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Urs, there’s an MO question with your name written all over it (practically literally); you might want to have a look. David Carchedi has already given an answer, and hoped you’d chime in.
Thanks. I was offline all day, having a bit of a cold, too. So thanks for the heads-up. Dave Carchedi also had kindly notified me by email.
I have now posted a brief reply at MO.
Eventually I hope to bring the lecture notes at geometry of physics into shape such that they answer all these kinds of questions. Probably from mid of March on I’ll be lecturing about cohomology and fiber oo-bundles in the Netherlands. Then I will expand these notes further.
Saw your answer; thanks! Get well soon.
from the discussion in the comment boxes and the other replies I gather that many people are not wondering so much about principal bundle theory here, but about the question: why higher differential geometry?
So I posted another reply answering that question.
in the tiny-comment-box-section of LaGatta’s question the discussion has now become “What are examples where higher differential geometry helps prove stuff in plain differential geometry?”.
I have briefly listed some stuff there. But we should start an Lab page with a more comprehensive discussion
I’ll try to look into this tomorrow afternoon, when I have some free time again. But if anyone feels like going ahead, please don’t hesitate.
I quickly started a list at
But now I should really be looking into something else. More tomorrow.
One general remark up front: orbifolds and foliations are standard objects of classical differential geometry, notwithstanding what some commenters on MO claim (foliation theory goes all the way back to Cartan himself). The fact that we may think of them equivalently as Lie groupoids should not count as “statement of the problem with mentioning higher suff” in this game, unless the rules of the game are unreasonable.
But anyway, here is one example that I like (I hope we can agree that Poisson geometry is part of differential geometry):
the solution of the quantization problem of Poisson manifolds is given by passing through their Poisson Lie algebroids (for the formal deformation quantization) or their Lie integrated symplectic Lie groupoids (for the strict deformation quantization).
At least the first case is a standard case of a “celebrated theorem”. Won a Fields medal. The second case should be more celebrated still, since it is stronger, but for some reason is not so widely known.
I’ll further expand that list tomorrow, when I have a bit of time.
Ah, and I should mention our -publication:
whose introduction is effectively designed as an answer to your question.
Okay, had a spare minute now to edit just a little bit further at
Added a pointer also to Haefliger’s theorem.
On many of the points once could cite loads of further references. Maybe in another quiet minute, have to rush now.
added paragraph on how symplectic groupoids solve the symplectic realization problem by Lie theory. Needs more (references etc). But I have to run now to catch a train.
since in the comments here it now says that the question was about motivation for higher differential geometry, I have accordingly expanded a bit more:
I changed the title of that entry to
and then added something on the motivaton from Lagrangian local QFT. Still plenty of room for further expansion and improvement, but for the moment I have to leave it at that.
Good work, Urs!
the solution of the quantization problem of Poisson manifolds is given by passing through their Poisson Lie algebroids
Probably I am not knowledgeable enough to say this, but I think Maxim Kontsevich did not use (at least explicitly) Poisson Lie algebroids and Poisson Lie groups in his original solution; he mainly used the homological algebra, or, implicitly, rational homotopy theory, of course with plethora of new ideas, using and formalism. But, of course, the things are convoluted and you are of course very right at some level.
Won a Fields medal.
The statement was that it was awarded for solutions of “four problems in geometry” concerning the subjects: 1) intersection theory on compactified moduli spaces and the solution to the Witten conjecture, 2) knot invariants, 3) deformation quantization and 4) quantum cohomology/mirror symmetry. Thus the Kontsevich formality is just one of the four.
I integrated some of these comments into the entry Maxim Kontsevich.
but I think Maxim Kontsevich did not use (at least explicitly) Poisson Lie algebroids and Poisson Lie groups in his original solution
I have heard the story like this:
First, Kontsevich publishes his solution with that mind-blowing series over graphs giving the star product and no hint for how he found that. Then next Cattaneo and Felder think “this must be the Feynman series of something” and eventually they identify it as the perturbation series of the Poisson sigma-model, identifying Kontsevich’s star product as the point-limit of the 3-point function of open topological strings propagaing on the Poisson Lie algebroid. They go to Kontsevich and show him this insight. He just shrugs and says “sure”.
That’s how I heard it. Can’t guaranetee for it. But I think even if Kontevich got his expression originally by sheer inspiration, it is still a good application of higher geoemtry to explain its origin naturally.
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