## 4!-torsor a la George Hart

20 April 2015

As a project with a certain 4-year-old relative of mine, we constructed the proof I described before that the outer vertices of George Hart’s 12-Card Star form a 4!-torsor.  (I guess I didn’t say it that way before, but it’s true!)  Here’s our proof:

Last time I suggested using a deck of 12 cards like this:

But instead, we used four solid colors, three cards of each.  So, our “star” permutes the colors red, white, black, and silver:

You can get any permutation of these colors in our Star by exactly one symmetry taking outer vertices to outer vertices.  The “exactly one” in this isomorphism is what makes the set of outer vertices a 4!-torsor rather than just a 4!-set.

Here’s what it looks like when you put three pieces together, from both sides:

## 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

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

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!)

Originally posted on Theoretical Atlas:

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 2,961 more words

## Observer Space: new paper and ILQGS talk

3 October 2012

Steffen Gielen and I just put our new paper on “observer space” on the arxiv:

S. Gielen and D. Wise, Lifting General Relativity to Observer Space

Then, today I gave the International Loop Quantum Gravity Seminar on the same topic. This a seminar between various institutions, mainly in North America and Europe, where people work on loop quantum gravity and related topics. It’s run the old-fashioned way, as a conference call.

I was a bit uneasy about volunteering for such a talk. I don’t like phones. I’m happy to speak in front of any audience I can see — but an audience I can’t see is a little intimidating, even if I do probably know most of them. Besides, on the phone, you never know whether someone might be recording your conversation, hoping to use it against you later. And in this case, they were! Here’s the audio to my talk in aiff or wav format. If you decide to listen to that, you might also want to look at the slides to my talk.

Seriously, I think the talk turned out rather well — except for the part where my Skype connection to the phone bridge cut out, and I didn’t even know it. Fortunately, though, as I only found out after the talk, Steffen took over, explaining to the audience the same stuff that I was simultaneously, unwittingly, explaining into a black hole. Steffen had never seen the slides, and described this as his first experience with live Powerpoint karaoke. I think he did an excellent job of filling in.

I’ll have to explain a bit more about “observer space” on this blog sometime later…

## Teleparallel gravity and Poincaré symmetry

19 April 2012

Lately John Baez and I have been thinking a bit about teleparallel gravity, from a somewhat esoteric point of view based on 2-groups.  We’re just about to finish up a paper on that.

Right now, though, I just have a few thoughts about one of the more usual ways of thinking about teleparallel gravity.

If you asked me what what teleparallel gravity is about, the first thing I’d tell you is that it is a rewriting of general relativity so that torsion takes the lead role, rather than curvature. But, not everyone motivates it in that way. One often hears, in particular, this statement:

Teleparallel gravity is a gauge theory for the translation group.

What does this mean? The isometries of Minkowski spacetime form the Poincaré group, and the “translation group” means the subgroup consisting of just translations by vectors. Let’s call this group $\mathbb{R}^{3,1}$, just to emphasize that we’re working in three “space” dimensions and one “time” dimension, but it’s really just the abelian group underlying the vector space $\mathbb{R}^4$.

From a certain point of view, it’s understandably tempting to try describing gravity as a gauge theory for the group of translations of Minkowski spacetime. After all, the tangent bundle is the bundle with the lead role in general relativity, but a principal $\mathbb{R}^{3,1}$ bundle on (3+1)-dimensional spacetime can start to look a lot like the tangent bundle, at least once you pick a section, so that all of those affine Minkowski fibers become vector spaces.

If you believe that Cartan geometry underlies any “geometric gauge theory” of gravity, as I do, then this suggests you are modeling gravity using the homogeneous space $G/H$ with $G=\mathbb{R}^{3,1}$ and $H = 0$, the trivial subgroup. This works OK, but it’s a bit strange geometrically: by ignoring the Lorentz transformations we’re treating Minkowski spacetime as being completely anisotropic. Reducing the symmetry from the Poincaré group to just the translation group is like adding some sort of structure that lets us distinguish absolute directions in space.

But Minkowski space itself doesn’t have preferred directions. The key property of Minkowski space that we want to mimic is its “distant parallelism”—the ability to compare vectors at distant points and decide whether they are parallel—which is something that’s preserved not only under translations but also under Lorentz transformations. So, it seems weird to throw out the Lorentz symmetry from the outset! What’s going on here, geometrically?

What I want to discuss now is this: Even though you can start off thinking of teleparallel gravity as a gauge theory for the translation group, if we think about the geometry a bit, and listen to the lessons of gauge theory, Lorentz symmetry is easily restored.

I guess now I should just come out and say what people actually do to think of teleparallel gravity as a gauge theory for the translation group. It’s pretty clever.

Say we start with a principal $\mathbb{R}^{3,1}$ bundle and pick a section $y$, which specifies a reduction to the trivial subgroup. If we’ve got a connection, say $A$, then we can compose it with the differential $dy$ of the section to get a map

$e= A\circ dy \colon TM \to \mathbb{R}^{3,1}$.

The connection $A$ has a curvature which we will denote by $T$. One can then write down the teleparallel gravity action, which begins like this:

$\displaystyle \int d^4x \det(e)\; T^a{}_{\mu\nu} T_a{}^{\mu\nu} + \cdots$

An attractive feature of this is that it looks roughly like Yang-Mills theory with gauge group $G=\mathbb{R}^{3,1}$, at least if you squint until those T’s start to look like F’s. I’ll say why we used “T” in a minute.

Of course, it’s not really Yang-Mills theory, and not just because the field strength is called $T$. In Yang-Mills, there’s a background metric, which could just as well be described by a coframe field $e$, and the volume form corresponding to this metric looks like $d^4x \det(e)$. But here, $e$ isn’t a background field, but a dynamical field—it is equal to the “connection” in the alleged Yang-Mills theory! Plus, there are more terms in the action, which I haven’t written, that can’t be written down in an ordinary Yang-Mills theory. These terms can only be written because of the peculiar double role of the connection as a coframe field. So, the resemblance to Yang-Mills is actually somewhat superficial. But, it’s still cute.

Anyway, on with the story.

While $e$ is really just the translation group connection, written in a particular gauge, it’s related to a certain connection on the tangent bundle called the “Weitzenböck connection”. For this, we note that $e$ can be viewed as a trivialization of $TM$, i.e. a vector bundle isomorphism

$TM \to M \times \mathbb{R}^{3,1}$

The Weitzenböck connection is just the pullback of the standard flat connection on the trivial bundle $M \times \mathbb{R}^{3,1}$. The reason we use $T$ for the curvature of $A$ is it is naturally identified with the torsion of the Weitzenböck connection.

The action for teleparallel gravity can then be written using just the following ingredients:

• the determinant of the coframe, $\det(e)$
• the metric: the pullback of the obvious metric on the trivial $\mathbb{R}^{3,1}$ bundle
• the torsion of the Weitzenböck connection

The first two of these things are invariant under local Lorentz group gauge transformations acting on $\mathbb{R}^{3,1}$. But what about the third? The torsion of the Weitzenböck connection (i.e. the curvature of the original translation group connection $A$) is invariant not under arbitrary Lorentz gauge transformations, but only covariantly constant gauge transformations.

In other words, as we’ve described it so far, teleparallel gravity has a “global Lorentz symmetry” that is not a “gauge symmetry”.

The lesson of gauge theory, though, is that we should generalize any global symmetry we find to a local gauge symmetry that can vary from point to point. How do we do this?

The trick is fairly obvious from my description of the coframe field as a vector bundle isomorphism. The reason the Weitzenböck torsion isn’t obviously invariant under Lorentz gauge transformations is that the connection is the pullback of a fixed connection on the trivial $\mathbb{R}^{3,1}$ bundle. Of course, saying it this way makes it sound a bit silly: if we’re transforming everything else by a gauge transformation, why are we not also transforming this connection on $M\times \mathbb{R}^{3,1}$? Once we do that, everything behaves much better under Lorentz gauge transformations.

In fact, there’s really no a priori reason to think of the coframe as setting up a trivialization. It’s more natural to think of a coframe as a vector bundle isomorphism

$TM \to \mathcal{T}$

where $\mathcal{T}$ is some vector bundle, which clearly must be isomorphic to $TM$, but not in any canonical way, and not necessarily trivial, in general. John and I like to call $\mathcal{T}$ a “fake tangent bundle”, a name I probably picked up from him, long ago.

If $\mathcal{T}$ is equipped with both a metric and a connection, these pull back to a metric and connection on $TM$. If the connection on $\mathcal{T}$ is flat, then so is its pullback, and this pullback is every bit as good for teleparallel gravity as the Weitzenböck connection, so we might as well call it the Weitzenböck connection—this is what we do in that paper we’re finishing up.

But, this version of the Weitzenböck connection is invariant under local Lorentz gauge transformations, since such gauge transformations act on both the coframe and the connection on the fake tangent bundle.

Lorentz gauge symmetry in teleparallel gravity is restored.

In fact, we then get teleparallel gravity, not as a gauge theory for the translation group, but rather as a gauge theory for a Cartan connection modeled on Minkowski space. That is, Cartan geometry based on the Poincaré group with the Lorentz group as stabilizer subgroup. Some of this is implicit in the new paper with John Baez (update: that paper is now done), though there the emphasis is rather on Cartan 2-geometry. I should perhaps write up the 1-geometry version more explicitly elsewhere.

## Workshop pictures from Bad Honnef

15 March 2012

Here are some of the folks that were at the workshop “Exploring Quantum Spacetime” in Bad Honnef last week (click for larger version):

I didn’t take this picture, but I did take some pictures of my own. Mostly, I took a lot of bad pictures, due to insufficient lighting and me not wanting to be any more obnoxious than necessary by using a flash. But, I’ll post a few here that turned out … well, OK, at least.

I spent a bunch of time at the workshop talking to Sean Gryb. Here he is talking to Steffen Gielen:

Steffen is one of my main collaborators right now, and I’m starting to talk seriously about doing some work with Sean, so it was nice to see both of them at the workshop.

As you might expect, there were a lot of other people having conversations about physics, like Dario Benedetti and Renee Hoekzema here, for example:

or Benjamin Bahr and Etera Livine:

Here are some shots taken at lunch one day (I won’t bother naming people in these):

Finally, here’s a self portrait I took in a hallway of the Physikzentrum one evening:

Actually, it’s not much of a portrait, but at least the mirror is nice. Mirrors I’ve seen in other physics institutes have been much more utilitarian.

## Quantum Spacetime in Bad Honnef

6 March 2012

Right now, I’m staying in this mansion:

This is the Physikzentrum in Bad Honnef, Germany. It’s a great place to spend time talking about physics, with an atmosphere that tastefully blends old and new. Inside, there’s a nice modern lecture hall:

but also—and probably even more important—comfortable spaces to talk … or to sit and write after most everyone else has gone to bed, like the room I’m in now:

Anyway, the reason I’m here is a workshop called

This is the kind of quantum gravity conference I like best—one that brings people together who have different viewpoints, and different approaches to the same questions.

Today we heard talks by Jan Ambjørn, Daniel Litim, Petr Hořava, Dario Benedetti, and Gianluca Calcagni. There were some common themes running through several of these talks, especially concerning renormalization group flow and asymptotic safety, and it was nice to see different perspectives. But for me, one the most interesting things was hearing a bit about the relationship between causal dynamical triangulations (which was the subject of Ambjorn’s talk) and Hořava-Lifshitz gravity (the subject of Hořava’s talk, though he modestly didn’t call it that himself).

Both causal dynamical triangulations and Hořava-Lifshitz gravity make a sharper distinction between space and time than general relativity does. The first introduces a fixed slicing of spacetime into discrete time-steps, while the second discards Lorentz symmetry at short distance scales. Since such theories that treat space and time anisotropically seem to be popping in from various starting points (another is “shape dynamics,” which I’ve mentioned here before) it is natural to wonder about relationships between them.

And, it’s good to see that some people are doing more than just wondering:

C. Anderson, S. Carlip, J. H. Cooperman, P. Horava, R. Kommu, P. R. Zulkowski, Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations.

Some of these folks are friends of mine from when I was at UC Davis.

Anyway, today was just the first day of talks, and I’m looking forward to the rest of the conference. Right now, I should probably get some sleep so I don’t doze off during Stefano Liberati’s talk in the morning!