April 20, 2012

Teleparallel Gravity as a Higher Gauge Theory

Posted by John Baez

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

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.

Posted at April 20, 2012 2:54 AM UTC

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Re: Teleparallel Gravity as a Higher Gauge Theory

another 2-group is born, the teleparallel 2-group.

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?

Posted by: Urs Schreiber on April 20, 2012 8:22 AM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Several of my colleagues consider lifting bundle gerbe for non-central, normal extensions by abelian groups in this paper:

The Faddeev-Mickelsson-Shatashvili anomaly and lifting bundle gerbes (arXiv:1112.1752)

It is the first application of some forthcoming work by Michael Murray. This may be useful in this situation.

Posted by: David Roberts on April 22, 2012 10:49 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Several of my colleagues consider lifting bundle gerbe for non-central, normal extensions by abelian groups in this paper:

The Faddeev-Mickelsson-Shatashvili anomaly and lifting bundle gerbes (arXiv:1112.1752)

You are talking about the construction on p. 9. Here is how I would formulate this intrinsically (without using components and explicit cocycles, which might be more transparent):

An extension of Lie groups

$A \to \hat H \to H$

induces the corresponding fiber sequence of delooped Lie groupoids = moduli stacks

$\mathbf{B}A \to \mathbf{B}\hat H \to \mathbf{B}H \,.$

Now we consider an $H$-principal bundle $P \to X$ classified by a morphism of smooth groupoids / stacks $g : X \to \mathbf{B}H$ and the lifting problem is whether we can find $\hat g$ in

$\array{ && \mathbf{B}\hat H \\ & {}^{\mathllap{\hat g}}\nearrow& \downarrow \\ X &\stackrel{}{\underset{g}{\to}}& \mathbf{B}H }$

(there is an implicit 2-cell filling this triangle, which is a diagram in Smooth∞Grpd $:= Sh_\infty(SmthMfd)$).

By the universal property of the (homotopy) pullback, this $\hat g$ exists if and only if there is a section of the (homotopy) pullback 2-bundle on the left of

$\array{ P \times_H \mathbf{B}A &\to& \mathbf{B}\hat H \\ \downarrow & & \downarrow \\ X &\stackrel{}{\underset{g}{\to}}& \mathbf{B}H } \,.$

That’s what prop. 3.1 in the article you point to asserts (or rather, that prop is a re-expression of the above in terms of explicit cocycle representatives).

This is something that one can always do. But it does not support the kind of functorial twisting which I was mentioning.

On the other hand, since in the present context we are talking about a lift through a split extension anyway (projection out of semidirect product) this would not be interesting, even if we were talking about a central extension.

So, hm, I don’t yet see a way to make invariant sense of the construction that John and Derek consider.

Posted by: Urs Schreiber on April 22, 2012 11:51 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

@Urs:So, hm, I don’t yet see a way to make invariant sense of the construction that John and Derek consider.

Uh oh, hope someone can do that for if it has not invariant meaning…

Also some of the discussion here seems to be in terms of ordinary classifying SPACES - does the stackified pov hep at all?

Posted by: jim stasheff on April 23, 2012 2:10 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Also some of the discussion here seems to be in terms of ordinary classifying SPACES - does the stackified pov hep at all?

I’d suggest that that stackified pov is the only one that gets one anywhere here. At least it helps a lot to make transparent the structures and see what’s going on. Everything becomes clearer, less error-prone and hence and less scary. What I said above was all in terms of this pov.

Posted by: Urs Schreiber on April 23, 2012 3:13 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Should we have a special Cartan 2-geometry page on nLab, or is it fine as it is as a subcase of higher Cartan geometry?

Posted by: David Corfield on April 20, 2012 10:54 AM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Should we have a special Cartan 2-geometry page on $n$Lab

Maybe what is needed is a page on gauge theory / Cartan geometry internal to the 1-category of smooth groupoids. This is what John and Derek consider.

This is different from gauge theory / Cartan geometry internal to the $(2,1)$-topos of smooth groupids, or eventually in the $(\infty,1)$-topos of smooth $\infty$-groupoids, which is what is considered on that $n$Lab page.

Posted by: Urs Schreiber on April 20, 2012 12:40 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

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.

John, I’m not sure if you meant generic moving frames as used in Cartan’s equivalence method or something more specificly related to gravity. Cartan’s method is actually older than GR itself, as it already appeared in this early article. The general ideas behind that method appear to be much older still and have appeared notably in the Frenet-Serret formulas classifying space curves.

Some historical references are on this Wikipedia page. A much more extensive discussion can be found in the introduction to one of the chapters of E. Cartan’s collected works.

Posted by: Igor Khavkine on April 20, 2012 9:16 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

>”teleparallel gravity is locally equivalent to general relativity, at least in the presence of only spinless matter!”

This sounds as if for example the teleparallel analog of the Shwartzenchild solution may not be asymptotically flat at infinity?

Posted by: student on April 21, 2012 6:51 AM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

teleparallel gravity is locally equivalent to general relativity, at least in the presence of only spinless matter!

This sounds as if for example the teleparallel analog of the Shwartzenchild solution may not be asymptotically flat at infinity?

The Schwarzschild spacetime is a vacuum solution to gravity, meaning that, apart from the field of gravity self-interacting with itself, there are no other fields around, in particular no fermion fields.

Also, when we speak about teleparallel gravity, we need to be careful with what we mean by saying that a spacetime is “flat”. Because it may be flat with respect to the usual formulation, where it means that the curvature of the Levi-Civita connection of the metric vanishes, or flat with respect to the teleparallel formulation, where it means that the curvature of the Weitzenböck connection vanishes – and the latter is identically the case by definition!

So every spacetime in teleparallel gravity is regarded as “flat”, asymptotically or not. This is the hallmark of the whole approach. But one has to understand that it is a different notion of flatness than what is usually understood when saying things like “the Schwarzschild spacetime is asymptotically flat”.

A discussion of specifically the Schwarzschild spacetime both from the point of view of ordinary and of teleparallel gravity is on the last page of

Gheorghe Zet, Schwarzschild solution with torsion (arXiv:gr-qc/0308078)

Posted by: Urs Schreiber on April 21, 2012 12:10 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

I would like to point out that there is an alternative approach to formulating GR as a higher-gauge theory:
[1] arXiv:1110.4694,
Poincare 2-group and quantum gravity
Aleksandar Mikovic, Marko Vojinovic

The idea of this paper is to reformulate GR as a constrained BFCG theory. A BFCG theory is a BF theory for Lie 2-group, given by an action whose Lagrangean is a 4-form + , where F is the fake curvature and G is the curvature 3-form for a 2-connection (A,b), see
[2] arXiv:0708.3051,
Topological Higher Gauge Theory - from BF to
BFCG theory,
F. Girelli, H. Pfeiffer, E. M. Popescu
Journal-ref: J. Math. Phys. 49: 032503, 2008

and
[3] arXiv:1006.0903,
Lie crossed modules and gauge-invariant actions
for 2-BF theories
Joao Faria Martins, Aleksandar Mikovic,
Journal-ref: Adv. Theor. Math. Phys. 15, 2011.

The result of [1] is that the GR action in the Cartan formulation can be written as a constrained BFCG action for the Poincare 2-group, where the constraint is B = (C \wedge C)* and the 1-form Lagrange multiplier fields C can be identified with the tetrads.

The difference with respect to the Baez-Wise approach, is that the 2-connection in [1] is not flat and hence one obtains the Cartan formulation of GR, as opposed to the teleparallel formulation obtained by Baez and Wise. From the point of view of physics, the Cartan formulation is more natural.

Since the Poincare 2-group 2-connection in [1] is not flat, it does not correspond to a 2-holonomy; however, if one wants to define a 2-holonomy one can impose the 2-flattnes condition. On the other hand, if one insists that the 2-connection for GR has to be flat always (off-shell, in physics terminology) than one cannot use the Poincare 2-group, and as Baez and Wise have shown, one needs a bigger 2-group, which is the teleparallel 2-group.

Posted by: Aleksandar Mikovic on April 22, 2012 11:06 AM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

one needs a bigger 2-group, which is the teleparallel 2-group.

This is misleading to say. What is called the “teleparallel 2-group” in def. 24 is equivalent to just the ordinary group $O(p,q)$.

If one is really looking at higher gauge theory, one has to be careful not to be evil and distinguish between equivalent incarnations of the same (“gauge”) equivalence class. The canonical morphism $Tel(p,q) \to O(p,q)$ is an equivalence of 2-groups.

Of course it is not an isomorphism of the underlying crossed modules. Up to page 27 of their article, John and Derek work with crossed modules and their principal bundles up to isomorphism. One can do this, but it is not really higher gauge theory. It is gauge theory internal to the 1-category of smooth groupoids.

If one does work in higher gauge theory, the equivalence $Tel(p,q) \simeq O(p,q)$ reduces the discussion up to equivalence to that of ordinary flat $O(p,q)$ gauge theory. As John and Derek discuss in 4.6.

Posted by: Urs Schreiber on April 22, 2012 5:40 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Hi Aleksandar,

I have a comment on the references that you list.

In [ 1 ] it might be worthwhile highlighting that the action (18) is indeed (higher!) gauge invariant if, let’s see… if the third Lagrange multiplier $\phi$ is taken to be a 2-form with values in $\Lambda^2 \mathbb{R}^4$, and taken to transform as expected under 1-gauge transformations and trivially under the higher gauge transformations.

Right?

What I would like to see is discussion of the collection of these three Langrange multipliers in a way intrinsic to higher gauge theory: currently you can say that the fields $(A,\beta)$ are components of a 2-connection, but you cannot make a similarly conceptual statement for $(B,C,\phi)$, can you? They are just introduced by hand, with their transformation laws determined by hand, it seems.

How about this: we know that ordinary BF-theory is already nicely understood as a gauge theory for a 2-connection, where both the “fields” and the “Lagrange multiplier” are parts of one and the same 2-connection (in fact with “cosmological constant” included it is a higher Chern-Simons functional for a 2-connection, as described 4.6.4.1 here).

So by direct analogy, one would except that BFCG theory or some variant of it would be theory actually of a single 3-connection, with fields and Lagrange multipliers all unified. Has anyone thought about something like this?

Posted by: Urs Schreiber on April 23, 2012 11:09 AM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Hi Urs,

Yes, the gauge transformations of the Lagrange multipliers B,e, and \phi are determined such that the 4-form
L = B\wedge R + e\wedge G + \phi\wedge [B - (e\wedge e)*]
is invariant under the 2-group gauge transformations of the 2-connection (w,b).

I do not know how to put together the fields (B,e,\phi) into a 2-connection; however, the result for the BF theory is very interesting. One can try to interpret the 4-form L as a BFCGDH theory for a 3-group, where the 3-connection is (w,b,c) and the Lagrange multipliers are (B,C,D), a set consisting of a 2,1 and a 0-form, but then it is difficult to see how (B,C,D) can be a 3-connection, which is a set consisting of a 1,2 and a 3 form.

Posted by: Aleksandar Mikovic on April 23, 2012 2:40 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

Hi Aleksandar,

for your convenience, let me insert a quick administrative comment on how to format comments here on this blog:

• to typeset math, choose right over the edit pane in the pulldown menu “Text Filter” something like “Markdown with iTex to MathML”. With that done, Latex-style math code enclosed in single (inline) or double (displayed) dollar signs will render as expected

• to make hyperlinks appear: with the “Markdown” text filter chosen you get a hyperlink by

square-bracket-open link-text square-bracket-close round-bracket-open url round-bracket-closed

Posted by: Urs Schreiber on April 23, 2012 3:07 PM | Permalink | Reply to this

Re: Teleparallel Gravity as a Higher Gauge Theory

In my previous comment I used the term “flat 2-connection”, and by that I meant that the 1-form component A and the 2-form component B have vanishing fake curvature
F’ = dA + A ^ A - t(B),
where “t” is the crossed module homomorphism.

Posted by: Aleksandar Mikovic on April 23, 2012 9:41 AM | Permalink | Reply to this

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