algebra, which is in turn connected to the theory of symmetric functions. The methods of cobordism theory have not been systematically applied to the study of configuration spaces, but questions in a variety of areas suggest that this merits investigation. It is particularly important to extend the results mentioned above to topological gravity coupled to a sigma-model, to encompass quantum cohomology along the lines suggested by T. Eguchi and his school. The interest of physicists in two-dimensional gravity has shed more light on the topology of the moduli space of Riemann surfaces than all previous research in this very classical subject combined, but their work answers questions that are not, at first sight, what a mathematician might want to ask, and it has been difficult to understand these new results in terms of the traditional mathematical approach to this subject. It is becoming increasingly clear that configuration spaces and cobordism techniques are at the geometric heart of the physicists' constructions, and that their intersection lies in a rich part of algebraic topology that has till now escaped systematic study. In physics, it is always taken to be an encouraging sign when `experimental' applications suggest new `theoretical' questions for investigation; this is what seems to be happening here in mathematics. ***