The n-Category Café

Obstructions to n-Bundle Lifts Part II: The BIG Diagram

Posted by Urs Schreiber

In Obstructions for $n$-Bundle Lifts and Obstructions, Tangent Categories and Lie $N$-tegration I mentioned some aspects of how to use the differential version of $n$-transport in terms of “$n$-Cartan”-connections (described first here and now with more details in the section String- and Chern-Simons $n$-transport in the slide show of the same name) together with the usage of weak cokernels (see also this) gives a direct way to demonstrate that rationally (i.e. at the level of deRham cohomology, refining this to integral classes will take more work) the obstruction to lifting a $G$-bundle through a String-like extension

$$0 \to \Sigma^{n-1}u(1) \to g_\mu \to g \to 0$$

of the Lie algebra $g = \mathrm{Lie}(G)$ to the Baez-Crans type Lie $n$-algebra $g_\mu$ coming from a Lie algebra $(n+1)$-cocylce is given (rationally) by the characteristic deRham class of the $G$-bundle with respect to the invariant polynomial $k$ corresponding to $\mu$.

So for $g_\mu$ the String Lie 2-algebra this says that the obstruction of lifting a Spin-bundle to a String-bundle is given by the Pontryagin class of that bundle. (But this argument cannot distinguish, at the moment, the torsion components and hence does not see the distinction between the Pontryagin class $p_1$ of the underlying $SO(n)$-bundle and the true class $\frac{1}{2}p_1$ of the $\mathrm{Spin}(n)$-bundle.)

While the essential diagram can be seen in section String and Chern-Simons $n$-Transport, subsection Obstructing $n$-bundles: differential picture, I had mentioned a big hand-drawn diagram which also contains all the details for how to exactly evaluate all the arrows as morphisms of the Koszul-dual quasi-free differential algebras.

This here is to provide this big diagram, in case anyone is interested.

Diagram illustrating String-like lifts and their Chern-Simons-like obstructions (beware: 6 MB)

This was drawn while I visited Hisham Sati in Yale. With Danny we are working on a generalization of this statement, which however I am not allowed to mention in public for the moment.

Recall that the main idea is this:

Given a principal $n$-transport for the $n$-group $B$ and/or the corresponding $n$-Cartan connection for the corresponding Lie $n$-algebra, and given a sequence

$$K \to G \stackrel{t}{\to} B$$

of Lie $n$-groups and/or Lie $n$-algebras, we ask whether and when we may lift the transport with values in $B$ to a transport with values in $G$.

To determine this, we compute the postcomposition of the $n$-transport with the cokernel of $t$. The nontriviality of this should be the obstruction to the lift.

Since we are dealing with higher stuff, we want to be sophisticated about this cokernel.

The idea is to first form the mapping cone $(K \to G)$, use that this is equivalent to $B$, and then compute the cokernel of the inclusion $G \hookrightarrow (K \to G)$.

At least for the case at hand, String-like extensions, that’s tractable and leads to the described result.