An Invitation to Higher Gauge Theory

March 1, 2010

An Invitation to Higher Gauge Theory

Posted by John Baez

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John Huerta and I are writing an expository paper based on the notes of the course I taught in Corfu last summer:

John Baez and John Huerta, An invitation to higher gauge theory (DRAFT VERSION).

As with the course, the goal here is to give mathematicians and physicists a little taste of higher gauge theory, just enough to whet their appetite. So, the category-theoretic prerequisites have been reduced to the bare minimum. You don’t even need to know what a category is! You just need to promise not to run out of the room screaming when you see the definition.

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We’ve tried to explain some applications. Since the Corfu summer school was about quantum gravity, we’ve tried to balance applications to string theory with applications that would be interesting to people working on spin foam models. Since spin foam models are largely based on $B F$ theory, and $B F$ theory in 4 dimensions is a higher gauge theory, this is easy to do! The Poincaré 2-group spin foam model is also interesting in this respect — Aristide Baratin gave a talk on that at Corfu, and I recommend his paper with Derek Wise for more details. I’m also interested in the ‘gravity 3-group’ idea, which is based on some ideas that Urs had. Unfortunately I haven’t had time to develop it very far.

Indeed, even in a very minimal sense, the paper is not done yet. But I’d already be happy for you to report typos and mistakes. Unfortunately we don’t have time for more ambitious rewrites. In theory, the paper was due today! But I plan to beg the editor for more time.

Another thing: I’ve been getting reports from people lately saying that when they try to post comments to the $n$ -Café, they get an error message that doesn’t actually list any errors. This bug may be gone now. But if you get this error message, just hit ‘preview’ again and hope it goes away. That always used to work for me. If your comment still doesn’t go through, you can email it to me.

Posted at March 1, 2010 4:48 AM UTC

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16 Comments & 1 Trackback

Re: An Invitation to Higher Gauge Theory

Thanks for this paper! This is a remarkable piece of exposition.

I need to take a closer look at chapter 4, and I’m looking forward to chapter 5, but lest I forget I’ll post what caught my eye so far:

A group is a category with one object: When I first heard this statement I asked myself, what role does the object play? If you identify the objects with their identiy morphism, the answer is obvious, but since you encourage your readers to distinguish the two (objects are states, morphisms are processes), a little hint that the object is inserted only to provide the morphisms = group elements with a domain and a codomain could turn out to be helpful.
A groupoid is not the first example of a category where arrows aren’t functions in the usual sense that comes to my mind. Probably because my mind makes group elements act on the group by left-/right multiplication or conjugation automatically. The very first example I come up with are cobordisms.
lazy paths: that is a particularly good example of how names of new notions should be chosen: short, intuitive, easy and unique.

p.25:

But for a finite-sized surface, this formula is no good, since it involves adding up B at different points, which is not a gauge-invariant thing to do.

Since we are talking about principal bundles, gauge-invariant means invariant with respect to the action of the gauge group on the fibers, right? But then the addition of “B at different points” is simply not defined, it has nothing to do with gauge-invariance, or has it?

p.30:

Suppose we have a representation of a Lie group G on a vector space H. We can regard H as an abelian Lie group…

Filling in some dots: H is supposed to be finite dimensional, we equip it with the unique topology that turns it into a topological vector space, which turns it into a Lie group with regard to vector addition, correct?

p.31:

In short, it appears that the 2-category of representations of the Poincare 2-group gives a background-free description of quantum field theory on 4d Minkowski spacetime.

I have to admit that I do not understand the preceding paragraph, so let’s simply say that I’m surprised to read about a background-free description on a specific background. (I’m aware that we are in immediate danger of opening a bag of worms here).

some typos:

superfluous “but”, p.9: “Any Lie group is a smooth groupoid, but and so is the path groupoid of any smooth manifold.”

missing article “a”, p.10: “Conversely, suppose we have smooth functor hol.”

same, p.13: “In a 2-category, we visualize the 2-morphisms as little pieces of 2-dimensional surface:”

same, p.20: “it is map between 2-categories that preserves everything in sight.”

should be “paper”, p.29: “see the papery by Gotay”

[John Baez: Thanks for catching all these typos! I also added a sentence about the role played by the one object in a group, at the place where I first raise this idea. There is a lot to say about this, but I’m trying to keep the presentation very brisk, so I confined myself to the most elementary observation, as you suggest: “The morphisms of this category are the elements of the group. The object is there just to provide them with a source and target.”

Your comment about how we regard a finite-dimensional vector space as an abelian Lie group is correct. Inserting the finite-dimensionality assumption led me to spot a more serious mistake, which I also fixed.

Since at this point in the exposition we are dealing with a trivial principal G-bundle, it would make sense to add elements lying in different fibers of an associated vector bundle. But this is not a gauge-invariant thing to do: it depends on the choice of trivialization. It would cease to make sense as soon as we move to nontrivial bundles. Since it gets tiresome to explain precisely why an idea is dumb, I will change the sentence in question to: “But for a finite-sized surface, this formula is no good, since it involves adding up B at different points, which is not a smart thing to do.”

I may release a number of drafts, but the latest up-to-date version will always be here.]

Posted by: Tim van Beek on March 1, 2010 12:03 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

The very first example I come up with are cobordisms.

or for me, paths in a topological space

Posted by: jim stasheff on March 1, 2010 1:35 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

Jim wrote:

or for me, paths in a topological space

That’s acutally the first example in the paper, the “group as a groupoid with one object” is the second one. Smooth paths in a smooth manifold turn into the flow of a differential equation in my mind, which in turn is close to a function again…anyway, as a barely interesting personal anecdote, the moment it made “click” was when I saw a cobordism…

Posted by: Tim van Beek on March 1, 2010 1:53 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

What rôle does the object play in a group?

When I was first taught about groups as an undergraduate, from a very classical perspective, we were given the usual examples of the symmetry groups of polygons, rigid maps of Euclidean space, etc., and we were encouraged to think of groups as a generalisation of “the symmetries of something” (to use a phrase of John Baez’s).

Then of course we got used thinking of them as sets-with-structure, and there’s no obvious interpretation of the object there. But thinking of a group as “the symmetries of something”, it’s clear: the object is that something! So the object isn’t just playing a formal rôle…

Posted by: Peter LeFanu Lumsdaine on March 1, 2010 2:26 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

Peter wrote:

…thinking of a group as “the symmetries of something”…

This is of course a very fruitful way to think about a group, my remark is headed to a different direction, however. Let’s take an arbitrary abstract group from the shelf, what we get is

a neutral element,
an associative composition and
an inverse for every group element.

Now we would like to build ourselves a category out of this material: In order to achieve this, we need to promote the group elements to arrows that point from some object, the domain, to some object, the codomain. Of course, we could shift our viewpoint, see the group as the symmetry group of something and take this something as domain and codomain.

My point is: We don’t need to, any object will suffice. You can take whatever object you find in your garage and define this to be the domain and codomain of every arrow = group element, and voilà: there is your category.

Posted by: Tim van Beek on March 1, 2010 3:00 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

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A group is a category with one object: When I first heard this statement I asked myself, what role does the object play?

Actually, a group is a pointed groupoid with one object.

If by “An X is a Y.” we mean: “The natural category of Xs is equivalent to the natural category of Ys”.

The passage from

the notion of group = set with product operation

the notion of one-object groupoid

is not exactly an identification . It is rather the operation of delooping in $\infty Grpd$ .

I claim that it pays to care about this distinction: many nice structures, especially in higher gauge theory, will remain opaque without this.

To amplify this, I like to write $G$ for the group regarded as a set with product structure, and $\mathbf{B}G$ for the corresponding 1-object groupoid.

This is useful. For instance a classifying map for a smooth $G$ -bundle on a manifold $X$ is precisely a morphism

$g : X \to \mathbf{B}G$

is $\infty Grpd_{smooth}$ . It is a non-accident that this looks reminiscent of the underlying map of topological spaces

$X \to \mathcal{B}G$

to the classifying space of $G$ , which classifies the underlying topological bundle. The fat $\mathbf{B}$ reminds us that we have internalized the classifying space construction from the structure-less $\infty$ -topos $\infty Grpd$ to the structure-rich $\infty$ -topos $\infty Grpd_{smooth}$ .

A differential refinement $(g,\nabla)$ of such a smooth $G$ -cocycle to a $G$ -bundle with connection is then an extension of this along the constant path inclusion $X \to \mathcal{P}_1(X)$

$\array{ X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow & \nearrow_{\mathrlap{(g,\nabla)}} \\ \mathcal{P}_1(X) } \,.$

Posted by: Urs Schreiber on March 1, 2010 6:38 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

Urs wrote:

The passage from

* the notion of group = set with product operation

* the notion of one-object groupoid

is not exactly an identification . It is rather the operation of delooping in ∞Grpd.

Jim responds: That may be true technically but it also is subject to his next comment:

I claim that it pays to care about this distinction: many nice structures, especially in higher gauge theory, will remain opaque without this.

Jim: that is in the topological category the original identification is reasonavle but the delooping is something else entirely.

Posted by: jim stasheff on March 1, 2010 8:50 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

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that is in the topological category the original identification is reasonavle but the delooping is something else entirely.

It is exactly the same!

Under the equivalence $\infty Grpd \simeq Top_{cg}$ the one object groupoid $\mathbf{B}G = \{ \bullet \stackrel{g \in G}{\to} \bullet\}$ is identified with the classifying space $\mathcal{B}G$ .

Posted by: Urs Schreiber on March 1, 2010 9:01 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

I know you think that way, but notice it is an _identification_

Under the equivalence you mentio and I can’t process in this response the one object groupoid BG is identified with the classifying space.

depending on the _equivalence_

especially in the draft we are discussing
that equivalence is not available

Posted by: jim stasheff on March 1, 2010 9:11 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

John wrote:

I may release a number of drafts, but the latest up-to-date version will always be here.

If you broadcast a ping once you got a preliminary final version, I’ll know when to do my second sweep.

[John Baez: I’ll do that! Thanks.]

Posted by: Tim van Beek on March 2, 2010 10:03 AM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

What is exactly that 3-gravity idea?

Posted by: Daniel de França MTd2 on March 1, 2010 12:41 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

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What is exactly that 3-gravity idea?

In the

D’Auria-Fre formulation of supergravity

a field configuration of supergravity is regarded as what the authors called a “soft group manifold”. But as explained at that entry, a “soft group manifold” is really a collection of $\infty$ -Lie algebra-valued differential forms.

So the D’Auria-Fré formalism really regards supergravity as a higher gauge theory for gauge groups certain $\infty$ -groups that integrate certain $\infty$ -Lie algebras.

Specifically for maximal $N=1$ , $d=11$ -supergravity, they identify a certain Lie 3-algebra $\mathfrak{sugra}(10,1)$ as the corresponding structure $\infty$ -Lie algebra.

[John Baez: I’m still unable to post comments from home, so let me take the liberty of responding here, as an addendum to Urs’ comment. When I mentioned the “gravity 3-group” I was not referring to the D’Auria-Fré $\mathfrak{sugra}(10,1)$ business. I haven’t gotten around to discussing that yet — I still need to add it to the paper.

I was referring instead to the idea of describing general relativity in 4 dimensions as a higher gauge theory governed by a 3-group, as discussed on page 37 of this draft.]

Posted by: Urs Schreiber on March 1, 2010 4:18 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

A small typo on page 11: The last diagram (showing composition) on the page has the arrow fg going from x to y, but it should be from x to z.

[John Baez: Fixed! Thanks!]

Posted by: Vishal Lama on March 1, 2010 6:32 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

Cool paper!

One historical comment. On page 29, you write:

“In 1999, Freed and Witten [45] showed that the B-field should be seen as a connection on a U(1) gerbe”

I think this statement is already in Gawedzki’s 1986 paper “Topological Actions in two-dimensional Quantum Field Theories”. He doesn’t use the words “gerbe” or “B-field”, but he constructs the string action from Deligne 2-cocycles.

Posted by: Konrad Waldorf on March 1, 2010 7:00 PM | Permalink | Reply to this

Re: An Invitation to Higher Gauge Theory

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Hi Konrad!

You point out:

One historical comment. On page 29, you write:

In 1999, Freed and Witten [45] showed that the B-field should be seen as a connection on a U(1) gerbe.

I think this statement is already in Gawedzki’s 1986 paper “Topological Actions in two-dimensional Quantum Field Theories”. He doesn’t use the words “gerbe” or “B-field”, but he constructs the string action from Deligne 2-cocycles.

My search function may be broken, but it looks like also the Freed-Witten article does not use the word “gerbe”. It does not even seem to use the word “Deligne cocycle”. It just writes one down. :-)

I added the Gawedzki-reference that you mention to

Kalb-Ramond field – References

The reference list there includes also some of the later developments on identification of the right differential cohomology of string backgrounds, but much more needs to be added.

[John Baez: Indeed, John Huerta and I noted that Freed and Witten don’t actually come out and say the word ‘gerbe’. They merely say “Mathematical foundations for the low degree case needed here are developed in [B], though we do not use that language.” [B] is Brylinski’s book on gerbes — which of course does not have ‘gerbe’ in the title. So, Freed and Witten are subtly alluding to the theory of gerbes, in such a way that only people who know about gerbes will recognize this.

Thanks to Konrad, we’ve now added a reference to Gawedski as well.]

Posted by: Urs Schreiber on March 1, 2010 9:52 PM | Permalink | Reply to this

What is a connection?

By page 20 a connection `IS” a holonomy functor.
I’ll agree to parallel transport = holonomy but only in the principle bundle case. Otherwise you are hiding the classical geometry of parallel transport of tangent vectors.

You seem to avoid the phrase `connection form’ in favor of a much longer description. Both of these
seem to be distinct from `a connection on a trivial bundle’ which I don’t see defined. You do say what a connection does but not what it is?

I gather horizontal distribution and covariant dervative are either irrelvant in higher setting or not yet done?

p.9 If we drop this condition - suggest adding or even continuity

p.10 suggest upgrading Lemma to at least a Prop

That parallel transport (without homotopies) leads of a connection `form’
shoudl generalize to the topological setting for a fibration where the 1-form becomes a twisting cochain
but I haven’t pushed it through yet.

Posted by: jim stasheff on March 1, 2010 9:01 PM | Permalink | Reply to this

Read the post Rational Homotopy Theory in an (oo,1)-Topos
Weblog: The n-Category Café
Excerpt: On rational homotopy theory and differential cohomology from the point of view of oo-topos theory.
Tracked: March 2, 2010 12:40 PM

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March 1, 2010