2-vector bundles, loop spaces, and elliptic objects

我在这里描述的项目的目的是建立一个几何角度椭圆上同调，类似于如何向量丛表示拓扑K-理论类。这种透视的存在与E.维滕和G. Segal，由S. Stolz和P. Teichner，并抵挡住了N. Baas等人，大约20年前。在这个项目中，我们考虑了椭圆上同调类几何代表的一个新的有前途的候选（“Schreiber 2-向量丛”），该候选部分由我自己与M. Ludewig和P. Kristel，并计划使用另一种方法来椭圆上同调（“泰特K理论”），由于N。Kitchloo和J. Morava; N. Ganter和其他人。这种新方法通过一个版本的等变K-理论的循环空间。我们的目标是联合收割机这些新的想法，以进行一个新的攻击的老问题，找到一个几何透视椭圆上同调。核心思想是将2-向量丛的超越发展到lop空间，以这种方式超越的2-向量丛构成泰特K理论中的类。我们希望在这一方向的进展将有助于解决相关的，甚至更深层次的问题：什么样的微分算子作用于部分的2-向量丛，这样的运营商有一个指数值在椭圆上同调，什么样的作用发挥消失这些指数的几何流形？

The project that I describe here aims at establishing a geometric perspective to elliptic cohomology, analogous to how vector bundles represent topological K-theory classes. The existence of such perspective is related to a famous conjecture of E. Witten and G. Segal, further established by S. Stolz and P. Teichner, and withstood already an attack led by N. Baas et al. almost twenty years ago. For this project, we consider a new promising candidate ("Schreiber 2-vector bundles") for geometric representatives of elliptic cohomology classes, developed in part by myself, in collaboration with M. Ludewig and P. Kristel, and also plan to use another approach to elliptic cohomology ("Tate K-theory"), due to N. Kitchloo and J. Morava; further investigated by N. Ganter and others. This new approach goes via a version of equivariant K-theory of loop spaces. Our aim is to combine these new ideas in order to carry out a new attack on the old problem of finding a geometric perspective to elliptic cohomology. The core idea is to develop the transgression of 2-vector bundles to lop spaces, in such a way that transgressed 2-vector bundles make up classes in Tate K-theory. We hope that progress in this direction will contribute to the solution of related, even deeper problems: what kind of differential operators act on sections of 2-vector bundles, do such operators have an index valued in elliptic cohomology, and what role plays the vanishing of these indices for the geometry of manifolds?