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Stream: theory: mathematics

Topic: 2-Noether

David Michael Roberts (May 22 2025 at 09:18):

Does anyone ever recall any discussion about Noether's theorem in the presence of a (suitable) Lie 2-group action? It would have to be an action on something like a Lie groupoid, because otherwise it factors through an ordinary group action (modulo issues like the space of isomorphism classes not being a manifold).

David Michael Roberts (May 22 2025 at 09:19):

@John Baez

David Michael Roberts (May 22 2025 at 09:20):

In principle, maybe one could have a Lagrangian on a manifold M with values in a 2-term chain complex C (thanks to John for suggesting this in the dms) , which can be encoded by some nontrivial anafunctor, and then the 2-group action only preserves that up to isomorphism. I'm not sure if this collapses the action of a 2-group G to a 1-group action, because what G is acting on is really the "mapping space" [M,C], which is not a (structured) set, and we are looking for "homotopy fixed points" or something like that. This is all just off the top of my head, I just finished reading the rather good book "Einstein's Tutor" about Noether and her theorem, and it sparked the question in me.

John Baez (May 22 2025 at 09:21):

I haven't seen these ideas studied!

I imagine Urs say these ideas are somehow lurking in work on BRST quantization and such, but I'd like to see them studied directly in some simple example involving a classical mechanics problem with a finite-dimensional manifold (or Lie groupoid) of states.

John Baez (May 22 2025 at 09:23):

I remember when I was just starting to work on higher gauge theory, I was upset that while I knew some field theories with a strict 2-group of gauge symmetries, these were described by an ordinary Lagrangian, whose integral over spacetime gave an ordinary $\mathbb{R}$-valued action that's strictly invariant under the underlying 1-group of gauge symmetries.

John Baez (May 22 2025 at 09:25):

This felt like a degenerate case of something better. But I never got anywhere on this, in part because I didn't know a canonical generalization of $\mathbb{R}$ to some nontrivial groupoid that the action could be valued in. An obvious sort of groupoid to use would be a 2-term chain complex, since $\mathbb{R}$ is a 1-term chain complex. But I was wanting a canonical 'best one', for some psychological reason.

John Baez (May 22 2025 at 09:28):

If I were going to return to this problem, I'd probably just pick some chain complex like

$\mathbb{R}^2 \xleftarrow{d} \mathbb{R}$

with nontrivial differential $d$. Its homology is just our friend $\mathbb{R}$ in degree zero, so this is a 'more flabby' version of $\mathbb{R}$.

John Baez (May 22 2025 at 09:30):

Then I'd find some little Lie groupoid $Q$ whose tangent bundle is equipped with a map to this chain complex, the 'Lagrangian', which physicists would tend to call $L(q, \dot{q})$.

John Baez (May 22 2025 at 09:31):

Then I'd try to carry out my simple-minded proof of a simple version of Noether's theorem, jacked up to this higher situation.

John Baez (May 22 2025 at 09:33):

As you can see there's preliminary work to be done, like figure out generalizations of momentum, force, and the Euler-Lagrange equations.

All in all, a highly worthwhile project. Even if failed, one would learn something.

John Baez (May 22 2025 at 09:35):

I think it'd be really crucial to try a little example and not get distracted by trying to prematurely construct a general framework. Physicists are generally better at this than mathematicians - that's why physicists often break new ground and mathematicians come in later and systematize things.

David Michael Roberts (May 22 2025 at 09:38):

One might consider the VB-groupoid of a Lie groupoid as the appropriate 'tangent bundle'. This is where you naively apply the tangent bundle functor and get a new Lie groupoid sitting over the old one.

John Baez (May 22 2025 at 09:40):

That sounds good to me. Who calls that "the VB-groupoid"?

David Michael Roberts (May 22 2025 at 09:46):

Well, not 'the' VB-groupoid, but it's an example of a VB-groupoid, in some sense the prototypical one. There are a bunch of South American geometers who have studied such things, including "representations ip to homotopy ". The sections of a VB-groupoid and natural transformations between them organise themselves into a 2-term chain complex.

David Michael Roberts (May 22 2025 at 09:47):

A VB-groupoid is a groupoid internal to vector bundles, roughly.

David Michael Roberts (May 22 2025 at 10:09):

So an example of a Lie groupoid is an orbifold, say presented by an almost free action of a compact Lie group on a manifold...

David Corfield (May 22 2025 at 10:22):

There's the still not appeared

• Igor Khavkine, Urs Schreiber, Lie n-algebras of higher Noether currents, in preparation,

with some hints as to what it would have contained here.

David Corfield (May 22 2025 at 10:29):

Naturally, there's discussion of higher Noether currents in dcct.

John Baez (May 22 2025 at 10:31):

This concerns ordinary number-valued Lagrangians, not the wacky chain-complex-valued Lagrangians I was dreaming of. This is certainly more along the main lines of physics.

John Baez (May 22 2025 at 10:40):

I'd been starting to think of working out a toy example of a chain-complex-valued Lagrangian with David, but this lessens my desire to do that. I try to stay well away from anything has Urs worked on, because it's no fun trying to catch up with him.

This reminds me that @David Michael Roberts and I were once working on another little project, 'spin combinatorics', where we replace species by functors from a certain 'double cover' of the groupoid of finite sets to Set. We spent some time trying to understand the two double covers of $S_n$ and which was better. Then we got distracted and quit.

David Michael Roberts (May 22 2025 at 13:05):

Yes, I think that project deserves a revisit!
But somehow I think a very small very nice worked example that hints at a wider class of examples is not treading into Urs' territory. My last big paper and its sequels are essentially working out examples and the technique needed to do the calculation. From the lofty Schreiberian $\infty$-topos view, they are mere trifles, just things living in a 2-topos. But they are actual worked out, geometric examples, not "the space of solutions is a contractible space hence there is exists one"

John Baez (May 22 2025 at 13:08):

Okay, let's try to dream up a nice Lagrangian for '2-classical mechanics': a function from the tangent bundle of some Lie groupoid to some 2-term chain complex. We can worry about a setup of this sort having a 2-group of symmetries later.

John Baez (May 22 2025 at 13:14):

The hard part for me is getting an intuition for why a Lagrangian would take values in some 2-term chain complex. One idea could be this: the value of the Lagrangian seems like a real number, but it's not, it's just a real number up to translation because there's no god-given zero. So only differences between values matter. One might try to model this by a Lagrangian valued in the chain complex

$\mathbb{R} \xleftarrow{1} \mathbb{R}$

which is the translation groupoid of $\mathbb{R}$ acting on itself, seen as a chain complex. That's weird because this is contractible groupoid! But the idea is that given two 0-chains there's a unique 1-chain such that $d$ of it is their difference, so we can use that to talk about differences of values of the Lagrangian.

John Baez (May 22 2025 at 14:49):

Thus, a symmetry is allowed to add a constant to the Lagrangian.... (?)

Adittya Chaudhuri (May 22 2025 at 16:50):

David Michael Roberts said:

One might consider the VB-groupoid of a Lie groupoid as the appropriate 'tangent bundle'. This is where you naively apply the tangent bundle functor and get a new Lie groupoid sitting over the old one.

In this paper in section 4, there is a slightly differrent notion of tangent bundle over a Lie groupoid arising from a notion of a connection on a Lie groupoid (an appropriate notion of the action of a Lie groupoid $[X_1 \rightrightarrows X_0]$ on the tangent bundle $TX_0 \to X_0$). However, unlike Ehreshmann connection in the paper, their connections may not always exists, but existence is guranteed for étale groupoids. Tangent VB groupoid $[TX_1 \rightrightarrows TX_0] \to [X_1 \rightrightarrows X_0]$ equipped with linear unital cleavage (right horizontal lift) is same as a Lie groupoid equipped with an Ehreshmann connection (in the sense of paper ). This fact was discussed in the Example 3.9 of the paper and I think can be establised via a linear version of Grothendieck construction.

• Interestingly, although Ehreshmann connection exsits for all Lie groupoids, it does not define an action of a Lie groupoid $[X_1 \rightrightarrows X_0]$ on the tangent bundle $TX_0 \to X_0$, but defines a quasi-action (Definition 2.11 in paper), but on the other hand, connection in this paper defines an action, but unfortunately such connections do not exists in every Lie groupoid.

John Baez (May 22 2025 at 17:33):

Interesting! It's odd that in Definition 2.11 Abad and Crainic define an 'action' (a strict action of a group on a Lie groupoid) and a 'quasi-action' (where the usual action law is dropped completely), but not a 'weak action', where the action law holds up to coherent isomorphism.

Adittya Chaudhuri (May 22 2025 at 20:33):

Thanks. I think Abad and Crainic have stated the coherence conditions "implicitly" in Proposition 2.15.

In my paper, I obtained a similar result in Proposition 3.8 in the framework of "quasi-action" of a Lie groupoid $[X_1 \rightrightarrows X_0]$ on a principal $G$-bundle over $X_0$, where the Lie group $G$ is a part of a Lie crossed module $(G, H, \tau, \alpha)$. These coherence conditions allowed us to see a groupoid object in the category of principal bundles as a Lie 2-group torsor version of a Grothendieck construction ( which we constructed in Proposition 3.9). Our result is in the same spirit to "how a VB-groupoid can be seen as a smooth linear version of Grothendieck construction of a 2-term representations upto homotopy of Lie groupoids" (Example 3.16 in this paper).