The String Coffee Table

2-bundles

Posted by Urs Schreiber

I was recently trying to learn about nonabelian gerbes. That turns out to be quite hard. The canonical reference is

Breen & Messing: Differential geometry of gerbes (2001)

which I have still not entirely deciphered. A much more digestible presentation is given in

P. Aschieri & L. Cantini & B. Jurco: Nonabelian bundle gerbes, their differential geometry and gauge theory (2003)

which I am able to follow, though it took me two days to work through it.

Anyone wondering what this has to do with strings should have a look at

P. Aschieri & B. Jurco: Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory (2004) .

John Baez kept telling me that there is an alternative to gerbe language that might have some conceptual advantages: That’s 2-bundles, the categorification of ordinary fiber bundles. Unfortunately, 2-bundles were top-secret - until a couple of days ago. Now they have been revealed to the world in the paper

Toby Bartels: Categorified gauge theory: 2-bundles (2004) .

I very much like this approach for several of reasons. Here is the central idea in a single sentence:

While an ordinary bundle is a map $p: E \to B$ from the total space $E$ to the base space $B$, a 2-bundle is a 2-map $p: E \to B$ from the 2-space $E$ to the 2-space $B$.

That’s it. Now you turn the 2-category-crank and everything else just drops out. By going from groups to 2-groups and from open covers to 2-covers, etc. one gets the definition of locally trivial principal $G$-2-bundles by copying word for word the respective definitions for ordinary principal bundles. And indeed that’s what Toby Bartels does, parts of his section 2 must have been deliberately copy-and-pasted from section 1, with the prefix ‘2-’ inserted here and there.

So could you just write a paper on 3-bundles by copy-and-pasting the above paper and replacing 2s by 3s? Not quite. Categorification adds additional ‘logic’ at every step, expressed in terms of coherence laws which tell you that while all expressions which are the same are equal, some are less equal than others (so to say). Coherence laws tend to manifest themselves in terms of intimidating diagrams of ($n$-)morphisms that may have a striking resemblance to diagrams of compounds in organic chemistry, but actually when you stare at them long enough most of them become quite intelligible.

What is nice is that the precise nature of the coherence laws for 2-bundles is not needed to understand the concept and to write down examples, for instance. Anyone interested in the care and feeding of such laws should have alook for instance at

J. Baez & A. Lauda: Higher Dimensional Algebra V: 2-groups (2003)

and

J. Baez & A. Crans: Higher Dimensional Algebra VI: Lie 2-algebras (2003) .

So does a $G$-2-bundle for non-abelian $G$ give us everything a gerbe would give us?

Almost everything. Apart from aspects which I cannot overlook yet, one crucial difference between 2-bundles and gerbes is that a gerbe more or less automatically comes with some analog of a connection, while the above 2-bundles don’t.

The reason why gerbes and connections go hand in hand is that gerbes are characterized by elements in Deligne hypercohomology, which are objects that include a higher order generalization of transition functions as well as 1-forms and 2-forms that generalize the ordinary 1-form of a connection on a fiber bundle. But this automatism has a drawback, it seems: Nobody knows (at the moment) how, in the non-abelian case (which can be understood using twisted abelian 2-gerbes), you get a holonomy from such 1+2 forms. In the abelian case this is well understood and pretty nifty, but nonabelian gerbes are, truly, harder to understand.

The only known way to nonabelian surface holonomy at the moment seems to be the one that I have talked about a lot here, which involves ordinary locally trivial bundles with connection on the space of paths/loops. It would be nice if this approach could be related to gerbes or 2-bundles somehow.

I have only a very vague idea how the connection between path space bundles and nonabelian gerbes could be obtained. But for 2-bundles it is very easy to see and quite beautiful:

In order to see that one needs to know that a ‘2-space’ $X$ is essentially an ordinary space $X^1$ (consisting of points) together with a space $X^2$ consisting of ‘paths’ or ‘morphisms’ between points (see section 2.1.1 of Toby Bartle’s papers for more details). So for instance spacetime $\mathcal{M}$ together with the space of paths (strings!) $P\mathcal{M}$ in $\mathcal{M}$ gives us a 2-space $B$ with $B^1 = \mathcal{M}$ and $B^2 = P\mathcal{M}$.

Next a 2-map between 2-spaces is essentially two maps, one going between the point-spaces the other going between the morphism-spaces and such that source and target of any path/morphism is respected by the two maps.

A 2-group $G$ is a 2-space which is also a group, roughly, which in particular means that the ‘paths’ in $G^2$ are group elements that have a source and a target group element.

But this means that a, trivial say, 2-bundle

is essentially an ordinary $G^1$-bundle over $\mathcal{M}$

together with a $G^2$-bundle over $P\mathcal{M}$, the space of paths in $\mathcal{M}$:

plus some consistency relations between these maps. Here $p^1$ and $p^2$ are the two parts of the 2-map $p$ which act on the point-parts and the path-parts of the 2-spaces respectively, and the above is just a verbose version of the simple statement that a 2-bundle is a 2-map $p: E \to B$.

This means, unless I am mixed up, that the notion of 2-bundle automatically gives us an ordinary 1-bundle on the space of paths. That’s great, because it is here that nonabelian holonomy is best understood.

Hence the only part of Toby Bartel’s paper that I am not completely happy with is his statement on p.18 that physically interesting 2-bundles have base spaces $B$ which are essentially souped-up ordinary spaces, with only the trivial constant paths included. I believe that instead the physically interesting case is the one I mentioned above, where $X^2$ is $P\mathcal{M}$. That’s where nonabelian strings live.

If 2-bundles really capture the same information as nonabelian gerbes there should be some translation mechanism between the two concepts. This I would like to better understand.

For starters, where in the 2-bundle approach are the transition $\lambda_{ijk}$ functions on quadruple overlaps $U_{ijkl}$? Toby Bartels does mention the space $U^4$ of all quadruple overlaps of a given cover $U$. It appears in the coherence laws in ‘equation’ (84). Is any part of that diagram to be identified with the action of the $\lambda_{ijk}$ or something?