String structures and connections

27.04.2015, 16:00 Uhr  –  Raum 1.08.0.50
Oberseminar Analysis, Geometrie, Physik

Christian Becker (Potsdam)

Let $X$ be a compact Riemannian $n$-manifold, with $n \geq 3$.

The bundle of orthonormal frames is a principal $O_n$-bundle.

Several geometric structures on $X$ can be described in terms of lifts of the structure group along a group homomorphism $G \to O_n$:

orientations are lifts to the connected component $SO_n \to O_n$, Spin structures are further lifts to the simply connected cover $\mathrm{Spin}_n \to SO_n$.

String structures are lifts to a $3$-connected cover of $\mathrm{Spin}_n$.

This group is usually called $\mathrm{String}_n$.

It is defined only up to homotopy and cannot be realized as a finite dimensional Lie group.

String structures on $X$ can be described in terms of homotopy theory.

The obstruction to the existence of a String structure is a certain characteristic cohomology class on $X$.

Isomorphism classes of String structures correspond to certain cohomology classes on the Spin structure.

However, there also exists a model of the group $\mathrm{String}_n$ as an infinite dimensional Fr\'echet Lie group.

It is built as a subgroup of the gauge group of a certain bundle over $\mathrm{Spin}_n$.

In this talk, based on joint work in progress with Christoph Wockel, we describe actual lifts of the structure group of a Spin structure to the group $\mathrm{String}_n$.

These are built from gauge group bundles.

We also discuss the construction of connections on our String bundles.

These are built from connections on the underlying Spin bundles and additional data, so-called Higgs fields.