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

点击翻译按钮获取中文摘要

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.