Mathematical Sciences: Two-dimensional Topological Field Theories and Complex Cobordism

在DMS 8713718的支持下，一个最重要的问题是推广椭圆上同调，以包含高属曲线。Y. Shimizu和研究者找到了这个问题的解决方案（在有理数上），基于椭圆曲线上的仿射连接（不能推广到更高的格）应该被解释为投影连接（存在于所有黎曼曲面）的特殊情况的想法。这使它们能够以一种自然的方式与任何手性共形场论（在Segal的意义上）联系起来，由场论的投影连接（或应力-能量张量）定义的拓扑格。他们能够在带一个电荷的费米子的情况下识别出这个属，并证明它（在椭圆曲线的情况下）简化为椭圆属。这种构造的一个推论是，复共环支持正则二次微分，在某种程度上与Virasoro代数的基本玻色子表示所支持的二次微分密切相关。这两个二次微分之间的关系似乎具有根本的重要性，并且将是当前项目的主要重点。近年来，受用于解释亚原子粒子的理论所启发的数学结构，已被反复应用于纯几何问题。为了共同的利益，物理学家和数学家在探索这些联系方面的互动比以往任何时候都要多，而研究者是这个游戏的主要参与者。

Under support from DMS 8713718, a problem of paramount interest was to generalize elliptic cohomology so as to encompass curves of higher genus. Y. Shimizu and the investigator found a solution to this problem (over the rational numbers), based on the idea that affine connections on elliptic curves (which do not generalize to higher genus) should be interpreted as special cases of the projective connections (which exist for all Riemann surfaces). This allowed them to associate in a natural way to any chiral conformal field theory (in the sense of Segal), the topological genus defined by the projective connection (or stress-energy tensor) of the field theory. They were able to identify this genus in the case of the charge-one fermion and to show that it reduced (in the case of the elliptic curve) to the elliptic genus. A corollary of this construction is that the complex cobordism ring supports a canonical quadratic differential, closely related in some way to a quadratic differential supported by the fundamental bosonic representation of the Virasoro algebra. The relation between these two quadratic differentials seems to be of fundamental importance and will be a major emphasis of the current project. In recent years mathematical constructions inspired by theories used to make sense of subatomic particles have found repeated application to purely geometric problems. To their mutual profit physicists and mathematicians are interacting more than ever in exploring these connections, and the investigator is a major player in this game.