The n-Category Café

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.