Versions of 2-vector bundles as a framework for a construction of stringor bundles

该项目的工作标题是“作为构造stringor束的框架的2向量束的版本”，涉及几何和拓扑领域的一个主题。这是对弦流形指标理论发展的一个贡献。弦流形是具有消失第一分数庞特里亚金类的自旋流形。启发式地，弦流形的自由循环空间是一个自旋流形。该领域的最终目标是发展弦流形的指标理论。期望该指标理论对应于环空间上的自旋指标理论的一个版本，由Witten假设的环空间上的Dirac算子控制，其指标是流形的Witten格。在尝试定义操作符之前，必须回答的一个问题是，这个操作符应该作用于哪里。与普通狄拉克算子作用于旋量束的部分的事实类似，术语弦量束出现了，尽管在经典意义上，弦量束不能被期望实现为局部平凡的纤维束。该项目涉及一个框架，其中可以构造和研究stringor束。启发式地，弦流形M的弦量束是弦量束的一个分类版本。在对循环空间的越界下，它被认为是LM的一个旋量束的一个版本。提出的项目的目的是开发一个数学框架，其中可以定义，构造和研究弦束。该框架将基于以下领域的最新进展：(a)弦结构和弦连接(b)环空间上束的融合和(c)更高范畴束理论。这个被提议的项目被认为是为弦流形的指标理论的方向做出贡献的第一步。

The project has the working title "Versions of 2-vector bundles as a framework for a construction of stringor bundles" and concerns a topic in the fields of geometry and topology. It is a contribution for the development of an index theory for string manifolds. String manifolds are spin manifolds with vanishing first fractional Pontryagin class. Heuristically, the free loop space of a string manifold is a spin manifold. An ultimate goal in the field is to develop an index theory for string manifolds. It is expected that this index theory corresponds to a version of spin-index theory on the loop space, governed by a Dirac operator on the loop space postulated by Witten, whose index is the Witten genus of the manifold. One question that has to be answered even before trying to define an operator, is where this operator should act on. In analogy to the fact that the ordinary Dirac operator acts on sections of the spinor bundle, the term stringor bundle came up, despite of the fact that it is not expected that the stringor bundle can be realized as a locally trivial fibre bundle in the classical sense. The project is concerned with a framework in which stringor bundles can be constructed and studied. Heuristically, the stringor bundle of a string manifold M is a categorified version of the spinor bundle. Under transgression to the loop space, it is supposed to be a version of a spinor bundle of LM. The aim of the proposed project is to develop a mathematical framework in which stringor bundles can be defined, constructed and studied. This framework will be based on recent progress in the fields of (a) string structures and string connections (b) fusion in bundles over loop spaces and (c) higher categorical bundle theory. The proposed project is supposed to contribute a first step into the direction of an index theory for string manifolds.