Higher Gauge Theory

November 15, 2006

Posted by John Baez

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On Thursday I’m flying to Baton Rouge to give a talk on higher gauge theory and check out the nearby gravitational wave detector. You can see my slides here:

John Baez, Higher gauge theory, Mathematics and Physics & Astronomy Departments, Louisiana State University, November 14, 2006.

If you’re an expert on this business, perhaps the only thing you may not have seen yet is a discussion of how $B F$ theory in 4 dimensions is a higher gauge theory.

If you spot typos or other mistakes I would love to hear about them - especially before Thursday morning!

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Abstract: Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of “higher gauge theory” that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language forthis, we must “categorify” concepts from topology and geometry, replacing Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity.

I was invited to give this talk by Jorge Pullin, a long-time colleague of mine who works on loop quantum gravity. I’ll talk with him about that, and meet other mathematicians and physicists, including some working on quantum computation.

Also, on Saturday, I hope to visit the Laser Interferometer Gravitational-Wave Observatory in Livingston Louisiana! This thing is big:

It consists of two tubes, each about 2.5 kilometers long. The goal is to bounce lasers back and forth along these tubes to measure their length with an accuracy of 10^-19 meters, to see if gravitational ripples are distorting space. For comparison, the radius of a proton is a whopping 10^-15 meters. So, there’s a tremendous amount of work involved trying to eliminate all forms of error. Jorge’s wife Gabriela González works on this project, trying to understand and eliminate thermal noise.

I’ll try to take some photos and post them here!

Posted at November 15, 2006 5:27 AM UTC

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7 Comments & 2 Trackbacks

Re: Higher Gauge Theory

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Typo: on p. 11, $t$ should go from $H$ to $G$ . And again on p. 13.

Posted by: David Corfield on November 15, 2006 9:16 AM | Permalink | Reply to this

Re: Higher Gauge Theory

Thanks!

Posted by: John Baez on November 15, 2006 9:40 PM | Permalink | Reply to this

Re: Higher Gauge Theory

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I have made a comment on BF theory in the context of higher gauge theory here (in the thread on Dijkgraaf-Witten theory #).

Posted by: urs on November 15, 2006 10:02 AM | Permalink | Reply to this

Re: Higher Gauge Theory

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Great. You mention that the relation between 4d $B F$ theory and 3d Chern-Simons theory is new to you. You might be amused by this old review article of mine, back from 1994:

John Baez, Knots and quantum gravity: progress and prospects.

You can see from the chart on page 3 that I was fascinated by the “ladder of field theories” including the Wess-Zumino-Witten model in 2d, Chern-Simons theory in 3d and $B F$ theory in 4d. I was desperately trying to get 4d general relativity into the game somehow - one of my holy grails, on which I’ve wasted a lot of time.

I explain a bit about how this “ladder of field theories” fits together… but we’re getting to understand it much better now, thanks to your work!

One way to turn an $n$ -dimensional TQFT, say $Z$ , into an $(n-1)$ -dimensional one, say $\tilde Z$ , goes like this: $\tilde Z( - ) = Z(- \times S^1)$ This really corresponds to decategorification - the more-or-less obvious process of turning $n$ -Hilbert spaces into $(n-1)$ -Hilbert spaces by taking their “dimensions”.

This process gets us from 3d Chern-Simons theory down to the 2d G/G gauged WZW model, but it’s not how we get from 4d BF theory to 3d Chern-Simons theory.

It sounds like you want to think about a corresponding process involving $n$ -groups, but for some mysterious reason your process goes up instead of down!

Posted by: John Baez on November 15, 2006 10:01 PM | Permalink | Reply to this

Re: Higher Gauge Theory

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Ah, I didn’t know this either! So that’s why you mentioned the coset WZW model here.

Hm, hm, I would have to think about this.

I guess in terms of parallel transport functors it would amount to this:

say we have 3D Chern-Simons theory realized on a 3-manifold $X$ in terms of a theory of 3-functors

(1)

\mathrm{tra} : P_3(X) \to G_3

from 3-paths $P_3(X)$ to the Chern-Simons structure 3-group $G_3$ .

Next consider some 3-category that models the circle. Simon Willerton likes to use $\Sigma(\mathbb{Z})$ . Other models should do, too, I guess. Whatever you choose, call that category $S$ .

Then the “space of circles in $X$ ” is the (3-)functor (2-)category

(2)

[S,P_3(X)] \,.

Our transport functor (the Chern-Simons 3-connection) pulls back to this loop space by postcomposition

(3)

\mathrm{tra}_* : [S,P_3(X)] \to [S,G_3] \,.

Next, we may want to assume that $X = Y \times S^1$ and restrict attention to those maps in $[S,P_3(X)]$ that wind once around the $S^1$ -factor. Call the subcategory of these maps

(4)

[S,P_3(X)]_\mathrm{wnd} \,.

Then $\mathrm{tra}_*$ actually becomes a 2-functor on 2-paths in $[S,P_3(X)]_\mathrm{wnd}$ , if set up suitably. It still takes values in (possibly just a sub-category of) $[S,G_3]$ .

So I guess, in outline, to answer your question what the structure $n$ -group of the WZW coset model would be, one would have to go through the above steps carefully and identitfy the relevant subgroup in $[S,G_3]$ . Or something like that.

Posted by: urs on November 15, 2006 10:31 PM | Permalink | Reply to this

M Lie-3-algebra

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Concerning the discussion of the Chern-Simons and the M-theory Lie 3-algebra on the last couple of slides:

one should maybe point out that if there is indeed a M-theory structure Lie 3-algebra, it is most likely some sort of combination of the Chern-Simons Lie 3-algebra ### and the supergravity Lie 3-algebra #.

That’s because at the quantum level the supergravity 3-form (to be interpreted as the local connection 3-form with values in the Lie 3-algebra of the M-theory 3-group) is known to be the sum of two Chern-Simons forms - one for an $E_8$ -connection, one for a $\mathrm{SO}(10,1)$ -connection.

For that reason I was originally trying to add the terms that make up what I now call the Chern-Simons Lie 3-algebra

(1)

\mathrm{cs}(g)

to those terms that define the 11-d supergravity 3-algebra

(2)

\mathrm{sugra}_{11} \,.

I still expect this should be possible and should be done. But I haven’t done it yet.

As it stands, $\mathrm{sugra}_{11}$ nicely captures the classical aspects of 11-D supergravity, its supersymmetry and its 3-form. But it knows nothing about the quantization conditions.

On the other hand $\mathrm{cs}(g)$ knows about the structure of the Chern-Simons terms, hence about the topological part of the action of the M-theory membrane - but not about the rest.

As far as the $E_8$ -part is concerned, all we need might be just the direct sum Lie 3-algebra

(3)

\mathrm{sugra}_{11}\oplus \mathrm{cs}(\mathrm{Lie}(E_8)) \,.

But I think that analogously forming

(4)

\mathrm{sugra}_{11}\oplus \mathrm{cs}(\mathrm{Lie}(E_8)) \oplus \mathrm{cs}(\mathrm{Lie}(SO(10,1)))

is not what we want, since the $SO(10,1)$ -Chern-Simons form comes from precisely that $SO(10,1)$ -connection which is encoded in $\mathrm{sugra}_{11}$ . This constraint must somehwo be built into our M-theory Lie 3-algebra.

(And, by the way, possibly whenever I write $SO(10,1)$ I should for the moment rather be writing $SO(11)$ .)

Posted by: urs on November 15, 2006 10:25 AM | Permalink | Reply to this

Re: M Lie-3-algebra

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Thanks! My audience will be everyone in the math and physics departments who feels like attending. They’ll doubtless be too information-saturated by the end of my talk to absorb any details about M-theory… but, I changed my talk to make clear that M-theory as a higher gauge theory is more complicated than mere 11d supergravity… and I can say more if anyone wants to know.

This M-theory Lie 3-superalgebra should ultimately be something very beautiful and “integrated”, not a bunch of pieces tacked together, if M-theory is as Magnificent as it’s supposed to be.

However, for all the exceptional Lie algebras (and superalgebras) the simplest description starts by sticking together a bunch of pieces. Only at the end do you see that there are many ways to decompose the same beautiful thing into these pieces. For example, consider the Lie algebra $\mathfrak{e}_8$ . As a vector space, it’s a direct sum $\mathfrak{e}_8 \cong \mathfrak{so}(16) \oplus S_{16}^+$ where $S_{16}^+$ is the chiral spinor rep of $\mathfrak{so}(16)$ . But the bracket on this space is a bit subtle, so the above direct sum is not a direct sum of Lie algebras in any way (and indeed, $S_{16}^+$ isn’t even a Lie algebra). In the end, there turns out to be no “preferred” splitting of $\mathfrak{e}_8$ like this - just a lot of different but equivalent splittings. We’ve taken a beautiful jewel and described it only after cracking it down the middle in an arbitrary way, ruining its symmetry.

I hope the M-theory Lie 3-superalgebra is like this.

Posted by: John Baez on November 16, 2006 6:23 AM | Permalink | Reply to this

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November 15, 2006