## Archive for the ‘Mathematics’ Category

### George Hart’s “12-Card Star” and suit permutations.

7 March 2015

When I was at the JMM in San Antonio in January, I was happy to catch a workshop by George Hart:

I went to some great talks at the JMM, but a hands-on, interactive workshop was a nice change of pace in the schedule. Having seen some of George’s artwork before, I couldn’t resist. In the workshop, he taught us to build his sculpture/puzzle which he calls the 12 Card Star. Here’s what mine looked like:

He supplied the necessary materials: 13 decks of cards, all pre-cut (presumably with a band saw), like this:

We each took 13 card from these decks—the 13th, he said, was “just in case something terrible happens.”

He showed us how to put three cards together:

Then he gave us a clue for assembling the star: the symmetry group is the same as that of a …

Wait! Let’s not give that away just yet. Better, let’s have some fun figuring out the symmetry group.

Let’s start by counting how many symmetries there are. There are twelve cards in the star, all identically situated in relation to their neighbors, so that’s already 12 symmetries: given any card, I can move it to the position of my favorite card, which is the one I’ll mark here with a blue line along its fold:

But my favorite card also has one symmetry: I can rotate it 180$^\circ$, flipping that blue line from end to end around its midpoint, and this gives a symmetry of the whole star. (Actually, this symmetry is slightly spoiled since I drew the five of hearts: that heart right in the middle of the card isn’t symmetric under a 180$^\circ$ rotation, but never mind that. This would be better if I had drawn a better card, say the two of hearts, or the five of diamonds.)

So there are $12\times 2 = 24$ symmetries in total, and we’re looking for a group of order 24. Since 24 = 4!, the most obvious guess is the group of permutations of a 4-element set. Is it that? If so, then it would be nice to have a concrete isomorphism.

By a concrete isomorphism, I mean a specific 4-element set such that a permutation of that set corresponds uniquely to a symmetry of the 12-card star. Where do we get such a 4-element set? Well, since there are conveniently four card suits, let’s get a specific isomorphism between the symmetry group of Hart’s star and the group of permutations of the set

At the workshop, each participant got all identical cards, as you can see in the picture of mine. But if I could choose, I’d use a deck with three 2’s of each suit:

From this deck, there is an essentially unique way to build a 12-Card Star so that the isomorphism between the symmetry group and the group of permutations of suits becomes obvious! The proof is constructive,’ in that to really convince yourself you might need to construct such a 12-card star. You can cut cards using the template on George’s website. He’s also got some instructions there. But here I’ll tell you some stuff about the case with the deck of twelve 2’s. % and from these it will be clear that (if you succeed) your star will have the desired properties.

First notice that there are places where four cards come together, like this:

In fact, there are six such places—call these the six 4-fold rotation points—and it’s no coincidence that six is also the number of cyclic orderings of card suits:

Now, out of the deck of twelve 2’s, I claim you can build a 12-card star so that each of these cyclic orderings appears at one 4-fold rotation point, and that you can do it in an essentially unique way.

This should be enough information to build such a 12-card star. If you do, then you can check the isomorphism. Think up an permutation of the set of suits, like this one:

and check that you can rotate your 12-card star in such a way that all of the suit symbols on all of the cards in the 12-card star are permuted in that way.  The rest follows by counting.

Sometime I should get hold of the right cards to actually build one like this.

Of course, there are other ways to figure out the symmetry group. What George Hart actually told us during the workshop was not that the symmetry group was the permutation group on 4 elements, but rather that the symmetry group was the same as that of the cube. One way to see this is by figuring out what the convex hull’ of the 12-card star is. The convex hull of an object in Euclidean space is just the smallest convex shape that the object can fit in. Here it is:

This convex polyhedron has eight hexagonal faces and six square faces. You might recognize as a truncated octahedron, which is so named because you can get it by starting with an octahedron and cutting off its corners:

The truncated octahedron has the same symmetry group as the octahedron, which is the same as the symmetry group of the cube, since the cube and octahedron are dual.

Thanks to Chris Aguilar for the Vectorized Playing Cards, which I used in one of the pictures here.

### From the Poincaré group to Heisenberg doubles

25 September 2014

There’s a nice geometric way to understand the Heisenberg double of a Hopf algebra, using what one might call its “defining representation(s).” In fact, it’s based on the nice geometric way to understand any semidirect product of groups, so I’ll start with that.

First, consider the Poincaré group, the group of symmetries of Minkowski spacetime.  Once we pick an origin of Minkowski spacetime, making it into a vector space $\mathbb{R}^{3,1}$, the Poincaré group becomes a semidirect product

$\mathrm{ISO}(3,1)\cong\mathbb{R}^{3,1} \rtimes \mathrm{SO}(3,1)$

and the action on $\mathbb{R}^{3,1}$ can be written

$(v,g)\cdot x = v + g x$

In fact, demanding that this be a group action is enough to determine the multiplication in the Poincaré group.  So, this is one way to think about the meaning of the multiplication law in the semidirect product.

In fact, there’s nothing so special about Minkowski spacetime in this construction.  More generally, suppose I’ve got a vector space $V$ and a group $G$ of symmetries of $V$.   Then $V$ acts on itself by translations, and we want to form a group that consists of these translations as well as elements of $G$.  It should act on $V$ by this formula:

$(v,g)\cdot x = v + g x$

Just demanding that this give a group action is enough to determine the multiplication in this group, which we call $V \rtimes G$.   I won’t bother writing the formula down, but you can if you like.

In fact, there’s nothing so special about $V$ being a vector space in this construction.  All I really need is an abelian group $H$ with a group $G$ of symmetries.  This gives us a group $H\rtimes G$, whose underlying set is $H \times G$, and whose multiplication is determined by demanding that

$(h,g) \cdot h' = h + gx$

is an action.

In fact, there’s nothing so special about $H$ being abelian.  Suppose I’ve got a group $H$ with a group $G$ of symmetries.  This gives us a group $H\rtimes G$, built on the set $H \times G$, and with multiplication  determined by demanding that

$(h,g)\cdot x = h (g x)$

give an action on $H$.  Here $gx$ denotes the action of $g\in G$ on $x\in H$, and $h(gx)$ is the product of $h$ and $gx$.

For example, if $H$ is a group and $G=\mathrm{Aut}(H)$ is the group of all automorphisms of $H$, then the group $H\rtimes \mathrm{Aut}(H)$ is called the holomorph of $H$.

What I’m doing here is defining $H \rtimes G$ as a concrete group: it’s not just some abstract group as might be defined in an algebra textbook, but rather a specific group of transformations of something, in this case transformations of $H$.  And, if you like Klein Geometry, then whenever you see a concrete group, you start wondering what kind of geometric structure gets preserved by that group.

So: what’s the geometric meaning of the concrete group $H \rtimes G$?  This really involves thinking of $H$ in two different ways: as a group and as a right torsor of itself.  The action of $G$ preserves the group structure by assumption: it acts by group automorphisms.  On the other hand, the action of $H$ by left multiplication is by automorphisms of $H$ as a right $H$ space.  Thus,  $H \rtimes G$ preserves a kind of geometry on $H$ that combines the group and torsor structures.  We can think of these as a generalization of the “rotations” and “translations” in the Poincaré group.

But I promised to talk about the Heisenberg double of a Hopf algebra.

In fact, there’s nothing so special about groups in the above construction.  Suppose $H$ is a Hopf algebra, or even just an algebra, and there’s some other Hopf algebra $G$ that acts on $H$ as algebra automorphisms.  In Hopf algebraists’ lingo, we say $H$ is a “$G$ module algebra”.  In categorists’ lingo, we say $H$ is an algebra in the category of $G$ modules.

Besides the Hopf algebra action, $H$ also acts on itself by left multiplication.  This doesn’t preserve the algebra structure, but it does preserve the coalgebra structure: $H$ is an $H$ module coalgebra.

So, just like in the group case, we can form the semidirect product, sometimes also called a “smash product” in the Hopf algebra setting, $H \rtimes G$, and again the multiplication law in this is completely determined by its action on $H$. We think of this as a “concrete quantum group” acting as two different kinds of “quantum symmetries” on $H$—a “point-fixing” one preserving the algebra structure and a “translational” one preserving the coalgebra structure.

The Heisenberg double is a particularly beautiful example of this.   Any Hopf algebra $H$ is an $H^*$ module algebra, where $H^*$ is the Hopf algebra dual to $H$.  The action of $H^*$ on $H$ is the “left coregular action” $\rightharpoonup$ defined as the dual of right multiplication:

$(h\rightharpoonup \alpha)(k) = \alpha(kh)$

for all $h,k\in H$ and all $\alpha \in H^*$.

One could use different conventions for defining the Heisenberg double, of course, but not as many as you might think.  Here’s an amazing fact:

$H \rtimes H^* = H \ltimes H^*$

So, while I often see $\rtimes$ and $\ltimes$ confused, this is the one case where you don’t need to remember the difference.

But wait a minute—what’s that “equals” sign up there.   I can hear my category theorist friends snickering.  Surely, they say, I must mean $H \rtimes H^*$ is isomorphic to $H \rtimes H^*$.

But no.  I mean equals.

I defined $H \rtimes H^*$ as the algebra structure on $H \otimes H^*$ determined by its action on $H$, its “defining representation.”   But every natural construction with Hopf algebras has a dual.  I could have instead defined an algebra $H \ltimes H^*$ as the algebra structure on $H \otimes H^*$ determined by its action on $H^*$.  Namely, $H^*$ acts on itself by right multiplication, and $H$ acts on $H^*$ by the right coregular action.  These are just the duals of the two left actions used to define $H \rtimes H^*$.

That’s really all I wanted to say here.  But just in case you want the actual formulas for the Heisenberg double and its defining representations, here they are in Sweedler notation:

### Hopf algebroids and (quantum) groupoids (Part 2)

8 September 2014

Last time I defined weak Hopf algebras, and claimed that they have groupoid-like structure. Today I’ll make that claim more precise by defining the groupoid algebra of a finite groupoid and showing that it has a natural weak Hopf algebra structure. In fact, we’ll get a functor from finite groupoids to weak Hopf algebras.

First, recall how the group algebra works. If G is a group, its group algebra is simply the vector space spanned by elements of G, and with multiplication extended linearly from G to this vector space. It is an associative algebra and has a unit, the identity element of the group.

If G is a groupoid, we can similarly form the groupoid algebra. This may seem strange at first: you might expect to get only an algebroid of some sort. In particular, whereas for the group algebra we get the multiplication by linearly extending the group multiplication, a groupoid has only a partially defined “multiplication”—if the source of g differs from the target of h, then the composite gh is undefined.

However, upon “linearizing”, instead of saying gh is undefined, we can simply say it’s zero unless s(g)=t(h). This is essentially all there is to the groupoid algebra.  The groupoid algebra $\mathbb{C}[G]$ of a groupoid $G$ is the vector space with basis the morphisms of $G$, with multiplication given on this basis by composition whenever this is defined, zero when undefined, and extended linearly from there.

It’s easy to see that this gives an associative algebra: the multiplication is linear since we define it on a basis and extend linearly, and it’s associative since the group multiplication is. It is a unital algebra if and only if the groupoid has finitely many objects, and in this case the unit is the sum of all of the identity morphisms.

Mainly to avoid saying “groupoids with finitely many morphisms”, I’ll just stick to finite groupoids from now on, where the sets of objects and morphisms are both finite.

If we have a groupoid homomorphism, then we get an algebra homomorphism between the corresponding groupoid algebras, also by linear extension. So we get a functor

$\mathbb{C}[\cdot]\colon\mathbf{FinGpd} \to \mathbf{Alg}$

from the category of finite groupoids to the category of unital algebras.

But in fact, this extends canonically to a functor

$\mathbb{C}[\cdot]\colon\mathbf{FinGpd} \to \mathbf{WHopf}$

from the category of finite groupoids to the category of weak Hopf algebras.

To see how this works, notice first that there’s a canonical functor

$\mathbb{C}[\cdot]\colon\mathbf{Set} \to \mathbf{Coalg}$

from the category of sets to the category of coalgebras:  Every set is a comonoid in a unique way, so we just linearly extend that comonoid structure to a coalgebra.

In case that’s not clear to you, here’s what I mean in detail.  Given a set $X$, there is a unique map $\Delta\colon X \to X \times X$ that is coassociative, namely the diagonal map $\Delta(x) = (x,x)$. This is easy to prove, so do it if you never have.  Also, there is a unique map to the one-element set $\epsilon\colon X \to \{0\}$, up the choice of which one-element set to use.  Linearly extending $\Delta$ and $\epsilon$, they become a coalgebra structure on the vector space with basis $X$. Moreover, any function between sets is a homomorphism of comonoids, and its linear extension to the free vector spaces on these sets is thus a homomorphism of coalgebras.  This gives us our functor from sets to coalgebras.

So, given a finite groupoid, the vector space spanned by its morphisms becomes both an algebra and a coalgebra.  An obvious question is: do the algebra and coalgebra structure obey some sort of compatibility relations?  The answer, as I already gave away at the beginning, is that they form a weak Hopf algebra.  The antipode is just the linear extension of the inversion map $g \mapsto g^{-1}$.

(More generally, for those who care, the category algebra $\mathbb{C}[C]$ of a finite category $C$ (or any category with finitely many objects) is a weak bialgebra, and we actually get a functor

$\mathbb{C}[\cdot] \colon \mathbf{FinCat} \to \mathbf{WBialg}$

from finite categories to weak bialgebras.  If $C$ happens to be a groupoid, $\mathbb{C}[C]$ is a weak Hopf algebra; if it happens to be a monoid, $\mathbb{C}[C]$ is a bialgebra; and if it happens to be a group, $\mathbb{C}[C]$ is a Hopf algebra. )

This is nice, but have we squashed out all of the lovely “oid”-iness from our groupoid when we form the groupoid algebra? In other words, having built a weak Hopf algebra on the vector space spanned by morphisms, is there any remnant of the original distinction between objects and morphisms?

As I indicated last time, the key is in these two “loop” diagrams:

The left loop says to comultiply the identity, multiply the first part of this with an element $g$ and apply the counit. Let’s do this for a groupoid algebra, where $1 = \sum_x 1_x$, where the sum runs over all objects $x$.  Since comultiplication duplicates basis elements, we get

$\Delta(1) = \sum_x 1_x \otimes 1_x$

We then get:

$g\mapsto \sum_x \epsilon(1_x\cdot g) \otimes 1_x = 1_{t(g)}$

using the definition of multiplication and the counit in the groupoid algebra.  Similarly, the loop going in the other direction gives $g \mapsto 1_{s(g)}$, as anticipated last time.

So, we can see that the image of either of the two “loop” diagrams is the subspace spanned by the identity morphisms.  This is a commutative subalgebra of the groupoid algebra, and these maps are both idempotent algebra homomorphisms.  So, they give “projections” onto the “algebra of objects”.

In fact, something like this happens in the case of a more general weak Hopf algebra.  The maps described by the “loop” diagrams, are again idempotent homomorphisms and we can think of them as analogs of the source and target maps.  But there are some differences, too.  For instance, their images need not be the same in general, though they are isomorphic.  The images also don’t need to be commutative.  This starts hinting at what Hopf algebroids are like.

But I’ll get into that later.

### Hopf algebroids and (quantum) groupoids (Part 1)

1 September 2014

I’ve been thinking a lot about weak Hopf algebras and Hopf algebroids, especially in relation to work I’m doing with Catherine Meusburger on applications of them to gauge theory.  I don’t want to talk here yet about what we’re working on, but I do feel like explaining some basic ideas.  This is all known material, but it’s fun stuff and deserves to be better known.

First of all, as you might guess, the “-oid” in “Hopf algebroid” is supposed to be like the “-oid” in “groupoid”.  Groupoids are a modern way to study symmetry, and they do things that ordinary groups don’t do very well.  If you’re not already convinced groupoids are cool—or if you don’t know what they are—then one good place to start is with Alan Weinstein’s explanation of them here:

There are two equivalent definitions of groupoid, an algebraic definition and a categorical definition.  I’ll use mainly categorical language.  So for me, a groupoid is a small category in which all morphisms are invertible.  A group is then just a groupoid with exactly one object.

Once you’ve given up the attachment to the special case of groups and learned to love groupoids, it seems obvious you should also give up the attachment to Hopf algebras and learn to love Hopf algebroids.  That’s one thing I’ve been doing lately.

My main goal in these posts will be to explain what Hopf algebroids are, and how they’re analogous to groupoids.  I’ll build up to this slowly, though, without even giving the definition of Hopf algebroid at first.  Of course, if you’re eager to see it, you can always cheat and look up the definition here:

but I’ll warn you that the relationship to groupoids might not be obvious at first.  At least, it wasn’t to me.  In fact, going from Hopf algebras to Hopf algebroids took considerable work, and some time to settle on the correct definition. But the definition of Hopf algebroid given here in Böhm’s paper seems to be the one left standing after the dust settled.  This review article also includes a brief summary of the development of the subject.

To work up to Hopf algebroids, I’ll start with something simpler: weak Hopf algebras. These are a rather mild generalization of Hopf algebras, and the definition doesn’t look immediately “oid”-ish. But in fact, they are a nice compromise between between Hopf algebras and Hopf algebroids.  In particular, as we’ll see, just as a group has a Hopf algebra structure on its groupoid algebra, a groupoid has a weak Hopf algebra structure on its groupoid algebra.

Better yet, any weak Hopf algebra can be turned into a Hopf algebroid, and Hopf algebroids built in this way are rich enough to see many of features of general Hopf algebroids. So, I think this gives a pretty good way to understand Hopf algebroids, which might otherwise seem obscure at first. The strategy will be to start with weak Hopf algebras and consider what “groupoid-like” structure is already present. In fact, to emphasize how well they parallel ordinary groupoids, weak Hopf algebras are sometimes called quantum groupoids:

So, here we go…

What is a Weak Hopf algebra?  This is quick to define using string diagrams.  First, let’s define a weak bialgebra.  Just like a bialgebra, a weak bialgebra is both an associative algebra with unit:

and a coassociative coalgebra with counit:

(If the meaning of these diagrams isn’t clear, you can learn about string diagrams in several places on the web, like here or here.)

Compatibility of multiplication and comultiplication is also just like in an ordinary bialgebra or Hopf algebra:

So the only place where the axioms of a weak bialgebra are “weak” is in the compatibility between unit and comultiplication and between counit and multiplication.  If we define these combinations:

then the remaining axioms of a weak bialgebra can be drawn like this:

The two middle pictures in these equations have not quite been defined yet, but I hope it’s clear what they mean. For example, the diagram in the middle on the top row means either of these:

since these are the same by associativity.

Just as a Hopf algebra is a bialgebra with an antipode, a weak Hopf algebra is a weak bialgebra with an antipode.  The antipode is a linear involution $S$ which I’ll draw like this:

and it satisfies these axioms:

Like in a Hopf algebra, having an antipode isn’t additional structure on a weak Hopf algebra, but just a property: a weak bialgebra either has an antipode or it doesn’t, and if it does, the antipode is unique.  The antipode also has most of the properties you would expect from Hopf algebra theory.

One thing to notice is that the equations defining a weak Hopf algebra are completely self-dual.  This is easy to see from the diagrammatic definition given here, where duality corresponds to a rotation of 180 degrees: rotate all of the defining diagrams and you get the same diagrams back.  Luckily, even the letter $S$ is self-dual.

There’s plenty to say about about weak Hopf algebras themselves, but here I want to concentrate on how they are related to groupoids, and ultimately how they give examples of Hopf algebroids.

To see the “groupoidiness” of weak Hopf algebras, it helps to start at the bottom: the antipode axioms.  In particular, look at this one:

The left side instructs us to duplicate an element, apply the antipode to the copy on the right, and then multiply the two copies together.  If we do this to an element of a group, where the antipode is the inversion map, we get the identity.  If we do it to a morphism in a groupoid, we get the identity on the target of that morphism. So, in the groupoid world, the left side of this equation is the same as applying the target map, and then turning this back into a morphism by using the map that sends any object to its identity morphism.  That is:

$g \mapsto 1_{t(g)}$

where $t$ is the map sending each morphism to its target, and $1_x$ denotes the identity morphism on the object $x$.

Likewise, consider the dual of the previous axiom:

In the groupoid world, the left hand side gives the map

$g \mapsto 1_{s(g)}$

where $s$ denotes the map sending each morphism to its source.

So… if weak Hopf algebras really are like groupoids, then these two loop diagrams:

must essentially be graphical representations of the target and source maps.

Of course, I only said if Hopf algebras are like groupoids, and I haven’t yet explained any precise sense in which they are.   But we’re getting there.  Next time, I’ll explain more, including how groupoid algebras give weak Hopf algebras.

Meanwhile, if you want some fun with string diagrams, think of other things that are true for groupoids, and see if you can show weak Hopf algebra analogs of them using just diagrams.  For example, you can check that the diagrammatic analog of $1_{s(1_{s(g)})}=1_{s(g)}$ (“the source of the source is the source”) follows from the weak Hopf algebra axioms.  Some others hold make a trivial rephrasing: while the obvious diagrammatic translation of $1_{t(S(g))} = 1_{s(g)}$ does not hold,  if you draw it instead starting from the equation $1_{t(S(g))} = S(1_{s(g)})$, then you get an equation that holds in any weak Hopf algebra.

### Blog post on Observer Space by Jeffrey Morton

17 October 2012

DW: This is a very nice blog post by Jeffrey Morton about observer space! He wrote this based on my ILQGS talk and my papers with Steffen Gielen. (In fact, Jeff has written a lot of other nice summaries of papers and talks, as well as stuff about his own research, on his blog, Theoretical Atlas — check it out!)

This entry is a by-special-request blog, which Derek Wise invited me to write for the blog associated with the International Loop Quantum Gravity Seminar, and it will appear over there as well.  The ILQGS is a long-running regular seminar which runs as a teleconference, with people joining in from various countries, on various topics which are more or less closely related to Loop Quantum Gravity and the interests of people who work on it.  The custom is that when someone gives a talk, someone else writes up a description of the talk for the ILQGS blog, and Derek invited me to write up a description of his talk.  The audio file of the talk itself is available in .aiff and .wav formats, and the slides are here.

The talk that Derek gave was based on a project of his and Steffen Gielen’s, which has taken written form in a…

View original post 2,961 more words

### Göttingen talk on Cartan geometrodynamics

28 February 2012

I’m in Göttingen now, at a meeting of the German Physical Society (DPG). Here are the slides to the talk I just gave this evening:

D. Wise, Cartan geometrodynamics, Göttingen, Feb. 2012.

My goal in this talk was simply to present the geometric ideas behind my recent work with Steffen Gielen, without going into the details of the resulting reformulation of general relativity.

Of course, if this piques your interest to that point that you want to learn those details, I recommend our paper. :-)

### The geometric role of symmetry breaking in gravity

30 January 2012

I suffered a blogging derailment at the end of 2011. Now I’m eager to get back on track, and I’ve got a bunch of stuff I want to write about.

Right at the end of the year, I wrote a short conference proceedings article:

The point of the paper is that the mathematics physicists are most familiar with because of "spontaneous symmetry breaking" plays a somewhat different role in gravitational physics, as a key ingredient of Cartan geometry.

Anyway, this paper is at a level of detail that could work just as well as a blog article, and it fits with what I’ve been talking about on this blog, so I’m putting a version here.

The success of spontaneous symmetry breaking in condensed matter and particle physics is famous. It explains second order phase transitions, superconductivity, the origin of mass via the Higgs mechanism, why there must be at least three generations of quarks, and so on. These applications are by now standard material for modern textbooks.

Much less famous is this: broken symmetry links the geometry of gauge fields to the geometry of spacetime. This, in my view, is the main role of symmetry breaking in gravity.

An early clue came in 1977, when MacDowell and Mansouri wrote down an action for general relativity using a connection for the (anti-) de Sitter group, but invariant only under the Lorentz group. Though their work was surely inspired by spontaneous symmetry breaking, it was Stelle and West who first made their action fully gauge invariant, breaking the symmetry dynamically using a field $y$ locally valued in (anti-) de Sitter space.

Whether one breaks the symmetry dynamically or by hand,’ the broken symmetry of the MacDowell–Mansouri connection plays the geometric role of relating spacetime geometry to the geometry of de Sitter space. This is best understood using Cartan geometry, a generalization of Riemannian geometry originating in the work of Élie Cartan, in which the geometry of tangent spaces is generalized—in this case, they become copies of de Sitter space. But to explain how this works, and how symmetry breaking is involved, it helps to back up further.

In geometry, inklings of spontaneous symmetry breaking date from at least 1872, in the work of Felix Klein. Ironically, to study a homogeneous space $Y$, with symmetry group $G$, one first breaks its perfect symmetry, artificially giving special significance to some point $y\in Y$. This gives an isomorphism $Y \cong G/G_y$ as $G$-spaces, where $G_y$ is the stabilizer of $y$, allowing algebraic study of the geometry. While $Y$ itself has $G$ symmetry, this description of it is only invariant under the subgroup $G_y$. Different algebraic descriptions of $Y$ are however related in a $G$-equivariant way, since $G_{gy} = gG_y g^{-1}$.

This is strikingly similar to spontaneous symmetry breaking in physics. There, one really has a family of minimum-energy states, related in a $G$-equivariant way under the original gauge group $G$. Singling out any particular state $|0\rangle$ as the’ vacuum breaks symmetry to $G_{|0\rangle}$.

Cartan took Klein’s ideas a dramatic step further, getting an algebraic description of the geometry of a nonhomogeneous manifold $M$, by relating it infinitesimally’ to one of Klein’s geometries $Y$. Just as Klein geometry uses broken symmetry to get an isomorphism $Y\to G/G_y$, in Cartan geometry, the broken symmetry in a $G$ connection induces an isomorphism $e\colon T_xM \to \mathfrak{g}/\mathfrak{g}_y$ for each tangent space. This is just the coframe field, also called the soldering form since identifying $T_xM$ with $\mathfrak{g}/\mathfrak{g}_y\cong T_yY$ effectively solders a copy of $Y$ to $M$, at each point $x$. These copies of $Y$ are then related via holonomy of the Cartan connection, which can be viewed as describing rolling $Y$ along $M$ without slipping’ (see my paper on MM gravity, and also Appendix B of Sharpe’s book).

Physics history unfortunately skips over Cartan geometry. The Levi-Civita connection is adequate for the standard metric formulation of general relativity, and more general kinds of connections played no vital role in physics until some time later. When these eventually were introduced in Yang–Mill theory, they served a purpose far removed from spacetime geometry. Yang–Mills gauge fields are really just the principal connections of Ehresmann, who, building on Cartan’s ideas, liberated connections from their bondage to classical geometry. Ehresmann’s definition, which lacks the crucial broken symmetry’ in Cartan’s original version, has just the flexibility needed for gauge fields in particle physics, which are concerned only with the geometry of an abstract internal space’—a bundle over spacetime, rather than spacetime itself. On the other hand, Cartan’s original version is better when it comes to studying gravity.

Concretely, a Cartan geometry may be thought of as a connection on a principal bundle (with Ehresmann’s now standard definition) together with a section $y$ of the associated $Y$ bundle. As an example, let us write a version of the MacDowell–Mansouri action, using de Sitter space $Y\cong G/H = \mathrm{SO}(4,1)/\mathrm{SO}(3,1)$ as the corresponding Klein geometry:

$I[A,y] = \int \mathrm{tr}(F_y \wedge {\star}_y F_y)$

The Cartan connection $(A,y)$ consists of an $\mathrm{SO}(4,1)$ connection $A$ and a locally de Sitter-valued field $y$. $F$ is the curvature of $A$, calculated by the usual formula, and $F_y$ is its $\mathfrak{g}_y$-valued part, where $\mathfrak{g}_y \cong \mathfrak{so}(3,1) \cong \Lambda^2\mathbb{R}^{3,1}$ has Hodge star operator $\star_y$.

I have described additional examples of Cartan-geometric formulations of various gravity theories elsewhere, and there are many more. But besides the diversity of specific examples, there are deep reasons that gravity, or any related "gauge theory of geometry," should be framed in the language of Cartan geometry. This is the subject of geometric "equivalence theorems."

In fact, if one believes semi-Riemannian metrics are fundamental in classical gravity, one is forced to accept Cartan connections as equally fundamental. The reason for this is Cartan’s method of equivalence, a process for proving that specified kinds of "raw geometric data" are equivalent to corresponding types of Cartan geometry. In the case of Riemannian geometry, solving the "equivalence problem" leads to the following theorem:

Theorem: A Riemannian metric determines a unique torsion-free Cartan geometry modeled on Euclidean space; conversely, a torsion-free Cartan geometry modeled on Euclidean space determines a Riemannian metric up to overall scale (on each connected component).

Proof: See Sharpe’s book.

Physically, the "overall scale" in the converse just represents a choice of length unit. One can also show that deformed versions (or "mutations") of Euclidean geometry, namely hyperbolic and spherical geometry, lead to Cartan geometries that carry the same information. The Lorentzian analogs of these results are the real reason de Sitter and anti de Sitter geometries work in MacDowell–Mansouri gravity.

Riemannian geometry is but one application of the equivalence method. There are analogous theorems, for example, in conformal geometry or Weyl geometry, relating various types of conformal structures to Cartan geometries that take the model $Y$ to be an appropriate kind of homogeneous conformal model. Sharpe’s book contains some such theorems, and some significant work has been done on applications of conformal Cartan geometry—which often goes by the name "tractor calculus"—in physics. (See, e.g. this paper and references therein.)

For now, I just want to describe one more application of Cartan geometric thinking in gravitational theory. Besides spacetime geometry, one can also use Cartan’s ideas to describe the geometry of space.

Wheeler’s term "geometrodynamics" originally referred to the of evolution of spatial geometries in the metric sense. This has sometimes been contrasted with "connection dynamics" (see e.g. here or here). In light of the above equivalence theorem, however, there seems little point in establishing any technical distinction between geometrodynamics and connection dynamics, at least if we mean connections in the Cartan-geometric sense. The metric and connection pictures have their own advantages, but the equivalence theorem suggests we should be able to translate exactly between the two.

In recent work with Steffen Gielen, we take an explicitly Cartan-geometric approach to evolving spatial geometries. In this case, the symmetry breaking field $y$ lives in 3d hyperbolic space $\mathrm{SO}(3,1)/\mathrm{SO}(3)$, and can be interpretated as a field of observers, since the spacetime coframe field converts it into a unit timelike vector field. This can be dualized via the metric to a unit covector field, which we might call a field of co-observers. Just as observers determine a local time direction, co-observers determine local space directions, by taking their kernel. Our strategy in the Hamiltonian formulation is to fix a field of co-observers—the infinitesimal analog of picking a spacetime folitation—but let the field of observers be determined dynamically, as part of determining the metric.

The result is a model in which the observer field plays a two part symmetry breaking role: first splitting spacetime fields into spatial and temporal parts, but then also acting as the symmetry breaking field in Cartan geometry of space. This gives a Cartan-geometric Hamiltonian framework in which the spatial fields fit neatly and transparently into their spacetime counterparts and transform in an equivariant way under local Lorentz symmetry.

Thanks to the equivalence theorem, this may be viewed as a concrete link between connection dynamics and geometrodynamics in the original sense.

It is conceivable that gravity descends from a more fundamental theory with larger gauge group, and so fits into the tradition of symmetry breaking in gauge theories. Such ideas are clearly worth pursuing (see, e.g. papers by Percacci or Randono). At the same time, we should not ignore the lesson of Cartan geometry: broken symmetry is the means to establishing exact correspondence between geometric structures living on tangent spaces on one hand and connections on the other.

Thanks to John Baez, Julian Barbour, James Dolan, Andy Randono and Steffen Gielen for helpful discussions.

(By the way, someone might still remember that in the last post on this blog, I promised to explain some things. I still plan to do that later… )

### Explaining Cartan geometry

26 September 2011

I recently got back from a week’s visit with Julian Barbour, which we spent talking about geometric foundations of Machian physics in general, and shape dynamics in particular.

Much of what Julian and I are discussing isn’t to the level of blog-worthy material yet, but one of the key ideas is Cartan geometry, particularly in its various “conformal” incarnations. So, one thing I did during the visit was to explain a bit of Cartan geometry to him. Explaining Cartan geometry is always fun for me: it’s an incredibly beautiful subject, can be understood on an intuitive level, and yet still seems to be rather underappreciated. We thought it would be fun to record part of our discussion here for others to read.

Essentially, Cartan geometry is a way of studying geometry by rolling one object around on another: the geometry of the one serves as a “prototype” for the geometry of the other, and the nontriviality of the rolling—i.e. the failure to come back to the same configuration after rolling around a loop—measures the geometric deviation from that of the prototype. I’ll explain this in more detail in a moment.

Physically speaking, Cartan geometry is all about gauge theory of geometry. Gravity is a kind of gauge theory, but unlike the gauge theories of particle physics, it is a gauge theory that determines the geometry of the space it lives on. This is precisely what Cartan geometry is good for. I’ve explained this elsewhere, where I’ve used the idea of “rolling without slipping” to study spacetime geometry by rolling a copy of, say, de Sitter space along it.

Julian isn’t immediately interested in spacetime geometry. As I mentioned before, he’s interested in physics where “time” plays no primary role: there is only space, or more precisely, only configuration space. But, in broad terms, he is currently studying a kind of “geometric gauge theory.” Cartan geometry should still be the most natural language for it.

So, we recently spent a day struggling to form some sort of synthesis of my work and his, using Cartan geometry and rolling without slipping to understand physics in a world without time. Here is what we came up with:

Oh, OK, so we got a bit further than that…

### Cartan versus Levi-Civita

We really did use the globe as a prop for discussing Cartan geometry. Unfortunately, Julian’s book is too slick and too bulky to effectively simulate rolling a plane on a sphere without slipping, but this drink coaster with cork backing worked beautifully:

Here, Julian is experimentally verifying my claims about Cartan geometry. By rolling the coaster around a cleverly chosen loop on the globe, carefully avoiding any slipping or twisting, he can get back to a configuration with the same point of tangency on the globe, but where the coaster has been both rotated and translated.

This “rolling without slipping” of the coaster along a path on the globe by is one of the simplest examples of a Cartan connection. It’s a rule for moving a homogeneous space like a plane, represented here by the coaster, around on a not necessarily homogeneous space, represented here by the globe. (While Julian’s globe looks pretty spherical at the scale of the picture, it’s actually a relief globe, which made it convenient to remember which space was supposed to be the homogeneous one.) The deviation from the homogeneous geometry is measured by the failure to come back to the same configuration after going around some loop using this rule.

Notice that this kind of “parallel transport” that is qualitatively quite distinct from that done by the more familiar Levi-Civita connection. In particular, while the Levi-Civita connection transports tangent vectors in a linear way, “rolling” the tangent plane gave us translations, which are not linear transformations.

To perform the Levi-Civita parallel transport using our drink coaster model, place one finger in the middle of the coaster, right at the point of tangency with the sphere:

Then move the coaster around with just that finger. To do this, you of course have to slide the coaster—a forbidden maneuver in the Cartan version—but you should still be careful not to twist the coaster relative to the globe.

Actually, you can think of the Levi-Civita transport via rolling without slipping, if you keep making corrections as you go. Suppose we fix the origin on our drink coaster so that we can think of it as a vector space. Then draw a vector on it. To transport the vector along a path, first break the path up into small steps. After rolling along the first bit, the coaster’s origin will no longer be at the point of tangency with the globe. This is unacceptable, since we’re supposed to be carrying our vector along by a linear transformation! So, make a correction: fortunately, there is a canonical way to slide the coaster without rotating it, maintaining the point of contact on the globe, so that the origin goes back where it should be.

Now make these little corrections after each little step along the total path. If your steps were sufficiently small, once you finally arrive at your destination, you’ll have a very good approximation to the Levi-Civita transport. In the limit of infinitely many infinitesimal steps, you get the Levi-Civita transport exactly.

So, the Cartan connection knows about the Levi-Civita connection, but it also knows more: in fact, the additional information we’ve suppressed in forcing the origin to remain in contact with the globe is enough to reconstruct the metric on the sphere, up to a constant global scale, or, if you prefer, up to a global choice of unit of length. For details, see Proposition 3.2 in Sharpe’s book on Cartan geometry for details. The equivalence of a Riemannian metric (up to global scale) with this type of Cartan geometry is one of the most basic applications of Cartan’s method of equivalence.

In any case, rolling a plane around on a sphere is just one kind of Cartan geometry—there’s really a different flavor of Cartan geometry for each kind of Klein geometry. For example, we could also talk about spherical Cartan geometry by rolling a ball on Julian’s globe:

Even though the globe and the ball are both spheres here, the “rolling distribution” is nontrivial (and would be even if the globe were perfectly spherical), because they are spheres of different diameter. Rolling around a loop on the globe, we can get any transformation of the ball we wish.

If, on the other hand, the globe were a perfect sphere and we had a ball that was an exact mirror image of it, parallel transport by rolling would would be completely trivial: starting out in a configuration where, say, Nairobi, Kenya touches its mirror image, and going around any loop from Nairobi back to itself, no matter how convoluted, the two Nairobis always come back in contact in the end.

The easy intuitive proof of this fact is to imagine rolling the globe on an actual mirror.

### Hamster geometry

By now you may be wondering if I’ll get beyond the picture of rolling homogeneous spaces and tell you more precisely what a Cartan geometry is. I will—at least up to a few details that you can look up.

I’ll assume you already know some Klein geometry, or that you at least have vague impressions of it and can fake the rest. Briefly, a (smooth) Klein geometry is a manifold $Y$ equipped with a Lie group $G$ of symmetries acting transitively: there is at least one $g\in G$ taking me from any point in $Y$ to any other point. Picking any point $y \in Y$, we can identify $Y$ with the coset space $G/H$. So, abusing terminology a bit, we often refer to “A Klein geometry $G/H$,” forgetting the name of the original homogeneous space $Y$.

Now, if $M$ is a manifold of the same dimension as the Klein geometry $G/H$, then a Cartan geometry on $M$, “modeled on $G/H$,” has two basic ingredients:

1. a principal $H$ bundle $P \to M$,
2. a $\mathfrak{g}$-valued 1-form on $P$ (the Cartan connection)

satisfying some properties that I won’t bother writing down here. Instead, I just want to describe the geometric meaning of these ingredients. I’ve explained this in my papers using what I call “hamster geometry,” and it is perhaps worth reiterating that explanation here.

In the example of a ball rolling on a surface, the ball has symmetry group $G= \mathrm{SO}(3)$ and point stabilizer $H= \mathrm{SO}(2)$. So, Cartan geometry for this model involves an $\mathfrak{so}(3)$-valued 1-form on a principal ${\rm SO}(2)$ bundle over a 2d manifold, namely a surface. To understand the geometric meaning of these things, think of the ball as being controlled by a hamster inside of it. Here is a hamster in a hamster ball on a clearly non-homogeneous torus:

Forgetting about the ball itself for the moment, a hamster can be placed at any point on the surface, facing in any of an $\mathrm{SO}(2)$‘s worth of directions. So, the configuration space of a hamster on a surface is a principal $\mathrm{SO}(2)$ bundle over the surface. That is the geometric meaning of the bundle.

Now what about the “Cartan connection”? In this case, it should be an $\mathfrak{so}(3)$-valued 1-form on our hamster configuration space.

The key to understanding this one form is to realize that, so long as there is no slipping of the ball on the surface, the motion of the ball is completely determined by the motion of the hamster. The $\mathfrak{so}(3)$-valued 1-form just describes the rotation of the ball as the hamster moves: it takes tangent vectors to hamster configuration space—”infinitesimal changes” in hamster configuration—and gives elements of $\mathfrak{so}(3)$—”infinitesimal rotations” of the model sphere.

Most importantly, you can “integrate” these infinitesimals (using the path-ordered exponential) to get actual rotations of the sphere from actual paths through hamster configuration space. It’s all just a precise setup for describing how the hamster drives the ball around.

For more general Cartan geometries, I often imagine a “generalized hamster” running around on my base manifold, pushing a copy of $G/H$ around as he goes. You may have a hard time visualizing a generalized hamster, but with a little practice, you can do it! I’ll explain how in an upcoming post.

When I do that, I also want to discuss some more particular examples: mainly various versions of conformal Cartan geometry, which I’ve been thinking about more lately, partly because of the discussions with Julian. In fact, this post was supposed to be about conformal Cartan geometry until I got carried away. It will have to wait for another time.