Narrow Results

We give a stochastic analog of the construction of McLaughlin of the string bundle on the loop space by using tools of algebraic topology.

We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. If (L → M → N) is a Lie 2-crossed module and Y → X is a surjective submersion then an (L → M → N)-bundle 2-gerbe over X is defined in terms of a so-called (L → M → N)-bundle gerbe over the fiber product Y^[2] = Y × _XY, which is an (L → M)-bundle gerbe over Y^[2] equipped with a trivialization under the change of its structure crossed module from L → M to 1 → N, and which is subjected to further conditions on higher fiber products Y^[3], Y^[4] and Y^[5]. String structures can be described and classified using 2-crossed module bundle 2-gerbes.

Studying the topological aspects of M-branes in M-theory leads to various structures related to Wu classes. First we interpret Wu classes themselves as twisted classes and then define twisted notions of Wu structures. These generalize many known structures, including Pin^- structures, twisted Spin structures in the sense of Distler–Freed–Moore, Wu-twisted differential cocycles appearing in the work of Belov–Moore, as well as ones introduced by the author, such as twisted Membrane and twisted String^c structures. In addition, we introduce Wu^c structures, which generalize Pin^c structures, as well as their twisted versions. We show how these structures generalize and encode the usual structures defined via Stiefel–Whitney classes.