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Jacques Carette (Feb 16 2022 at 18:17):Originally, Lie Groups arose out of the study of symmetries of PDEs, and they remain quite useful for integrating them and for finding "good coordinate systems" in which to study a given PDE. Then they (Lie Groups) took on a life of their own.
My question: is there a similar story for Lie Groupoids and PDEs? Or Lie Groupoids and X, where X somehow generalizes PDEs? I feel like I've actually asked that question before, but I can't dig it up. I don't think it got an answer that made me go 'aha, that makes sense!'.
John Baez (Feb 16 2022 at 18:38):One big use of Lie groupoids has been to study "gauge theories", which are PDE with infinite-dimensional groups of "gauge symmetries" - roughly, symmetries such that two solutions related by a symmetry are considered "physically equivalent".
John Baez (Feb 16 2022 at 18:40):An example is the PDE saying that a connection on a bundle is flat. There's a "stack" of solutions of this equation - that is, roughly, a Lie groupoid of solutions!
John Baez (Feb 16 2022 at 18:41):The morphisms in this Lie groupoid are the gauge symmetries.
John Baez (Feb 16 2022 at 18:43):I chose a relatively simple example of how Lie groupoids show up in PDE, and there's a lot you can read about them here:
John Baez (Feb 16 2022 at 18:44):In this work, you can think of stacks as a particular attitude toward working with Lie groupoids - sometimes informed by algebraic geometry (rather than merely differential geometry), and taking a careful attitude toward what count as the correct morphisms between Lie groupoids.
Jacques Carette (Feb 16 2022 at 21:35):That google search gives me lots of things on moduli spaces of flat connections (notice the stack -> space switch, done by Google). What I don't find is any mention of PDEs, or of the use of symmetries to 'integrate'. [I have no doubts you're right, I just couldn't connect the dots given this material.]
Plus, in the classical methods, you get a Lie group of symmetries, not of solutions. So that's got me quite puzzled.
John Baez (Feb 17 2022 at 04:50):What I don't find is any mention of PDEs
A connection is 'flat' if it obeys a specific PDE. But often you won't find any mention of the phrase 'PDE', and they probably won't even write equation down since everyone studying this subject knows it.
or of the use of symmetries to 'integrate'.
That's not how Lie groupoids are used here; instead the space of solutions of this particular PDE is much better than a mere space, it's a Lie groupoid called a 'moduli stack'. (As I mentioned, the morphisms in this Lie groupoid are called 'gauge symmetries'.)
A moduli space is a poor man's version of a moduli stack: it's the space of isomorphism classes of objects in the Lie groupoid.
John Baez (Feb 17 2022 at 04:52):Plus, in the classical methods, you get a Lie group of symmetries, not of solutions. So that's got me quite puzzled.
Yeah, the concept of 'connection' shows up in 'gauge theories', which exhibit a lot of interesting mathematical structure that you don't see in other PDEs. So this is a different ball game here!
Jacques Carette (Feb 17 2022 at 12:30):So I'm left with the impression that in the intersection between "Lie Groupoids" and "PDEs", there might be completely different things - you give me one, that is cool indeed, but of a rather different nature (it seems) then the classical occurrence as 'symmetries'. Is that fair?
John Baez (Feb 17 2022 at 15:47):Now that you're making me think about it, they are not completely different. Consider the 'classical case' of a PDE with a Lie group of symmetries. Let be the space of solutions of the PDE. acts on , so we can form the 'weak quotient', or 'action groupoid' . This is often an infinite-dimensional Lie groupoid since is often an infinite-dimensional manifold.
John Baez (Feb 17 2022 at 15:49):The exact same pattern is being used in the forming the moduli space of flat connections, except here is the space of solutions of the 'flatness' PDE, the symmetry group , called the 'group of gauge transformations', is an infinite-dimensional Lie group, and the weak quotient is equivalent to a finite-dimensional Lie groupoid: the 'moduli stack of flat connections'.
John Baez (Feb 17 2022 at 17:29):Basically making infinite-dimensional allows the weak quotient to be equivalent to a finite-dimensional Lie groupoid even though is infinite-dimensional. Making bigger makes smaller (have fewer isomorphism classes of objects).
Jacques Carette (Feb 17 2022 at 23:37):Always glad when I make people think :grinning_face_with_smiling_eyes: Thanks, that helps. Something to ruminate on.