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October 12, 2007

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

Posted by Urs Schreiber

In Obstructions for nn-Bundle Lifts and Obstructions, Tangent Categories and Lie NN-tegration I mentioned some aspects of how to use the differential version of nn-transport in terms of “nn-Cartan”-connections (described first here and now with more details in the section String- and Chern-Simons nn-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 GG-bundle through a String-like extension

0Σ n1u(1)g μg0 0 \to \Sigma^{n-1}u(1) \to g_\mu \to g \to 0

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

So for g μg_\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 1p_1 of the underlying SO(n)SO(n)-bundle and the true class 12p 1\frac{1}{2}p_1 of the Spin(n)\mathrm{Spin}(n)-bundle.)

While the essential diagram can be seen in section String and Chern-Simons nn-Transport, subsection Obstructing nn-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 nn-transport for the nn-group BB and/or the corresponding nn-Cartan connection for the corresponding Lie nn-algebra, and given a sequence

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

of Lie nn-groups and/or Lie nn-algebras, we ask whether and when we may lift the transport with values in BB to a transport with values in GG.

To determine this, we compute the postcomposition of the nn-transport with the cokernel of tt. 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 (KG)(K \to G), use that this is equivalent to BB, and then compute the cokernel of the inclusion G(KG)G \hookrightarrow (K \to G).

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

Posted at October 12, 2007 4:35 PM UTC

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