Klein 2-Geometry XII

August 17, 2011

Posted by John Baez

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Back in May 2006, David Corfield wrote a blog entry called Klein 2-geometry, saying:

As a small experiment in collective, public thinking, I’m going to devote a post to the attempt to categorify Kleinian geometry, and update the date so it doesn’t slip off the radar of ‘Previous Posts’.

His question was:

What prevents an Erlangen program for 2-groups?

The Erlangen program, is, of course, Felix Klein’s plan to study highly symmetrical spaces by thinking of them as quotient spaces $G/H$ where $G$ is a group and $H$ a subgroup. If you’ve heard of this program but never really read about it, you might like his recent review article:

Felix Klein, A comparative review of recent researches in geometry, arXiv:0807.3161.

Generalizing this idea to 2-groups (or beyond) seemed like a great idea, and David’s original post helped trigger the formation of the n-Category Café. The discussion went on and on, all the way to Klein Geometry XI. However, it never developed to the height of magnificence that I’d hoped, mainly because of the lack of a clear goal.

But then this summer I went to Erlangen, and talked to Derek Wise…

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Derek has thought deeply about gravity and Cartan geometry. As I mentioned, Klein geometry studies beautiful, symmetrical spaces of the form $G/H$ . Cartan geometry is a generalization of Klein geometry that studies lumpy, bumpy spaces that are locally, approximately shaped like $G/H$ .

A famous example is Riemannian geometry. This studies manifolds that locally, approximately look like Euclidean space. Euclidean space is $G/H$ where $G$ is the Euclidean group and $H$ is the rotation group. Each point of a Riemannian manifold has a ‘tangent space’ that looks exactly like Euclidean space. That’s the idea of Riemannian geometry: it studies spaces that locally, approximately look like Euclidean space — and this approximation becomes perfect in the limit.

But there’s another example of Cartan geometry that studies manifolds where each point has a ‘tangent sphere’. And there’s another that studies manifolds where each point has a ‘tangent hyperbolic space’. And there’s another that studies manifolds where each point has a ‘tangent deSitter spacetime’. Hmm, that’s starting to sound a bit like our universe! Yes, we can say as usual that each point has a ‘tangent Minkowski spacetime’, but if the cosmological constant is nonzero, the ‘default’ for spacetime is not flat like Minkowski spacetime, but slightly curved, like deSitter’s expanding universe.

If you don’t know what I mean, don’t worry — these are just different choices of $G/H$ .

Anyway, one dark and stormy night Derek and I were walking around Erlangen, when we heard a ghostly voice, calling out:

You must categorify my program, and the work of Cartan!

We looked around, but we couldn’t see where the voice was coming from. Then I heard:

You must use the Poincaré 2-group… and don’t forget torsion…

I felt a cold hand on my shoulder. I twisted around, trying to elude its icy clutch. I didn’t see anyone.

My panic was so great I must have fainted. When I came to I was lying near the side of the road. Evidently a whole night had passed. When I looked up, this is what I saw:

Derek was already awake, scribbling frantically in a notebook in a nearby café. In the grip of inspiration, he hadn’t even bothered to revive me. He wordlessly handed me a cup of coffee, and then explained his ideas.

The Poincaré 2-group is an easy example of a strict Lie 2-group: it has the Lorentz group as its group of objects, and for each object, the translation group of Minkowski spacetime as its group of automorphisms. Crane and Sheppeard suggested using this 2-group to build a ‘state sum model’—that is, a kind of quantum field theory of sorts. After considerable work this was achieved, not for the Poincaré 2-group, but for the closely related ‘Euclidean’ 2-group, which is easier because the group of objects, now the rotation group, is compact. The result turned out to be closely related to Baratin and Freidel’s spin foam model of flat spacetime.

However, physical meaning of the Poincaré 2-group remains obscure. Why? Because we don’t know a classical field theory involving a 2-connection for the Poincaré 2-group! Such a 2-connection would look locally like a 1-form taking values in the Lie algebra of the Lorentz group, together with a 2-form taking values in the Lie algebra of the translation group!

Of course, we get a 1-form taking values in the Lie algebra of the Lorentz group whenever we’re studying gravity, because that’s what a metric-compatible connection on a Lorentzian manifold looks like, locally. But a 2-form taking values in the Lie algebra of the translation group? Where would we ever get a thing like that?

Wait — that’s exactly what torsion is! Given a connection $\nabla$ on the tangent bundle of a manifold, its torsion is defined by $T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$ You can think of $T$ as a 2-form taking values in the tangent bundle—or locally speaking, in the Lie algebra of the translation group.

On a Riemannian or Lorentzian manifold, the Levi-Civita connection is the unique metric-compatible connection with vanishing torsion, and the curvature of this connection describes gravity. But for manifolds meeting some restrictions on their topology, one can also find a metric-compatible connection with vanishing curvature, called the Weitzenböck connection, and then the torsion of this connection is nonzero, and interesting.

There is, in fact, a way to rewrite general relativity involving the Weitzenböck connection. This is called teleparallel gravity. This theory seems conceptually quite different than general relativity. For example, in this theory, unlike in general relativity, gravity is seen as a force. This force makes freely falling particles deviate from geodesics as defined by the Weitzenböck connection.

It sounds wacky! However, teleparallel gravity is completely equivalent to general relativity… at least in the absence of spinning matter… at least on manifolds with trivial tangent bundles.

So, without coming out and ‘endorsing’ teleparallel gravity, it’s still interesting to see if we can write it as a higher gauge theory, involving a 2-connection for the Poincaré 2-group. And, after a month or two of work, it seems to be working:

John Baez and Derek Wise, Teleparallel gravity as a higher gauge theory I. (Draft version.)

Moreover, we even seem to be seeing strong clues that teleparallel gravity can nicely understood using Cartan 2-geometry! In part I here, we take you to the brink of believing that… and then explain why it’s not as easy as it sounds. In part II we’ll show how to do it. So, this paper ends on a bit of a cliffhanger.

Nonetheless, there’s enough to see here that we’d like you take a look and say what you think! We make no claims as to the ultimate importance of these ideas, but they do seem kind of fun.

Posted at August 17, 2011 7:57 AM UTC

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41 Comments & 1 Trackback

Re: Klein 2-Geometry XII

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When I had a trackback mentioning the word ‘teleparallel’ in my inbox, I felt sure there was some crank about. How pleasant to find it’s back to Klein 2-geometry.

Let’s see if predictions about what Cartan 2-geometry might involve are close.

Posted by: David Corfield on August 18, 2011 12:15 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Does the word ‘teleparallel’ make you think ‘crank’? If so, why? It would be good for us to know. Maybe we should change the title of our paper.

My impression of teleparallel gravity had long been that it’s an unorthodox, probably unimportant, but not insane idea. Phrases like ‘distant parallelism’ and ‘Fernparallelismus’ spring to mind. It goes back to Einstein’s work on unified field theories:

Tilman Sauer, Field equations in teleparallel spacetime: Einstein’s Fernparallelismus approach towards unified field theory.

However, our ambitions are far less grand in scope: we picked this one out of the many almost-equivalent formulations of general relativity because the Weitzenböck connection and its torsion combine to give a Poincaré 2-group 2-connection. This is one of the first hints as to what the Poincaré 2-group might actually be good for!

Posted by: John Baez on August 18, 2011 2:52 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Tilman Sauer should clearly collaborate with this fellow.

Posted by: Tom Leinster on August 18, 2011 10:06 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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It was just a prejudice on my part. I fleetingly imagined things like telepathy with people in parallel universes.

But back to the more important business, when will you tell us what’s happening in your paper in terms of categorified hamsters rolling about in categorified balls?

Posted by: David Corfield on August 18, 2011 4:10 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Oh I see. That’s the cliffhanger you leave us on at p.21. What do hamster 2-balls look like?

Posted by: David Corfield on August 18, 2011 4:55 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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One very common theme of the series was that Klein 2-geometry had something to do with bundles on a homogeneous space or homogeneous orbifolds. A Euclidean 2-space was to have a Euclidean space of objects with a vector space of morphisms at a point. Is there a Riemannian 2-geometry which sees the hamster roll out such a Euclidean 2-space, and then a Cartan 2-geometry which allows the full range of homogeneous 2-spaces to be rolled out?

Hmm, then your p. 21 note that you appear only to have a point with fibre to model space might suggest the need to expand that point into some homogeneous space.

Posted by: David Corfield on August 19, 2011 8:01 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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David wrote:

What do hamster 2-balls look like?

Well, sort of like this, only categorified:

That’s what the second part will talk about! I think we’ll give a general definition of Cartan 2-geometry (at least in the limited ‘strict’ setting we’re exploring here), and then show how it applies to our example.

So far we just want to make this point:

1) Teleparallel gravity can be seen as involving a 2-connection with 2-group $G = \mathbf{Poinc}(p,q)$ , the Poincaré 2-group. However, it’s only invariant under gauge transformations taking values in the smaller 2-group $H = \mathrm{O}(p,q)$ , which actually just a 1-group, namely the Lorentz group.

2) There are many theories of gravity that involve a connection with a group $G$ , but are invariant only under gauge transformations living in some smaller group $H$ . These can often be formulated using Cartan geometries that are locally modeled on the space $G/H$ . The simplest example is Palatini gravity, where $G$ is the Poincaré 1-group and $H$ is the Lorentz group. Then $G/H$ is Minkowski spacetime. In Palatini gravity, spacetime is described as a manifold where each point has a tangent space equipped with the structure of a Minkowski spacetime.

3) Unfortunately, if $G = \mathbf{Poinc}(p,q)$ and $H = IO(p,q)$ , the quotient $G/H$ has just one object, with the translation group of Minkowski spacetime as automorphisms of this object. So, a 2-space that’s locally modelled on this $G/H$ would need to have a discrete set of points (that is, objects), each with a lot of translation symmetries (that is, morphisms).

So, in terms of the hamster ball explanation of Cartan geometry, our hamster’s 2-ball has no room to roll, though the hamster could do the cha-cha inside all night long (my silly metaphor for the presence of lots of morphisms).

I don’t want to give away the solution to this dilemma, at least not until more people have read Part I. But indeed, your guess is right.

Posted by: John Baez on August 19, 2011 8:29 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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When might part II appear?

Something I’d like to see is whether it might be useful to have the model Klein 2-geometry with compact fibres, e.g., the Hopf fibration. And we never sorted out how to involve a metric to categorify the metric Klein geometry as described by Derek in definition 2 of his paper.

Posted by: David Corfield on August 19, 2011 11:01 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Hi David -

Part II should appear “soon”. I’m not sure we want to commit ourselves beyond that. :-)

I don’t think you can get the Hopf fibration to be a “2-space”. You would need a section that picks out the identity morphism in each fiber. But the Hopf fibration doesn’t have any continuous sections.

But, thinking about the Hopf fibration does remind me of one nice example of the construction in our paper, that’s not mentioned in our paper. Take any semisimple Lie group G. G has a metric given by the Killing form. It also has a flat metric-preserving connection, given by left (or right) translation. Using the stuff from our paper, this means there is a canonically defined 2-bundle with 2-connection on any semisimple Lie group, where the structure 2-group is the Poincare 2-group of the same signature as the Killing form.

(I guess I didn’t mention what this has to do with the Hopf fibration… But, if you do the SU(2) case of this, you’ll see.)

Posted by: Derek Wise on August 19, 2011 4:42 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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I don’t think you can get the Hopf fibration to be a “2-space”.

Oh yes. But I would so like a hamster rolling in one ball and a smaller hamster rolling in an epicyclic way in a smaller ball on the other.

Posted by: David Corfield on August 19, 2011 9:43 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Does this have anything to do with: A. N. Lasenby, C. J. L. Doran and S. F. Gull. Gravity, gauge theories and geometric algebra?

Posted by: Robert Smart on August 18, 2011 10:10 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

It has ‘anything’ to do with it, but not a whole lot: this paper tries to use ‘geometric algebra’ to formulate a new theory of gravity. ‘Geometric algebra’ is a buzzword for ‘Clifford algebras’—a buzzword used by a certain community of people, fans of a guy named ‘Hestenes’, who love Clifford algebras yet don’t always know all the deep work that other mathematical physicists have done using Clifford algebras. So the phrase ‘geometric algebra’ sets off alarm bells for me, a bit like the alarm bells that went off for David when he saw the word ‘teleparallel’… although I believe his alarm system is a lot less finely tuned than mine, when it comes to unorthodox theories of physics.

Anyway, I can easily point you to lots of papers that are much more related to our paper than this one. But the easiest thing is to go to the References at the end of our paper and start clicking links: most of the references are free online.

Posted by: John Baez on August 19, 2011 8:40 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Ah, putdowns in this area aren’t what they used to be since Pertti Lounesto drowned. Well I guess I was just wondering if teleparallel was the same “flat space-time equivalent to Einstein” formulation, or different. These guys got involved in a wider group thinking about Bayesian reasoning, maximum entropy, etc. One of that wider group went off to save the world…

Posted by: Robert Smart on August 19, 2011 11:26 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Robert wrote:

Well I guess I was just wondering if teleparallel was the same “flat space-time equivalent to Einstein” formulation, or different.

Teleparallel gravity is pretty well-known among people who are serious about general relativity and its variants; there’s probably a couple hundred papers about it, starting with Einstein’s, so if these authors were doing something equivalent to teleparallel gravity, you’d hope they’d mention it—but the word ‘teleparallel’ doesn’t show up at all.

Indeed, as best as I can tell, their theory is just general relativity written in the language of geometric algebra. Equation 4.12 on page 30 writes down the all-important action, and it’s just the usual integral of the Ricci scalar, minus the matter Lagrangian.

I really don’t want to continue talking about their paper…

Posted by: John Baez on August 19, 2011 12:08 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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On p. 10 is currently says:

The fake flatness condition was first introduced by Breen and Messing

This is not true: they did introduce what they called “fake curvature” but had no reason to demand it to vanish (becuase they did not consider parallel transport over surfaces). The term “fake flatness” originates later. I’d think we introduced it in our articles.

Posted by: Urs Schreiber on August 21, 2011 12:22 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Whoops! Strange — I thought I just copied this passage from “An invitation to higher gauge theory”, but now I see that there I correctly gave Breen and Messing credit for “fake curvature” but not the fake flatness condition. Good, so I can just fix it here.

Yes, our triumph was understanding the role of fake flatness.

Posted by: John Baez on August 21, 2011 12:34 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

In speaking to Breen (a few years back now!), he regretted introducing the term ‘fake curvature’; he would have preferred, in hindsight, to have used 2-curvature, in analogy with the 3-curvature one also gets. In his defence he said that he and Messing didn’t understand its role when they found it.

Posted by: David Roberts on August 21, 2011 10:44 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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he regretted introducing the term ‘fake curvature’;

Yes, and also “fake flatness” is an unfortunate term.

It should be called (vanishing) “2-form curvature” or “the 2-form component of the curvature”. Maybe “the degree 2 component of the curvature”.

In general, for $\mathfrak{g}$ a Lie $n$ -algebra, $\mathfrak{g}$ -valued forms have curvature datum that consists of a 2-form, a 3-form, etc. up to an $(n+1)$ -form. If $\mathfrak{g}$ is even a Lie $n$ -algebroid, there is in addition a curvature 1-form.

Posted by: Urs Schreiber on August 22, 2011 9:29 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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The discussion in your article makes me think the following:

a $\mathfrak{io}(p,q)$ -connection has locally two components, corresponding to the two factors $\mathbb{R}^{p+q}$ and $\mathfrak{so}(p,q)$ :

a 1-form $\omega \in \Omega^1(U, \mathfrak{so}(p,q))$ .
a 1-form $E \in \Omega^1(U, \mathbb{R}^{p+q})$ – the vielbein ;

Accordingly the curvature 2-form has two components:

the Riemannian curvature $R \in \Omega^2(U, \mathfrak{so})$ ;
the torsion $\tau \in \Omega^2(U, \mathbb{R}^{p+q})$ .

Torsion is a part of $\mathfrak{io}$ -curvature!

Now, as to every Lie algebra, also here there is the inner automorphism Lie 2-algebra $inn(\mathfrak{io})$ . This is such that a flat $inn(\mathfrak{io})$ -valued differential form data consists of all of the above data: 1-forms $E$ and $\omega$ and 2-forms $R$ and $\tau$ , constrained to be the Riemannian curvature and torsion of $E$ and $\omega$ , respectively.

A general $\mathfrak{io}$ -connection is therefore locally a flat $inn(\mathfrak{io})$ -2-connection.

You discard part of the components of $inn(\mathfrak{io})$ such as to force $R$ to vanish. It seems to me.

Posted by: Urs Schreiber on August 21, 2011 12:35 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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All this is in our second paper. Stop reading our minds!

Posted by: John Baez on August 21, 2011 12:40 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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All this is in our second paper.

Wait. All this except the last line is in the existing literature!

Concerning the last line…

You discard part of the components of $inn(\mathfrak{io})$ such as to force $R$ to vanish.

… I am not sure what to think about it. Discard by hand? Or for some reason? By some mechanism?

I know of one good intrinsic mechanism for discarding certain components of a (higher) connection: form differential c-structures .

These homotopically kill certain curvature characteristics. For instance killing $tr(R \wedge R)$ corresponds to differential string structures .

I was at times wondering if there is any interest in considering the twisted differential structures induces by the torsion invariant $tr (\tau \wedge \tau)$ .

Stop reading our minds!

I wish I could! For instance I am wondering: if you are interested in gravity as a higher gauge theory, why not follow up on the ideas that you were giving talks about a few weeks back, in relation to John Huerta’s thesis? Higher supergravity is always higher gauge theory. And there the identification of the varous concepts is clear. We know what happens to the torsion, how the higher gauge fields come in, what the corresponding higher parallel transport means and all that. I’d enjoy seeing higher nonabelian gauge theorists get more involved in this interesting application.

Even though it requires the extension to supergravity, I find it less speculative than what you are trying to do with Derek Wise here. But of course that’s just a ver subjective impression.

Posted by: Urs Schreiber on August 21, 2011 7:15 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Urs wrote:

Wait. All this except the last line is in the existing literature!

I know. This is just a small portion of our paper, not the main point.

For instance I am wondering: if you are interested in gravity as a higher gauge theory, why not follow up on the ideas that you were giving talks about a few weeks back, in relation to John Huerta’s thesis?

That’s an obviously worthwhile project. So, it will get done regardless of whether I do it — and since supergravity and superstrings are popular subjects, it will probably get done quite soon.

I never feel happy helping a crowd of people push a big snowball that’s already rolling down a steep hill. I can never tell if I’m helping it go faster or just running to keep up! If I could run faster than other people I might enjoy this activity… but I can’t. So, I’d prefer to work on something that nobody cares about, run the risk of it not working out, but have the hope that I might discover something that would remain hidden for a long time otherwise.

You may think this is a perverse attitude, but this attitude is the reason I got interested in $n$ -categories back in 1993. I got lucky then, and perhaps getting lucky once in a lifetime is enough. My main highly abstract activity now is applying category theory to complex systems starting from electrical circuits and stochastic Petri nets and leading up to examples from biology and ecology. That’s where I could use some luck now.

Even though it requires the extension to supergravity, I find it less speculative than what you are trying to do with Derek Wise here.

Right. The combination of teleparallel gravity and Cartan 2-geometry is a bit unusual, but that’s why I like it.

Interestingly, both teleparallel gravity and Cartan geometry owe a lot to Eli Cartan. So, Derek and I sometimes feel like we are ‘categorifying Cartan’.

Posted by: John Baez on August 22, 2011 2:37 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

if you are interested in gravity as a higher gauge theory,…

Just to clarify our motivation a bit, this project John and I are working on didn’t start because we were interested in writing gravity as a higher gauge theory. It began with trying to figure out how to interpret 2-connections for the Poincare 2-group. We wanted to do this partly because of the Poincare 2-group state sum model. This state sum might correspond to some higher gauge theory, but we don’t know the corresponding continuum theory, if there is one.

But, when we realized that Poincare 2-connections can be built from a flat connection and its torsion, we got to thinking about another context in which the torsion of a flat connection plays a key role: teleparallel gravity.

Anyway, the main mathematical point of our paper wouldn’t necessarily have to be applied to teleparallel gravity, but I think the gravity aspect makes it more fun. And, thinking of teleparallel gravity in these terms also helped us better understand the geometric picture of Poincare 2-connections. So, for us, the mathematical and physical sides fit nicely together.

…why not follow up on the ideas that you were giving talks about a few weeks back, in relation to John Huerta’s thesis? … Even though it requires the extension to supergravity, I find it less speculative …

The idea of supergravity as a higher gauge theory is interesting to me too, and I might even get involved in that story at some point. But, it’s just a different story, with different motivation.

Posted by: Derek Wise on August 25, 2011 9:26 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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We wanted to do this partly because of the Poincaré 2-group state sum model.

When hearing and reading about these constructions, I always have the following questions. Maybe you can help me.

What evidence is there that these sum-expressions are related to gravity?

(In your article you proceed from the Freidel-model. When we discussed that last time I got away with the impression that the conceptual understanding of what this is actually about is lacking.)

What are the amplitudes given by these state sums? Have they been computed? The 4d pair of pants induces an algebra structure. Which algebra is that?

Why go via state sums, given a representation $n$ -category? We know that the first thing to check is if that $n$ -category is fully dualizable. That then implies everything else.

What I meant when I said “I wish I could read your minds” was: I never quite understand what motivates interest in these constructions. I find much more compelling approaches elsewhere. Of course that’s okay, we are all different. But that’s why I mentioned it.

Posted by: Urs Schreiber on August 26, 2011 10:28 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Urs wrote:

What evidence is there that these sum-expressions are related to gravity?

If this is a general question, then I refer you to the entire spin foam literature. That’s a bigger discussion than I want to get into here.

On the other hand, if it is a question about the Baratin-Freidel model, which is the one that can be interpreted as a Poincaré 2-group state sum model, then, this model explicitly excludes gravity from the outset. You could view it as an experiment to see if Feynman diagrams in 4d can be written as a state sum model in the style of spin foams.

(In your article you proceed from the Freidel-model…

I didn’t read the whole discussion you linked to, but from scanning through it quickly it seems like this was mostly about a different model (in 3d) by Freidel and collaborators.

Why go via state sums, given a representation n-category?

Well, historically, this is backwards: we didn’t “go to state sums, given a representation n-category”. We had a state sum (where by “we”, I mean Baratin and Freidel – I didn’t work on that part) that came from considerations that had nothing to do, a priori, with 2-group representations.

John and I, along with Jeff Morton, started discussing things with them because, for some mysterious reason, their state sum model seemed to be related to representations of the Poincaré 2-group. The reason remains mysterious to me. Anyway, certainly most of the abstract work on this model, starting from the representation 2-category, has just not been done yet.

What I meant when I said “I wish I could read your minds”…

Actually, when you said “I wish I could!”, right after John told you to “stop reading our minds!”, it sounded like you meant “I wish I could stop reading your minds!” As in, “Hey, stop thinking so loudly! I can hardly hear myself talk!” :-)

Posted by: Derek Wise on August 29, 2011 10:31 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Actually, when you said “I wish I could!”, right after John told you to “stop reading our minds!”, it sounded like you meant “I wish I could stop reading your minds!”

Ah, I see. I didn’t say it well. I meant that, contrary to what John seemed to be suggesting (maybe he wasn’t), I can’t really see where your approach here is motivated from and where it is headed.

I mean, I see these references you give me, but somehow I must still be missing something. Maybe I understand it later. Right now I have to look into other things.

Well, historically, this is backwards: we didn’t “go to state sums, given a representation n-category”. We had a state sum (where by “we”, I mean Baratin and Freidel – I didn’t work on that part) that came from considerations that had nothing to do, a priori, with 2-group representations.

Okay, but now that you have the proposal that this model wants to be about the 2-category of representations of some 2-group I’d be tempted to urge you: figure out to which extent this is fully dualizable. Try doing the 2-category analog of (an aspect of) this theorem which says that what makes fusion categories induce state sum models for 3d TQFTs is their fully-dualizablility.

Posted by: Urs Schreiber on August 29, 2011 8:08 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Urs wrote:

Okay, but now that you have the proposal that this model wants to be about the 2-category of representations of some 2-group I’d be tempted to urge you: figure out to which extent this is fully dualizable.

That’s a good puzzle. My guess is that the symmetric monoidal 2-category of (measurable) unitary representations of the Poincaré 2-group is not fully dualizable.

In fact I’d already guess that the symmetric monoidal category of (measurable) unitary representations of the Poincaré group is not fully dualizable.

In fact I’d already guess this for the symmetric monoidal category of (measurable) unitary representations of $SU(2)$ . This category gives an interesting state sum model, the Ponzano-Regge model — but this category is not modular, and Ponzano-Regge model is not a TQFT, as far as anyone can tell, because the state sums diverge in many cases. The problem is that $SU(2)$ has infinitely many irreducible unitary representations.

Similarly, I doubt the Poincaré 2-group state sum model gives a TQFT.

I don’t think divergent state sum models are necessarily useless. After all, I don’t think Feynman diagrams with loops are useless, even though they diverge. But, theories with divergences require a lot of extra thought.

There could be a relation between divergent Feynman diagrams and the divergences in the Poincaré 2-group state sum model, since Baratin and Freidel appear to have evidence that this state sum model can be used to compute Feynman diagrams (so far with fixed masses and spin zero labelling the edges).

… I can’t really see where your approach here is motivated from and where it is headed.

I’ve been curious about the Poincaré 2-group ever since I discovered it: it didn’t seem to be related to any known physics, except in a very mysterious way via the work of Baratin and Wise.

So, I was pleased to see that it makes an appearance in teleparallel gravity. And I was even more pleased when trying to understand this forced Derek and I to think about Cartan 2-geometry — a topic we’d been interested in for years, but had never gotten around to working on.

That’s all — for me, at least. Just having fun tying up a few loose ends here and there.

Posted by: John Baez on August 30, 2011 5:26 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Hi John,

okay, I see the obvious point about the dualizability.

So if not a TQFT with “finite dimensional state spaces” then next one would hope that the model gives a “positive boundary TQFT” which may have “infinite-dimensional state spaces”, by discarding the cap (or the cup), i.e. by restricting to cobordisms that always have non-empty outgouing boundary. For 2-d TQFTs this includes all the physically interesting cases (A-model, B-model, etc.).

But from what you say it seems unclear how to get even that. The sums don’t ever converge?

This puzzles me. I mean, not that they don’t converge, but that you can accept that. The whole undertaking is motivated from supposedly being nonperturbative QFT, isn’t it? It seems wrong to then appeal to divergencies and to Feynman diagrams. Every appeal to Feynman diagrams that I have seen in this context I find very puzzling. When I try to make sense of the situation, I come to very different conclusions than I see stated by people working on this. (Which need not mean much, as I am not really spending much time with this, evidently.)

Of course maybe some magic produces something finite from divergent sums quite independently of perturbative QFT. But is there any idea what kind of magic that should be here?

What I am lacking in this business here is a supply of hints for what’s going on. I can live with speculations, but at some point I’d need a hint of something. What are the big hints here? Derek said “see the spin foam literature”. I looked at some of it here and there and didn’t get the kind of hints that I’d hope for.

But I’ll stop bugging you now with my worries about spin foam approaches to quantum gravity. I gather it’s a matter of waiting a few more years to see if either some big breakthrough happens, or not.

Instead, I’ll send another hopefully more constructive comment on Cartan connections. In a moment.

Posted by: Urs Schreiber on August 30, 2011 10:19 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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think about Cartan 2-geometry

I am wondering:

a Cartan geometry is a Cartan connection.

A $(H\hookrightarrow G)$ -Cartan connection is a special kind of $G$ -principal connection $\nabla$ .

Namely one such that the structure group reduces to $H$ and over each point $x$ in base space $X$ the composite (in any local trivialization)

$T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$

is an isomorphism.

Now, we know how principal connections generalize first to connections on a 2-bundle and then all the way to connection on an ∞-bundle.

We also know how these locally come from L-∞ algebra valued forms.

So given all this, there seems to be an evident definition of $\infty$ -Cartan geometry :

Let $\mathfrak{h} \hookrightarrow \mathfrak{g}$ be an inclusion of L-∞-algebras and write $H \hookrightarrow G$ for their Lie integration to smooth ∞-groups.

Then say an $(H \hookrightarrow G)$ - $\infty$ -Cartan geometry over a smooth ∞-groupoid $X$ is

a $G$ -∞-connection over $X$ ;
such that for each point $x : * \to X$ with tangent $L_\infty$ -algebra $T_x X$ the canonical morphism

$T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$

is a quasi-isomorphism.

See what I mean? The evident verbatim categorification of the definition of Cartan connection. Is that not what you’d be after, too? Am I overlooking something?

I’d say that, if we grant them their different language, then Castellani-D’Auria-Fre in their formulation of higher dimensional (super)gravity, look precisely at examples of this.

In their book they start out discussing 4-dimensional gravity in terms of $(O(d,1) \hookrightarrow Iso(d,1))$ -Cartan geometry as usual.

Then they pass (as you know, I am including links just for the record, in case anyone else reads this here) from the super Poincare Lie algebra to its higher central extensions, such as the supergravity Lie 6-algebra for the case of 11-dimensional supergravity, $\mathfrak{sugra}_{11}$

Their whole book is about how the field content of higher dimensional supergravity is – in my language here – an $\infty$ -Cartan connection for inclusions of the form $(\mathfrak{so} \oplus \bigoplus_i b^{n_i} \mathbb{R} \hookrightarrow \mathfrak{sugra}_{d})$ .

I think.

Posted by: Urs Schreiber on August 30, 2011 11:08 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Trying to relate this $\infty$ -Cartan geometry with our conclusion as to what $\infty$ -Klein geometry should be

…the homotopy theory of pairs of connected based spaces and fibrations between them–the corresponding homogeneous space then being the fiber of the fibration,

one thing is that we hadn’t mentioned smoothness, but another thing is that we assumed that

an “ $\infty$ -subgroup” just means any $\infty$ -group homomorphism $H \to G$ .

Your Lie integration of the inclusion of $L_{\infty}$ -algebras is more restricted, isn’t it?

Posted by: David Corfield on August 31, 2011 9:36 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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David writes:

Trying to relate this ∞-Cartan geometry with our conclusion as to what ∞-Klein geometry should be

…the homotopy theory of pairs of connected based spaces and fibrations between them–the corresponding homogeneous space then being the fiber of the fibration,

one thing is that we hadn’t mentioned smoothness,

Incorporating smoothness (or any other kind of cohesive structure) is immediate by using cohesive ∞-topos technology:

there are now some remarks about this in the $n$ Lab entry higher Klein geometry .

Your Lie integration of the inclusion of $L_\infty$ -algebras is more restricted, isn’t it?

No, I don’t think so. I could have (and should have) just talked about any $L_\infty$ -morphism $\mathfrak{h} \to \mathfrak{g}$ .

Notice that in a context with infinitesimal cohesion an $\infty$ -Lie algebra is a cohesive $\infty$ -group, one that happens to have infinitesimal extension.

Posted by: Urs Schreiber on August 31, 2011 2:27 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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By the way, the notation “ $IO(p,q)$ ” for the Poincaré-group seems to be unusual.

Am I guessing right that you dropped the “S” from the usual “ $ISO(p,q)$ ” because it involves not $SO(p,q)$ but $O(p,q)$ ?

But “ $ISO(p,q)$ ” is for “isometries”, not for “ $i$ -special orthogonal” ;-)

Posted by: Urs Schreiber on August 22, 2011 9:46 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Really? I’ve never thought of “ISO” as meaning “isometries”. I think the “I” stands for “inhomogeneous”. In fact, whenever I have a group G of transformations of a vector space V, I’d be happy to write IG for the inhomogeneous, or “affinized” version of the group. I realize this may not be standard.

So, if you use “ISO(p,q)” to mean the isometry group, then what would be the oriented isometry group? “SISO(p,q)”? But then I read the first S as “super”.

Posted by: Derek Wise on August 22, 2011 10:22 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Really?

Try googling “isometry group iso” for examples of the use of “ISO” “iso” and “Iso” for “isometry group”. Notice that several of the groups listed there do not involve any orthogonal group at all (special or not).

A random example of a reference that uses terminology and notation as I am suggesting it is being used is

Claudio Dappiaggi, Gandalf Lechner, Eric Morfa-Morales, Deformations of Quantum Field Theories on Spacetimes with Killing Vector Fields (pdf)

I think the “I” stands for “inhomogeneous”.

Can you point me to a reference that follows your usage of notation?

Posted by: Urs Schreiber on August 22, 2011 1:03 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Derek and I always thought “ $\mathrm{O}$ ” meant “orthogonal”, “ $\mathrm{SO}$ ” meant “special orthogonal”, and “ $\mathrm{ISO}$ ” meant “inhomogeneous special orthogonal”. Apparently we’re not alone in thinking that, but the number of Google hits is small enough that we could be wrong, along with those other authors.

However, it doesn’t matter so much what the letters are an acronym for! What matters more is whether “ $ISO(p,q)$ ” is supposed to include all 4 connected components containing 1, P, T and PT, or just the 2 connected components containing 1 and PT. Derek and I believed the latter. That is, we believed that

$ISO(p,q) = SO(p,q) \ltimes \mathbb{R}^{p,q}$

So, we introduced the larger group

$IO(p,q) = \mathrm{O}(p,q) \ltimes \mathbb{R}^{p,q}$

by analogy. But if most people call this group $ISO(p,q)$ , I’ll be happy to follow along!

Posted by: John Baez on August 22, 2011 1:41 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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not alone

But does anyone else write “ $IO$ ”? Except in computer science?

Posted by: Urs Schreiber on August 22, 2011 1:49 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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No, we’ve never seen anyone talk about ‘IO(p,q)’ — we were under the impression that we were inventing this notation.

Posted by: John Baez on August 22, 2011 1:55 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

Can you point me to a reference that follows your usage of notation?

For example, see the abstract of this paper which uses this “I” notation for the inhomogeneous extensions of a bunch of classical groups. IO(p,q) is missing from the list, even though they discuss ISU(p,q) and IU(p,q) separately.

I must admit, it is cute that in the case of ISO(p,q), the group elements are ISOmetries. Personally, though, I’ve always written Isom(X) for the group of isometries of X. That seems to be a common notation in web searches as well.

Using such notation, I could write Isom(R^n)=IO(n).

But anyway, John is right: the main thing is that people know precisely which group we’re talking about. Perhaps we should clarify that in the draft, lest other readers think we are using “iometries” instead of isometries.

Posted by: Derek Wise on August 22, 2011 3:21 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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For example, see the abstract of this paper

Thanks for this reference!

Posted by: Urs Schreiber on August 22, 2011 3:44 PM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Derek Wise has written about our work and also his conversations with Julian Barbour on his brand-new blog:

Derek Wise, The 2-Erlangen program and Mach’s principle.

Posted by: John Baez on September 6, 2011 8:16 AM | Permalink | Reply to this

Re: Klein 2-Geometry XII

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Then I heard:

You must use the Poincaré 2-group…

Aleksandar Mikovic and Marko Vojinovic appear to have heard a similar voice:

We show that General Relativity can be formulated as a constrained topological theory for flat 2-connections associated to the Poincaré 2-group. Matter can be consistently coupled to gravity in this formulation. We also show that the edge lengths of the spacetime manifold triangulation arise as the basic variables in the path-integral quantization, while the state-sum amplitude is an evaluation of a colored 3-complex, in agreement with the category theory results.

Do they answer your point

…we don’t know a classical field theory involving a 2-connection for the Poincaré 2-group?

Posted by: David Corfield on October 24, 2011 1:19 PM | Permalink | Reply to this

Read the post Teleparallel Gravity as a Higher Gauge Theory
Weblog: The n-Category Café
Excerpt: How to see Einstein's old theory of 'teleparallel gravity' as a theory involving a 2-connection.
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August 17, 2011