Twisted topological structures related to M-branes II: Twisted Wu and Wu^c structures

TL;DR

This paper explores the topological structures related to Wu classes in M-theory, introducing twisted Wu and Wu^c structures that unify and extend known geometric structures like Pin^- and Spin^c.

Contribution

It defines twisted Wu and Wu^c structures, generalizing many known topological structures relevant to M-branes and M-theory.

Findings

Introduces twisted Wu classes as twisted classes.

Defines twisted Wu and Wu^c structures generalizing Pin^- and Spin^c.

Shows these structures encode classical topological structures via Stiefel-Whitney classes.

Abstract

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.