The n-Category Café

Well said.

By the way, regarding puzzle pieces: did you notice how the Landman-fact seems to hang together with BV?

Every fiber-linear dual of a Lie algebroid is naturally a Poisson manifold. Now, those fibers really ought to be regarded as being in degree +1. After fiberwise dualization they ought to be regarded as being in degree -1. But then the Poisson structure needs to be regarded as a graded degree +1 Poisson structure. That’s precisely the starting point for BV!

In particular, take the archetypical case where the Lie algebroid to start with is the tangent Lie algebroid. Dualize to get the cotangent bundle. But now with the grading just described, regarded as the shifted cotangent bundle with the fibers in degree -1. Compare with the BV-BRST grading structure (for instance here). These are the antifields!

So what seems to happen is that you start with a Lie algebroid, dualize to get a Poisson manifold in the graded world. Then extend that in turn to a DGCA, which amounts to picking a differential, hence an odd vector field, hence the Hamiltonian generating it, hence the action functional satisfying the master equation. Hence, finally, another Lie $n$-algebroid structure on the extended dual of the original one.

Any report backs on Danny’s talk on The basic bundle gerbes on unitary groups, revisited, or Aldrovandi’s talk on 2-gerbes, or Wang’s talk on Differential Twisted K-theory and Applications?

Unfortunately I am commuting between Bonn and elsewhere and have trouble being here in the morning before 10am. As a result, I missed Danny’s and Aldrovandi’s talk and saw just the last few minutes of Wang’s.

But Danny says he will give me his notes Aldrovandi. (Hm, maybe I can make Aldrovandi give me his notes on Danny’s talk, then ;-)

I did take notes in the talk following that, Arthur Jaffe on supersymmetric quantum mechanics and its relation to cyclic cohomology by a generalization of the formula for the index of the supercharge to a certain QM correlator. You want to see those?

After Jaffe’s talk I ran back from the Max-Planck institute to the Hausdorff center to hear Masoud Khalkali continue his lecture on noncommutative geometry – also talking about cyclic cohomology (of course!).

After that I looked into our research with Hisham and Jim and with Konrad, then I had already missed most of tea and had to run already again for yet another part of Masoud’s lecture.

Now I just need to reply to my email and some blog comments, and then I am already ready to do some real work. :-)

Busy days, these days.