### Teleparallel Gravity as a Higher Gauge Theory

#### Posted by John Baez

It’s finally done! Let me say a bit about it:

- John Baez and Derek Wise, Teleparallel gravity as a higher gauge theory.

Abstract.We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold $M$, we first construct a principal 2-bundle with the Poincaré 2-group as its structure 2-group. Any flat metric-preserving connection on $M$ gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way if $M$ is simply connected and has vanishing 2nd deRham cohomology. Extending from the Poincaré 2-group to the teleparallel 2-group, a 2-connection includes an additional piece: a coframe field. Taking advantage of the teleparallel reformulation of general relativity, in which a coframe field, a flat connection and its torsion are the key ingredients, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.

We’ve already talked here about a preliminary version of this paper, but it’s bigger now, so I’ll say more. (There will not, as originally planned, be two parts. It’s all here.)

There are many kinds of geometry, so there are probably lots of interesting unexplored relations between geometry as traditionally studied and higher category theory — for example, higher gauge theory. This paper is about one.

Geometry and symmetry go hand in hand so often geometers take inspiration from some symmetry group, looking for geometries that have that symmetry, perhaps just ‘locally’ or ‘infinitesimally’. But this paper took its inspiration from a *categorical* group, or 2-group: that is, a category equipped with a multiplication and inverses just like a group. You can describe a lot of 2-groups starting with four pieces of data:

- A group $G$: the group of objects.
- A group $H$: the group of morphisms whose source is the identity object $1 \in G$.
- A homomorphism $t: H \to G$ sending any morphism in $H$ to its target.
- An action $\alpha$ of $G$ as automorphisms of $H$.

These data need to satisfy two equations to deserve the title of a crossed module — and if they do, you can assemble them into a 2-group.

You can try to build a crossed module by taking $t$ to be trivial, sending everything in $H$ to the identity in $G$. This only works if $H$ is abelian, but then it *always* works: $\alpha$ can be anything you want.

So: whenever you have a group acting on an abelian group, there’s a 2-group waiting to be born.

In physics an obvious example is this. Let $G$ be the group of Lorentz transformations, which acts on Minkowski spacetime in a way that preserves a chosen point. Let $H$ be the abelian group of translations of Minkowski spacetime. We have a group $G$ acting on an abelian group $H$. *Voilà:* a 2-group is born!

In physics, people usually use the same data to build a mere group, by forming the semidirect product $G \ltimes H$. This is called the Poincaré group, and it’s very important: it’s the full symmetry group of Minkowski spacetime.

So, when I first ran into the 2-group built from this data about a decade ago, I called it the **Poincaré 2-group**. It seemed interesting. But I could never out what it was good for!

When Derek and I were on a train from Nuremberg to Erlangen last summer, I explained one reason why. In ordinary gauge theory we study connections for a a Lie group $G$, and locally they’re just $\mathfrak{g}$-valued 1-forms. Similarly, in higher gauge theory we can study 2-connections for a given 2-group — at least if it’s a Lie 2-group, like the Poincaré 2-group. And if our Lie 2-group comes from a crossed module $(G,H,t,\alpha)$, a 2-connection consists locally of:

- a $\mathfrak{g}$-valued 1-form $A$ and
- a $\mathfrak{h}$-valued 2-form $B$ obeying
- the ‘fake flatness’ condition $F = \underline{t}(B)$, where $F = d A + A \wedge A$ is the
**curvature**of $A$ and $\underline{t} : \mathfrak{h} \to \mathfrak{g}$ is the differential of $t: H \to G$.

The last condition is needed if we want parallel transport over both curves and surfaces to be well-defined… and well-behaved.

I didn’t need to explain this stuff to Derek—he knows it. I just said: suppose you want a 2-connection for the Poincaré 2-group. Then $G = SO(3,1)$ is the Lorentz group and $H = \mathbb{R}^4$ is the group of translations of Minkowski spacetime, so we want

- a $\mathfrak{so}(3,1)$-valued 1-form $A$ and
- a $\mathbb{R}^4$-valued 2-form $B$ obeying
- the flatness condition $d A + A \wedge A = 0$.

where fake flatness has become honest flatness since $\underline{t} = 0$.

“But,” I said, “where do you see anything like this in physics???”

Somehow when I said this with Derek staring at me, we both instantly knew the answer: *teleparallel gravity.*

You see, Derek is an expert on different reformulations of general relativity, and precisely this sort of $A$ and $B$ show up in teleparallel gravity, where $A$ is called the ‘Lorentz connection’ and $B$ is called the ‘torsion’.

While teleparallel gravity is not very famous, it was invented by Einstein, and he studied it intensively from 1928 to 1931. He also had a significant correspondence on the subject with Élie Cartan. I’m not sure, but I think this helped Cartan develop his concept of moving frames, since these are fundamental to teleparallel gravity.

At first glance, teleparallel gravity seems conceptually quite different from general relativity. Unlike in general relativity, the spacetime of teleparallel gravity is *flat*. As a consequence, it is possible to compare vectors at distant points to decide, for example, whether the velocity vectors of two distant observers are parallel — hence the term ‘teleparallel’.

Flat spacetime clearly flies in the face of Einstein’s geometric picture of gravity as spacetime curvature: in fact, in teleparallel theories, gravity is a *force*. Indeed, teleparallel gravity sounds like such a throwback to the Newtonian understanding of gravity that it would be easy to dismiss, except for one fact: teleparallel gravity is *locally equivalent* to general relativity, at least in the presence of only spinless matter!

So, there are interesting puzzles here for the philosopher of physics. But Derek and I merely wanted to follow up on the clue we found, and develop a formulation of teleparallel gravity in terms of higher gauge theory.

When we tried, we found it works better if we slightly enlarge our 2-group. Let $G$ be the Poincaré group. Let $H$ be the group of translations of Minkowski spacetime. Let $t : H \to G$ be the obvious inclusion. Then $H$ becomes a normal subgroup of $G$, so let $\alpha$ be the action of $G$ on $H$ by conjugation. *Voilà:* another 2-group is born, the **teleparallel 2-group**.

By what I’ve said so far, 2-connection for the teleparallel 2-group consists locally of this stuff:

- a $\mathfrak{so}(3,1) \ltimes \mathbb{R}^4$-valued 1-form $A$ and
- a $\mathbb{R}^4$-valued 2-form $B$ obeying
- the ‘fake flatness’ condition $F = \underline{t}(B)$, where $F = d A + A \wedge A$ is the curvature of $A$ and $\underline{t} : \mathbb{R}^4 \to \mathfrak{so}(3,1) \ltimes \mathbb{R}^4$ is the obvious inclusion.

But in physics we usually break $A$ into two parts, and use the fake flatness condition to solve for $B$ in terms of $A$. Then we get:

- a $\mathfrak{so}(3,1)$-valued 1-form $\omega$ called a
**Lorentz connection**and - an $\mathbb{R}^4$-valued 1-form $e$ called a
**coframe field**and - a $\mathbb{R}^4$-valued 2-form $B$ obeying
- the condition $d\omega + \omega \wedge \omega = 0$ saying that the Lorentz connection is flat, and
- the condition $B = d e + \omega \wedge e$ saying that $B$ is the
**torsion**.

These are exactly the fields we find in a modern formulation of teleparallel gravity, and they obey both these conditions! So, we can write teleparallel gravity as a theory whose only field is a 2-connection for the teleparallel 2-group.

There’s a lot more in our paper which I’m not discussing here, including some actual theorems, and some remarks on how to see teleparallel gravity in terms of categorified Cartan geometry. But this should give you a taste of it.

## Re: Teleparallel Gravity as a Higher Gauge Theory

This 2-group, $Tel(3,1)$, is equivalent to the 1-group $O(3,1)$.

Accordingly, parallel 2-transport with values in $Tel(3,1)$ is equivalent to parallel 2-transport with values in $O(3,1)$, which in turn is equivalent to flat parallel 1-transport with values in $O(3,1)$. Which is trivial when $\pi_1$ of base space is trivial. This is why you find theorem 37.

One might try to get around this in a gauge invariant way, i.e. without breaking gauge invariance by hand as you do in the article, by considering “twisted” $O(3,1)$-cohomology:

The short exact sequence of smooth groups

$\mathbb{R}^4 \to IO(3,1) \to O(3,1)$

continues as a long fiber sequence of smooth groupoids

$\mathbb{R}^4 \to IO(3,1) \to O(3,1) \stackrel{\kappa}{\to} \mathbf{B} \mathbb{R}^4 \,,$

where on the right $\mathbf{B} \mathbb{R}^4$ is the 2-group corresponding to the crossed module $(\mathbb{R}^4 \to 1)$.

The equivalence $Tel(3,1) \stackrel{\simeq}{\to} O(3,1)$ serves to present the connecting homomorphism $\kappa$ here by a span of strict 2-groups – an anafunctor of strict 2-groups:

$\kappa \simeq ( O(3,1) \stackrel{\simeq}{\leftarrow} Tel(3,1) \to (\mathbb{R}^{3,1} \to 1) ) \,.$

Unfortunately $\mathbb{R}^{3,1}$ is not central in $IO(3,1)$, otherwise we could deloop this long fiber sequence, or in other words: otherwise it would be a long fiber sequence of 2-group homomorphism. This would have allowed to consider $\mathbf{B}\kappa$-twisted $O(3,1)$-connection, which would then be indeed strict $Tel(3,1)$-connections with fixed 2-form piece, this being no longer equivalent to $O(3,1)$-connections. But, since we don’t have a central extension, this does not work.

What else could one do about it?