## July 7, 2007

### Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

#### Posted by John Baez

I’m mainly here in Paris to talk about categories, logic and games with Paul-André Melliès of the Preuves, Programmes et Systèmes group at Université Paris 7. But, I was invited by Marc Lachièze-Rey of the AstroParticule et Cosmologie group to give a talk on physics.

So, I took this as an excuse to speak about the work of my student Derek Wise.

Derek just finished his thesis in June. This fall he’s going to U. C. Davis. He’s been talking to me about Cartan geometry and MacDowell–Mansouri gravity for several years now — but I’ve been keeping most of it secret, so nobody would scoop his thesis. It’s a great pleasure to finally say more about it!

Abstract and transparencies of the talk follow…

#### Cartan Geometry and MacDowell-Mansouri Gravity: the Work of Derek Wise

MacDowell and Mansouri invented a clever formulation of general relativity in which the Lorentz connection and coframe field are combined into a single connection with the DeSitter group SO(4,1) or anti-DeSitter group SO(3,2) as gauge group, depending on the sign of the cosmological constant. While this formulation may seem like a ‘trick’, it actually has a deep geometrical meaning. This is best understood in terms of Cartan’s approach to connections — an approach which was somewhat forgotten after his student Ehresmann developed a simpler approach which eventually became standard. Witten’s formulation of 3d gravity as a Chern–Simons theory is also clarified using Cartan geometry. However, in 3 dimensions the relevant Cartan connection is flat and gravity is a topological field theory, while in 4 dimensions this is true only in a certain limit. In this limit, point particles and certain string-like excitations can be nicely described as topological defects. This talk is an exposition of the work of Derek Wise.

• Cartan geometry and MacDowell-Mansouri gravity: the work of Derek Wise, in PDF.

Much of this work is contained in Derek’s thesis. While it’s basically done, he’s still polishing it a bit before making it public. For now you can read these papers of his:

You can also see the transparencies of these talks:

The following papers are very relevant as well:

Posted at July 7, 2007 9:55 PM UTC

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### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

The url-link of “Quantum gravity in terms of topological observables” needs fixing - remove empty space in paper number.

Posted by: nitin on July 8, 2007 2:21 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Thanks! Fixed!

I should add that while this blog entry is rather unfriendly — it basically says ‘leave the room and go read my talk!’ — Cartan geometry is very beautiful. First of all, Cartan geometry completes this commutative square:

$\array{ Euclidean geometry & \rightarrow & Riemannian geometry \\ \downarrow & & \downarrow \\ Klein geometry & \rightarrow & Cartan geometry }$

It’s like what category theorists call a ‘pushout’ — the ‘least common generalization’ of both Klein geometry and Riemannian geometry, which both generalize Euclidean geometry. So, if you like these other forms of geometry, you’re bound to like Cartan geometry!

Second of all, Cartan geometry is very down-to-earth: it amounts to describing a space by saying how you’d roll a Klein geometry on it. A Klein geometry is a homogeneous space: a plane, or a sphere, or a hyperbolic plane, or something fancier like that. The instructions for how to do the rolling are packed into a gadget called a ‘Cartan connection’.

Cartan geometry has been sadly neglected after his student Ehresmann came up with the more general but more abstract approach to connections on principal bundles that we use today. From what I gather, Cartan wanted Ehresmann to formalize the notion of Cartan connection, and Ehresmann did, but then he went further…

So, it’s a pleasant revenge that to understand MacDowell and Mansouri’s approach to gravity, you really need Cartan connections — as the picture in my blog entry hints!

Posted by: John Baez on July 8, 2007 4:14 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

So when will torsoroids see the light of day? Or has someone come up with a better name?

Posted by: David Corfield on July 12, 2007 9:40 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Derek and Jim have been having lots of conversations over the last few years, only a bit of which found their way into Derek’s thesis. Derek should start explaining this stuff to the world!

However, even now that it’s summer, he’s pretty busy. He’s not roaming Europe like us lazy guys with permanent positions! He’s teaching summer school, polishing up his thesis before putting it on the arXiv, and — to make a few extra bucks — improving the format of old editions of This Week’s Finds.

The idea of a torsoroid is pretty nice, so I should explain it before I forget it.

When a space $M$ is given a Cartan geometry modelled on the Klein geometry $G/H$, it must have the same dimension as $G/H$. Even better, it comes with a principal $H$-bundle $P \to M$ where $P$ has the same dimension as $G$.

We should think of $P$ as a space that mimics $G$, in a manner that’s completely successful in the $H$ directions, but only ‘infinitesimally’ successful in the $G/H$ directions.

At one extreme, if $H = G$, $P$ is a perfect copy of $G$! More precisely, it’s a $G$-torsor.

At the other extreme, if $H$ is trivial, $P$ is a lousy copy of $G$, except infinitesimally: it’s just any manifold with the same dimension as $G$.

This leads us to call $P$ a ‘$G$-torsoroid, flat in the $H$ directions’.

Derek and Jim formalized this a bit, roughly as follows.

Suppose $G$ is a Lie group, and let $Lie(G)$ be its Lie algebra. Let $P$ be a smooth manifold.

If $G$ acts smoothly on $P$ we can differentiate this action to get a Lie algebra homomorphism:

$\phi: Lie(G) \to Vect(P)$

Moreover, if $P$ is a $G$-torsor, the linear map

$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$

is an isomorphism for each $x \in P$.

We can also turn around these statements, at least when $P$ is compact and $G$ is simply connected, so every action of $Lie(G)$ as vector fields on $P$ generates an action of $G$ on $P$. Under these assumptions, if we have a Lie algebra homomorphism

$\phi: Lie(G) \to Vect(P)$

it always comes from differentiating an action of $G$ on $P$. If moreover

$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$

is an isomorphism for each $x \in P$, $P$ is a $G$-torsor!

We get the concept of ‘torsoroid’ — which is secretly almost the concept of Cartan geometry! — by weakening this setup a little.

Suppose $H$ is any Lie subgroup of $G$. We define a $G$-torsoroid to be a smooth manifold $P$ equipped with a linear map

$\phi: Lie(G) \to Vect(P)$

such that

$\array{ Lie(G) &\to& T_x P \\ v &\mapsto& \phi(v)_x }$

is a vector space isomorphism for each $x \in P$.

Note, we drop the condition that $\phi$ is a Lie algebra homomorphism!

However, if $\phi$ restricted to $Lie(H) \subseteq Lie(G)$ is a Lie algebra homomorphism, we say our $G$-torsoroid is flat in the $H$ directions. If $H$ is simply-connected and $P$ is compact, we then obtain an action of $H$ on $P$, and defining

$M = P/H$

we get a principal $H$-bundle

$P \to M$

But, we get more: if $G$ is simply-connected, we get a Cartan geometry on $M$!

Even better, we can define a curvature that measures the failure of $\phi$ to be a Lie algebra homomorphism on all of $Lie(G)$:

$curv(v,w) = \phi([v,w]) - [\phi(v),\phi(w)]$

This vanishes for all $v, w \in Lie(H)$ if and only if our $G$-torsoroid is ‘flat in the $H$ directions’.

Let me translate this into something more resembling English. A $G$-torsoroid is a manifold such that if you’re sitting at any point and someone hands you an element of $Lie(G)$, you know which direction to march in. So, you might think you were living on a copy of $G$.

However, if you attempt to march around following these instructions, you may not come back to where you started when you expect to!

We say the $G$-torsoroid is ‘flat in the $H$ directions’ if you do come back where you started whenever you expect to, as long as you only move in directions given by the Lie subalgebra $Lie(H) \subseteq Lie(G)$. The ‘curvature’ defined above then vanishes when $v$ and $w$ are in $Lie(H)$. And under some mild conditions, your torsoroid will then be a bundle of $H$-torsors. In other words, it becomes a principal $H$-bundle over $P/H = M$.

If I had half an hour more, and could draw you pictures, you’d see the beauty of this idea. I can do it when we meet in Delphi this Thursday!

Posted by: John Baez on July 14, 2007 2:00 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

If I had half an hour more, and could draw you pictures, you’d see the beauty of this idea. I can do it when we meet in Delphi this Thursday!

With a stylus and wax tablet, or with a stick in the sand?

Posted by: David Corfield on July 14, 2007 3:26 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

I was going to test out some new technology called a ‘quill’.

Posted by: John Baez on July 15, 2007 8:29 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

When we finally have Klein 2-geometry nailed down, Cartan 2-geometry should be a breeze, with a little help from 2-torsoroids.

Posted by: David Corfield on July 15, 2007 10:30 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Let me translate this into something more resembling English. A G-torsoroid is a manifold such that if you’re sitting at any point and someone hands you an element of Lie(G), you know which direction to march in.

This is a very pretty picture. I’ve now officially been won over to “torsoroids”! Any diagrams would be much appreciated.

Posted by: Bruce Bartlett on July 15, 2007 1:33 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Hey, Bruce Bartlett is back! It’s been sadly quiet around here lately.

Bruce wrote:

I’ve now officially been won over to “torsoroids”!

Great! The name is not good, though. First of all, people hate things like ‘weak pseudoquasioids’, since they sound like defective entities. Second of all, these so-called torsoroids aren’t examples of the general process of oidization.

I guess the right name for them may be something like ‘Cartan geometries’!

Any diagrams would be much appreciated.

If you get the idea, you’ve probably got the right mental picture already: a space $P$ with vector fields on it corresponding to elements of $Lie(G)$, which don’t integrate to give an action of $G$, but only of some subgroup $H$… so that $P$ looks like a bundle of copies of $H$ — or more precisely, a principal $H$-bundle.

If you know enough geometry, you can ‘see’ all this stuff without being shown the pictures. I wasn’t sure if David would see it.

Posted by: John Baez on July 15, 2007 4:16 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Lol at Urs’ desperate attempts to jack himself back into the matrix, if only for a moment.

Posted by: Bruce Bartlett on July 16, 2007 1:35 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

$\phi : \mathrm{Lie}(G) \to \mathrm{Vect}(P)$

am on vacation STOP but managed to escape to an internet café STOP am seeing tons of interesting stuff at our n-café STOP wish I were there STOP all this here is closely related to the second edge of the cube STOP greetings from Conil

:-)

Posted by: Urs Schreiber on July 15, 2007 7:26 PM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

So is it fair to think of torsoroids as a sort-of more “invariant” formulation of Cartan geometry… at least in the case when G is simply-connected?

I ask that because, when I first read through the slides of your talk on Cartan geometry and Derek’s work, I hit the sentence

(1)$X = G/H$

and I kind of sighed inwardly, a bit, and I gave up. I’m just an amateur, but I’ve always been uncomfortable when things get quotiented out, especially in geometrical contexts. It kind of messes with my geometric intuition slightly. Also, this is the n-category cafe, isn’t it? Quotienting is often considered poor form in these parts. Thirdly, it messes about with smooth structures. For instance, I prefer the groupoidy/stacky picture of orbifolds to the local charts picture.

But then in your post, you mentioned the formulation of this in terms of a “$G$-torsoroid, flat in the $H$-direction” . This I find very appealing, there’s no explicit quotienting going on, and geometrically I find it easier to understand. I suspect that’s a bit silly of me; an expert has no qualms switching from one picture to the other, I’m sure.

So are torsoroids a completely equivalent formulation of Cartan geometry? What about a Cartan connection… can you phrase it in torsoroid language? I couldn’t find the word “torsoroid” in the talks/papers you referenced above.

Posted by: Bruce Bartlett on July 16, 2007 1:57 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Bruce wrote:

So is it fair to think of torsoroids as a sort-of more “invariant” formulation of Cartan geometry… at least in the case when $G$ is simply-connected?

Yes, something like that. In particular, you can talk about a $G$-torsoroid without specifying the subgroup $H$; it will then tell you what $H$ is. In the traditional approach to Cartan geometry, you need to specify $G$ and $H$ from the very start.

So are torsoroids a completely equivalent formulation of Cartan geometry?

I hope so — modulo those nitpicky compactness and simply-connectedness assumptions, which I’m using to guarantee that any Lie algebra homomorphism $\phi: Lie(G) \to Vect(P)$ exponentiates to give an action of $G$ on $P$. There should be various ways to deal with this annoying issue.

What about a Cartan connection… can you phrase it in torsoroid language?

I just did! Or at least I think so.

A Cartan geometry is usually described as a principal $H$-bundle $P \to M$ equipped with a ‘Cartan connection’, namely a $Lie(G)$-valued 1-form on $P$ satisfying a list of 3 properties. A ‘$G$-torsoroid’ should be another way of talking about the same thing.

Actually I should make sure all 3 properties hold — I haven’t checked them all. I just wanted to report on this partially baked idea that Derek and Jim developed before we all forget it.

I couldn’t find the word “torsoroid” in the talks/papers you referenced above.

Right; that’s exactly the point. David asked when torsoroids would ever see the light of day, and I realized I’d almost forgotten what they were! So I decided to remember and jot it down here before I completely forgot.

Derek may develop this stuff more someday, but he might not. There are a lot more ‘practical’ aspects of Cartan geometry which he’s more urgently interested in, like the relation to MacDowell–Mansouri gravity.

You should not shrink from quotient spaces — if you do it too much, you’ll never enter the wonderful world of Klein geometries and symmetric spaces. Sure we $n$-category theorists want to eliminate all equations. But if you cringe at all parts of math that contain equations, you’ll be cringing too much to have fun.

Posted by: John Baez on July 16, 2007 8:25 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

If, as it has been said,

Torsors are everywhere. You have only to open your eyes and look!

how hard do we have to look for torsoroids? Is it enough just to

Any chance of them appearing in music theory? Maybe you only find $G$-torsors for discrete $G$ there.

So next question is there any sense in thinking of torsoroids for discrete $G$? If $G = \mathbb{Z}$ and $H = m \mathbb{Z}$, wouldn’t that make $M$ a cyclically ordered set with $m$ elements, and $P = \mathbb{Z}$ sits fibred above it with fibres isomorphic to $H$.

Posted by: David Corfield on July 16, 2007 9:30 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

I shouldn’t say $P = \mathbb{Z}$. It’s more like a spiral staircase, extending infinitely in each direction.

Your definition of torsoroids was in terms of Lie groups and Lie algebras, which appeared to rule out the discrete setting. Does Urs’ idea about the tangent space of any category allow for smooth transfer of definition?:

Let $C$ be any category. It’s tangent space at any object $x \in \mathrm{Obj}(C)$, which I’ll write

$T_x(C)$

ought to be the category whose objects are morphisms in $C$ starting at $x$ and whose morphisms are commuting triangles.

Posted by: David Corfield on July 17, 2007 10:16 AM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: An arrow-theoretic formulation of tangency and its relation to inner automorphisms.
Tracked: July 21, 2007 4:36 PM

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

John, thanks for your columns and blogs over the years, the little bits which I have understood have been very entertaining - it’s a great service you provide to the scientific community! But the revelation to me here (I’m a neurobiologist) was your statement “but I’ve been keeping most of it secret, so nobody would scoop his thesis”, ie. the sociology of mathematical physicists isn’t all that different from molecular biologists!

Posted by: Andrew Tan on April 18, 2008 6:25 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

I tend to be pretty open with drafts of my own papers, but a bit more protective when my students are involved.

I eventually got so impatient waiting for Derek to polish his thesis and put it on the arXiv that I put a version of it on my website. If you try to view this using your web browser, it may crash — at least, that’s what always happens with me! But this is not a fiendish trick to prevent people from stealing his results; I don’t know why it happens. If I download the PDF file and then view it using Adobe Acrobat, it works okay.

Derek says he’ll polish up his thesis and put it on the arXiv ‘soon’.

Posted by: John Baez on April 21, 2008 10:40 AM | Permalink | Reply to this

### Re: Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Derek Wise published a relly cool new article on this topic:

Symmetric space Cartan connections and gravity in three and four dimensions

Authors: Derek K. Wise (Submitted on 10 Apr 2009)

Abstract: Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant theories in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest that the language of Cartan geometry provides a guiding principle for elegantly reformulating any "gauge theory of geometry".

Posted by: Daniel de França MTd2 on April 13, 2009 4:32 AM | Permalink | Reply to this

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