The n-Category Café

Classical Solution of Charged $n$-Particles via Gradient Flow

The two morning lectures were about classical solutions to equations of motion of the charged $n$-particle, for $n=1$ and $n=2$.

First Iskander Taimanov reviewed a long list of (partially very old) results concerning the space of solutions of the Nambu-Goto particle coupled to an electromagnetic field, and propagating on manifolds of various sorts.

The method of choice here is to pick an arbitrary trajectory – one that doesn’t satisfy its equations of motions in that it doesn’t extremize the action functional, and then follow the gradient flow of the action functional, in the hope of reaching the physical solutions this way.

For instance one can estimate the size of the homology groups of the space of solutions this way, things like that.

In the second talk Dennis Koh from Potsdam talked about work on trying to generalize this to $n=2$, the charged string: what can we say about its classical solutions given an arbitrary Kalb-Ramond-gerbe it couples too?

Again, the main idea was to look at the gradient flow. But now of course everything becomes more complicated. At the moment people can only show that the gradient flow exists for short times, but it may blow up later on.

The talk ended with mentioning that one might have to add other terms to the action functional (background gravity + background Kalb-Ramond) in order to regularize the flow.

My obvious guess would be: turn on the dilaton, just like Perelman did. Only difference is that in this case we are not interested in the flow on target space coming from the beta-functional equations induced by the action functional, but in the worldsheet action functional itself.

Prof. Huisken seemed to be interested in that idea…

I was surprised to learn in Taimanov’s talk how much there is to say about the space of classical solutions of the (electromagnetically!) charged 1-particle already, how deep the math involved is – and how huge the open questions still are.

Just imagine what will happen here when we allow 1-particles charged under gauge groups other than $U(1)$, – and when we pass to $n=2$, – and when we do both…

My impression was, with Huisken being in the audience, that with the recent interest in the completion of Hamilton’s approach to the Poincaré conjecture by gradient flow methods also these old “toy models” for gradient flow (namely flows along the gradients of ordinary action functionals of classical particles) are receiving renewed attention.

And what I find striking is that on the one hand we have these world-volume action functionals themselves, those of the string notably, whose gradient flow itself is interesting, since it knows about the classical trajectories of these objects, while on the other hand from these very action functionals one obtains those beta-functional renormalization group flow equations, which are themselves a gradient flow, now of what is called in physics the “background field action”.

Of course from a physical standpoint this is a banality, at least since the discovery of gravity in string theory, but now in light of the fact of the mathematical interest that these gradient flows – on the worldsheet as well as on the corresponding target space – receive, this seems to be rather remarkable, once again.

Well, I have quite generally the impression that with the fast-forward progress of the string theory investigation apparently having slowed down a bit as of late, there are lots of bits and pieces that have long been laid ad acta by hep physicists and which still are awaiting their full investigation and development to sound mathematical structures in their own right.

Okay, enough rambling, On with the report.

Sections and Fibre Integration in terms of Deligne Cohomology

Christian Becker talked about work trying to clarify the notion of

a) sections

and of

b) fiber integration

for abelian $n$-gerbes with connection.

The work reported on is entirely based on using the Deligne cocycle description of abelian $n$-gerbes.

(A class here is nothing buth the descent data (transition data) of a locally trivialized $n$-gerbe with connection, with respect to the local structure $i = \mathrm{id}_{\Sigma^n U(1)}$.)

What is a “section” of such a Deligne class? I believe that the definition that Christian Becker presented is precisely what one gets when one takes the general definition of section of an $(n+1)$-bundle with connection and then restricts everything in sight to be abelian and finally passing to local trivializations and descent data.

So they find the section of an abelian gerbe to exist if and olny if that gerbe is trivializable (globally) in which case the section is the trivializing line bundle.

More generally, a section of an line bundle gerbe (= line 2-bundle, i.e. rank-1 2-vector bundle) exists also if the gerbe has a DD class which is pure torsion, in which case the section is the trivializing gerbe module. (This is what I discussed in my second talk in Toronto).

Of course, these higher rank gerbe modules are “nonabelian” in that they are not visible in terms of ordinary Deligen cohomology.

The second concept Christian Becker talked about was that of fiber integration. It amounts essentially to figuring out what transgression of $n$-bundles with connection means in terms of Deligne cocycles.

The definition Becker presented involved pulling back the Deligne class in the obvious way and then simply integrating all forms that appear in the cocycle separetely. This can’t be the general prescription, and in fact it isn’t. But it does work in simple special cases, some of them useful in applications.

In private discussion with others later on I dared to mention that various open questions here do have natural answers, but met with quite some resistance regarding the fact that this involves higher categories. Some are trying to fight it.

Bi-Branes and Gerbe Bimodules

The topic of the last talk I had mentioned before.

I am not sure yet if I follow the claim that the fusion product can be understood “geometrically” only using gerbe bi-modules. After all, ordinary gerbe modules carry a canonical fusion product just by themselves.

Moreover, in the degenerate case where the gerbe is trivial, we know that the product on its “modules” (which then are just ordinary vector bundles) is nothing but the ordinary product in ordinary (untwisted) K-theory, namely just tensor product of vector bundles.

On the other hand, after the dust has settled one finds that these bibranes that are related to the fusion product do crucially involve the space of flat connections on the 3-holed sphere. And that’s of course precisely the structure which is known to govern the fusion product of these twisted groupoid reps…

One nice application of this bibrane formalism might be, I am thinking, that it seems to provide a way to consistently formulate the sigma-model for the string propagating in “T-folds”, since it allows to formulate the WZW term in cases where parts of the worldsheet map into different, but T-dual, spacetimes.

A “T-fold” is like a manifold, only that one allows for gluing of patches not just diffeomorphism, but also T-duality transformations if both patches can be written as torus bundles. See for instance

C. Hull
A Geometry for Non-Geometric String Backgrounds

(P. P. Cook once made some comments on this work here.)