The n-Category Café

Twisted Higher Bundles in Münster

Posted by Urs Schreiber

Right now I am at the 17th NRW Topology Meeting. In a few minutes I will talk about Principal ∞-Bundles – Theory and Applications.

By coincidence it turns out that the previous speaker, Ulrich Pennig discussed, in a nice talk, such an application: twisted 2-vector bundles.

This is joint work of him and Brano Jurčo. They consider BDR 2-vector bundles which, by definition, are the objects classified by, roughly, the monoidal delooping of the monoidal category $GL_\bullet(Vect)$. Their starting point to consider twists of these structures is the discussion in Thomas Kragh’s Orientations and Connective Structures on 2-vector Bundles, who constructs, for each $n$, a fiber sequence

$$B OGl_n(Vect) \to B Gl_n(Vect) \stackrel{c}{\to} K(\mathbb{Z}_2, 3) \,,$$

and interprets the space on the left as that classifying “oriented BDR 2-vector bundles”, in higher analogy of orientation of vector bundle. Accordingly, the map $c$ induces a notion of twisted (twisted oriented) 2-vector bundles, with twist a class in $H^3(-,\mathbb{Z}_2) \hookrightarrow H^4(-, \mathbb{Z})$, hence with twisting $\infty$-bundles specific $B^2 U(1)$-principal 3-bundles (aka bundle 2-gerbes).

This is similar to the twisted String 2-bundles which are twisted by the fractional Pontryagin class in $H^4(-, \mathbb{Z})$.

Have to run now. More details later.