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October 13, 2006

Bundle Gerbes: Connections and Surface Transport

Posted by Urs Schreiber

A continuation of my proposed addition # to the Wikipedia entry on bundle gerbes.

By the way, if anyone reading this is versed in Wikipedia editing and feels like inserting these nn-Café entries nicely formatted into Wikipedia, I’d very much appreciate it. I can offer the source code on request, if that helps.

Connection on a bundle gerbe - General.

Like bundle gerbes are a categorification of transitions in fiber bundles, bundle gerbes with connection are a categorification of transitions in fiber bundles with connection.

Like a connection on a locally trivialized bundle is encoded in a Lie algebra-valued connection 1-form on YY, the connection on a bundle gerbe gives rise to a Lie-algebra valued 2-form on YY (this shift in degree is directly related to the step from second to third integral cohomology). This 2-form is sometimes addressed as the curving 2-form of a bundle gerbe.

But there is more data necessary to describe a connection on a bundle gerbe. The details of the definition - which is evident for line bundle gerbes but more involved for principal bundle gerbes - can be naturally derived from a functorial concept of parallel surface transport, just like connection 1-forms on bundles can be derived from parallel line transport.


Connection on a bundle gerbe - Definition.

Line bundle gerbes.

A connection (also known as “connection and curving”) on a line bundle gerbe BpY [2]YπX B \stackrel{p}{\to} Y^{[2]} \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X is

  • a 2-form on YY BΩ 2(Y) B \in \Omega^2(Y)
  • a connection \nabla on the line bundle BY [2]B \to Y^{[2]}
  • such that π 1 *Bp 2 *B=F \pi_1^*B \; -\; p_2^*B \;=\; F_\nabla
  • together with an extension of the bundle gerbe product μ\mu to an isomorphism μ :p 12 *(B,)p 23 *(B,)p 13 *(B,) \mu_\nabla \;:\; p_{12}^* (B,\nabla) \;\; \otimes p_{23}^* (B,\nabla) \;\to\; p_{13}^* (B,\nabla) of line bundles with connection.

Notice that this condition ensures that dBd B is a 3-form on YY which agrees on double intersections p 1 *dB=p 2 *dB. p_1^* d B \;\; = \;\; p_2^* d B \,. This means that dBd B actually descends to a 3-form on XX.

The curvature associated with the connection on a line bundle gerbe is the unique 3-form on XX HΩ 3(X) H \in \Omega^3(X) such that π *H=dB. \pi^* H = d B \,.

The deRham class [H][H] of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.


Principal bundle gerbes.

A connection on a GG-principal bundle gerbe is

  • a Lie(G)\mathrm{Lie}(G)-valued 2-form on YY BΩ 2(Y,Lie(G)) B \in \Omega^2(Y,\mathrm{Lie}(G))
  • together with a Lie(Aut(G))\mathrm{Lie}(\mathrm{Aut}(G))-valued 1-form on YY AΩ 1(Y,Lie(Aut(G))) A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G)))
  • and a certain twisted notion of connection on the GG-bundle BB
  • satisfying a couple of conditions that reduce to those described above in the case G=U(1)G = U(1).

For the case that F A+adB=0F_{A} + \mathrm{ad} B = 0, these conditions are nothing but a tetrahedron law on a 2-functor from 2-paths in YY to the category Σ(GBiTor)\Sigma(G\mathrm{BiTor}). This is discussed in math.DG/0511710.

For the more general case a choice for these conditions that harmonizes with the conditions found for (proper) gerbes with connection by Breen & Messing in math.AG/0106083 has been given by Aschieri, Cantini & Jurčo in hep-th/0312154.


Surface transport.

From a line bundle gerbe with connection one obtains a notion of parallel transport along surfaces in a way that generalizes the procedure for locally trivialized fiber bundles with connection.

Recall that in the case of fiber bundles, the holonomy associated to a based loop γ\gamma is obtained by

  • choosing a triangulation of the loop (i.e., a decomposition into intervals) such that each vertex sits in a double intersection U ijU_{ij} and such that each edge sits in a patch U iU_i
  • choosing for each edge a lift into Y= iU iY = \sqcup_i U_i
  • choosing for each vertex a lift into Y [2]= ijU iU jY^{[2]} = \sqcup_{ij} U_i\cap U_j
  • assigning to each edge lifted to U iU_i the transport computed from the connection 1-form a ia_i
  • assigning to each vertex lifted to U iU jU_i \cap U_j the value of the transition function g ijg_{ij} at that point
  • multiplying these data in the order given by γ\gamma .

For bundle gerbes this generalizes to a procedure that assigns a triangulation to a closed surface, that lifts faces, edges, and vertices to single, double and triple intersections, respectively, and which assigns the exponentiated integrals of the 2-form over faces, of the connection 1-form over edges, and assigns the isomorphism μ ijk\mu_{ijk} to vertices.

For the abelian case (line bundle gerbes) this procedure has been first described in

K. Gawedzki & N. Reis
WZW branes and Gerbes
hep-th/0205233

based on

O. Alvarez
Topological quantization and cohomology.
Commun. Math. Phys. 100 (1985), 279-309.

Further discussion can be found in

A. Carey, S. Johnson & M. Murray
Holonomy on D-branes
hep-th/0204199.

Gawedzki and Reis showed this way that the Wess-Zumino term in the WZW-model is nothing but the surface holonomy of a (line bundle) gerbe.

In terms of string physics this means that the string (the 2-particle) couples to the Kalb-Ramond field - which hence has to be interpreted as the connection (“and curving”) of a gerbe - in a way that categorifies the coupling of the electromagnetically charged (1-)particle to a line bundle.

The necessity to interpret the Kalb-Ramond field as a connection on a gerbe was originally discussed in

D. Freed and E. Witten
Anomalies in string theory with D-branes
Asian J. Math. 3 (1999), 819-851.
hep-th/9907189.

Underlying the Gawedzki-Reis formula is a general mechanism of transition of transport 2-functors. This applies to more general situations than ordinary line bundle gerbes with connection.

The generalization to unoriented surfaces (hence to type I strings) was given in

K. Waldorf, C. Schweigert & U. S.
Unoriented WZW Models and Holonomy of Bundle Gerbes
hep-th/0512283.

Posted at October 13, 2006 1:42 PM UTC

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14 Comments & 9 Trackbacks

Re: Bundle Gerbes: Connections and Surface Transport

Yeah, I can incorporate these into the Wikipedia article. I assume that you wrote these in iTeX; the source for that (or anything else in TeX) would be helpful. Do you want me to wait for them to be hashed out further through discussions here?

I hope that you also learn how to write Wikipedia articles (easy), including using TeX (trickier, due to some bugs and Bad Things in texvc) yourself! But contributing to Wikipedia through me is better than not contributing at all.

Posted by: Toby Bartels on October 13, 2006 10:48 PM | Permalink | Reply to this

n-coffee for Wikipedia

Yeah, I can incorporate these into the Wikipedia article.

Thanks a lot for offering help!

Thanks also for the links to Wikipedia tutorials that you offered. I will take a look at them.

Is there a standard way to generate (commuting) diagrams? From looking at the sources I see these diagrams are incorporated as .png files.

Do you want me to wait for them to be hashed out further through discussions here?

While I am hoping that somebody will criticize my exposition here and there, and while I am aware that I left out some aspects that ought to be included eventually (Deligne cocycles and morphisms of bundle gerbes, for instance), I guess it would be okay to transport this to Wikipedia already. Once a decently structured Wikipedia entry exists, it should be easier (for me, for instance) to expand/improve/correct it.

Posted by: urs on October 14, 2006 12:36 PM | Permalink | Reply to this

Re: n-coffee for Wikipedia

Urs, is anything here more your opinion (say about how bundle gerbes should be thought of or why they are good or bad) than the general opinion of (the hundred or so) knowledgeable people? Please read Wikipedia’s neutrality policy and ensure that you believe that what you’ve written satisfies it. Also please state for the record that you release it under the GNU FDL. Then I will post it.

Is there a standard way to generate (commuting) diagrams? From looking at the sources I see these diagrams are incorporated as .png files.

No, one need to compile these with one’s own copy of TeX, convert the PostScript to a PNG, and upload that (preferably with the TeX source in the description). The refusal to even consider covering this is is one of the more profound flaws of texvc.

Posted by: Toby Bartels on October 14, 2006 4:26 PM | Permalink | Reply to this

Re: n-coffee for Wikipedia

more your opinion (say about how bundle gerbes should be thought of or why they are good or bad) than the general opinion of (the hundred or so) knowledgeable people

There are several ways that bundle gerbes can be thought of, which all knowledgeable people should be able to agree on. One of these ways (categorified transition function) I stated in my first sentence, another (groupoid extension) I mention as a derived consequence.

While I don’t say that one should think about bundle gerbes one way or the other, I do think that an article trying to explain the raison d’être of bundle gerbes profits from putting it this way.

In part, the way I have described it is the way I wish somebody had explained it to me back when I found all things bundle gerbe ad hoc and mysterious.

(Though maybe what I wrote now will sound just as ad hoc and mysterious to people entering the subject, and all that has really changed is my internal state of understanding.)

All explicit statements I make should be shared by all experts (unless I was mixed up about details when writing them, which everybody is kindly invited to check). On the other hand, the way the article is structured reflects my personal conviction about how bundle gerbes are best explained in an encyclopedia article.

You are yourself among the knowledgable people. So if you have any concerete concern, please let me know.

Also please state for the record that you release it under the GNU FDL.

Yes, I do release it under the GNU FDL. I will insert such a statement into the entries themselves, if that helps.

Thanks again for your help (especially with generating the diagrams)!

Posted by: urs on October 15, 2006 3:57 PM | Permalink | Reply to this

Re: n-coffee for Wikipedia

I can’t say I believe in this ‘neutrality’ idea. I’ve just been discussing the Wikipedia entry for ‘epistemology’. That’s clearly not neutral, but rather expresses the mind-set of a certain strand of late twentieth/early twenty-first century Anglo-American philosophy. Now, they could flag this at the beginning of the article, and allow parallel entries. But then there would have to be similar flaggings on linked pages, forming an overlaid network of philosophies.

I suppose things are less contentious in math. But I dare said if everyone’s opinion were expressed, there would be a similar mess. If they had an entry on ‘spectrum’, would they allow a clause saying, according to Baez and Dolan, spectra should be seen as groupoids not sets?

Posted by: David Corfield on October 15, 2006 5:48 PM | Permalink | Reply to this

Re: n-coffee for Wikipedia

I can’t say I believe in this ‘neutrality’ idea.

Probably this is related to the caveats I mentioned #.:

whenever one begins describing a certain topic, one inevitably has to make some choices concerning the structure of the description. Some aspects have to be chosen to be named before other aspects or even instead of other aspects. Some aspects will be presented as fundamental, others as derived. Some as deep, others as shallow.

But I think the “neutrality” that the Wikipedia etiquette wants to ensure is something less subtle.

Posted by: urs on October 16, 2006 10:03 AM | Permalink | Reply to this

Re: n-coffee for Wikipedia

I can’t say I believe in this ‘neutrality’ idea.

The first thing you have to know to understand Wikipedia’s NPOV (neutral point of view) is that the name is a misnomer. The French have it right: Neutralité de point de vue. It’s essence is contained in Urs's comment:

There are several ways that bundle gerbes can be thought of, which all knowledgable people should be able to agree on.

Then you add the idea that, instead of deciding which way is correct, you report everybody's opinions and reasons (on which all agree).

Of course, that doesn't extend well to the large-scale organisation of articles, which causes no end of headaches (people tend to settle on whatever is perceived to be mainstream, not necessairly a good idea), but IMO it does an excellent job applied to small-scale disputes over specific claims.

The answer to your question

If they had an entry on ‘spectrum’, would they allow a clause saying, according to Baez and Dolan, spectra should be seen as groupoids not sets?

is a resounding “Yes!”. Specifically, it would be perfectly appropriate to add something like

[[John Baez]] and James Dolan have argued that spectra should be viewed as a ''<b>Z</b>-groupoids'', a generalisation of [[omega-groupoid|&omega;-groupoid]]s "with <i>j</i>-morphisms for all <i>j</i>" in the set <b>Z</b> of [[integer]]s, "all of which are equivalences" [http://arxiv.org/abs/math/9802029].

to [[Spectrum (homotopy theory)]]. I would add it now, to demonstrate, but too much of the necessary background —like the article [[Omega-groupoid]]— is still missing. (In particular, you can see that this article could use a lot of other, more important additions.)

I hope that you help fix Wikipedia’s article on epistemology. Using material translated from the French and German Wikipedia’s could be an easy way to help (and perhaps the least contentious!).

Incidentally, the first version of that article was written by the same epistemologist (Larry Sanger) who wrote the first version of Wikipedia’s NPOV policy. (I don’t agree with Larry’s philosophical opinions myself, but that doesn’t seem to be a problem!)

Posted by: Toby Bartels on October 17, 2006 10:32 PM | Permalink | Reply to this

Re: n-coffee for Wikipedia

David wrote:

If they had an entry on ‘spectrum’, would they allow a clause saying, according to Baez and Dolan, spectra should be seen as groupoids not sets?

Toby said “Yes!” and suggested something like this:

[[John Baez]] and James Dolan have argued that spectra should be viewed as a ”<b>Z</b>-groupoids”, a generalisation of [[omega-groupoid|&omega;-groupoid]]s “with <i>j</i>-morphisms for all <i>j</i>” in the set <b>Z</b> of [[integer]]s, “all of which are equivalences” [http://arxiv.org/abs/math/9802029].

It’s a bit of a digression to point it out here, but David was undoubtedly talking about the spectrum of a commutative ring, which is best seen as a groupoid. Toby’s proposed addition to Wikipedia is about a completely different sort of “spectrum”, which is the main subject matter of “stable homotopy theory”. This sort of spectrum is best seen as a \mathbb{Z}-groupoid.

I explained both kinds of spectrum in week199, but the idea that the spectrum of a commutative ring is a groupoid is probably due to Grothendieck, and I touch on it in week205.

Of course we can have fun by smashing the two ideas against each other. In brave new algebra, people \infty-categorify the concept of “commutative ring” and get the concept of “E E_\infty ring spectrum”. Just as a commutative ring has a spectrum which is a groupoid, an E E_\infty ring spectrum should probably have a spectrum that’s an \infty-groupoid.

Posted by: John Baez on October 18, 2006 4:44 AM | Permalink | Reply to this

Re: n-coffee for Wikipedia

I certainly don’t intend to knock Wikipedia. It’s a wonderful resource. But I can foresee problems in contentious disciplines such as philosophy if Wikipedia is developed into an encyclopedia with essay length entries, along the lines of the intentions of the original French and Scottish Encyclopaedists.

And, yes, John is right about the spectra I meant.

Posted by: David Corfield on October 18, 2006 8:14 AM | Permalink | Reply to this

Re: n-coffee for Wikipedia

All explicit statements I make should be shared by all experts (unless I was mixed up about details when writing them, which everybody is kindly invited to check). On the other hand, the way the article is structured reflects my personal conviction about how bundle gerbes are best explained in an encyclopedia article.

This is about all that can be expected in a Wikipedia article. Of course, people that disagree with you about the best way to put things may come along (and probably will, eventually), but that is what the [[Talk:]] pages are for. As long as what the article actually states is neutral, then that should be OK.

Posted by: Toby Bartels on October 17, 2006 9:25 PM | Permalink | Reply to this

Re: Bundle Gerbes: Connections and Surface Transport

I’ll put the comment here, but I’m talking about the previous part of the entry #.

The section `Interpretation in terms of groupoid extensions.’ is not terribly clear. The literature I have seen on such a treatment (Anything by Ping Xu, and some earlier ones, e.g. Moerdijk) puts it in a more down to earth way. Such as: “An extension of groupoids is a short exact sequence …”, “The kernel must needs be a bundle of groups” and so on. Then one can give the example of the suspension of an exact sequence of (nonabelian) groups. Most people are probably better versed in such basics of homological algebra and can work by analogy, than in enriched category theory. Saying that, I would keep the enriched viewpoint.

Also some examples might be nice - say the obstruction to lifting a principal bundle given a central extension of the structure group, or the canonical bundle gerbe, cooked up by transgressing the 3-form to loop space and then forming the line bundle the usual way.

Posted by: David Roberts on October 16, 2006 3:03 AM | Permalink | Reply to this

further content

David, thanks a lot for your comments! I perfectly agree with your remarks.

The bundle-gerbes-as-groupoid-extensions part certainly would benefit from further discussion, and in general examples are missing.

Other things are missing, too. Morphisms of bundle gerbes, the 2-category of bundle gerbes, modules for bundle gerbes, D-branes and bundle gerbes, bundle gerbes from gerbes, bundle gerbes from 2-bundles, lifting bundle gerbes, bundle gerbes and QFT, bundle 2-gerbes, bundle n-gerbes, Chern-Simons bundle 2-gerbes. Anything else? :-)

Maybe I find the time and energy to write further parts. Or maybe somebody else does. I’d certainly be motivated by a collaborative effort. Meaning: I’d probably write more myself when others chime in.

Posted by: urs on October 16, 2006 9:50 AM | Permalink | Reply to this

Re: further content

The best possible Wikipedia article on bundle gerbes would probably spend a lot of time explaining in a very simple-minded way why one should care about bundle gerbes in the first place. And even more of this sort of thing should be done in the article on gerbes! Bundle gerbes are just a way of making gerbes more easily digestible

A lot of math articles in Wikipedia are missing the few sentences at the beginning which would explain the basic idea to a nonmathematician reader.

Posted by: John Baez on October 18, 2006 6:16 AM | Permalink | Reply to this

Re: further content

A lot of math articles in Wikipedia are missing the few sentences at the beginning which would explain the basic idea to a nonmathematician reader.

I very much agree with this. That’s the reason why I began my “article” by explaining the “general idea” in plain English.

But I gather that this explanation itself might need to be supplied with more down-to-earth background discussion in order to be really of help for the complete layman.

Bundle gerbes are just a way of making gerbes more easily digestible

This statement I happen not to quite agree with. True, historically, that’s the way it happened.

Because for some reason, when people started categorifying gauge theory, of all the different kinds of description of an ordinary fiber bundle (which I listed), people chose to pick for categorification the one that is the most indirect: the point of view of sheaves of sections.

But that does not imply that the more easily accessible points of views are just a way of making the former point of view more digestible. They exist in their own right.

People can define, reason about and apply bundle gerbes successfully without ever mentioning locally nonempty transitive stacks in groupoids.

Just like they can use transition functions to talk about fiber bundles without ever mentioning sheaves.

Posted by: urs on October 18, 2006 9:55 AM | Permalink | Reply to this
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