Cobordism of Configuration Spaces and Its Applications

小行星9802616 在以前的工作中，作者证明了二维拓扑引力的维滕τ-函数具有同伦论的解释，它是一种取值于环谱上模范畴的拓扑场论。 这个环谱是稳定代数曲线的模空间的一个大亏格完备;它是由同伦理论方法从这种曲线上的光滑点的配置空间构造的。 它的上同调有许多让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。 配边理论的方法还没有被系统地应用到构形空间的研究中，但是在各个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耦合到σ模型的拓扑引力，以包含T.江口和他的学校。 物理学家对二维引力的兴趣，使人们对黎曼曲面模空间的拓扑结构有了更多的了解，这比以前在这一非常经典的学科中的所有研究加起来还要多，但是他们的工作回答了一些问题，乍一看，这些问题并不是数学家想要问的，而且用传统的数学方法来理解这些新的结果是很困难的。 越来越清楚的是，位形空间和配边技术是物理学家构造的几何核心，它们的交叉点存在于代数拓扑学的丰富部分，而这些部分迄今尚未得到系统的研究。 在物理学中，当“实验”应用提出新的“理论”问题供研究时，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. ***