The String Coffee Table

Hi Jens,

sorry that I didn’t come by your office again yesterday to say good-bye. But I’ll try to come next Thursday to your talk, even though it is only the second of two parts. This stuff sounds interesting.

So, let’s see. Say we have a $K(\pi,1)$ target space and a 2D HQFT. Assuming we have foliated our surface, then a homotopy class of maps from it into $K(\pi,1)$ assigns an element of $\pi$ to every leaf.

Hm, in which sense is that

a TFT for manifolds with principal $\pi$-bundles

?

I am not and have never been a student or researcher of String Theory

As for me, as long as you have something interesting to say about physics or mathematical physics, or maybe just math, I’d be interested. Right now the SCT is not threatened by being flooded by off-topic posts. If that should ever change we might want to talk about restricting the topics a little, but right now I do not see the need.

In particular, if you really take a lot of notes at the Steetfest and want to share them they should be very welcome here. While not explicitly about strings, maybe, there is a wealth of interesting relationships between the topics discussed in Sydney and string theory.

Ah, ok, now we are on the same page.

Sure, a smooth discrete groupoid is nothing but a copy of a Lie group $G$ at every point of some manifold $M$.

Similarly, a smooth discrete $p$-groupoid is nothing but a copy of a Lie $p$-group at every point of some manifold.

On the other hand, a principal $G$-bundle over $M$ has a $G$-torsor over every point of $M$. But locally it does look like a bundle of groups.

Similarly, a principal $G_p$-$p$-bundle $E$ over a categorically trivial base space really has a $G_p$-$p$-torsor $E_x$ associated to every point $x \in M$ and locally, over contractible patches $U \subset M$, it looks just like $M \times G_p$, which we can regard as a smooth discrete $p$-groupoid.

What is kind of interesting is that the description of $p$-bundles with $p$-connection and $p$-holonomy that I am talking about also involves morphisms between groupoids, but apparently in a somewhat different way than in the bundle gerbe context. I am inclined to say in a more transparent way, but I might be biased. ;-)

So actually it seems that a principal $G_p$-$p$-bundle $E$ with $p$-connection and $p$-holonomy over a base manifold $M$ is nothing but a smooth (I am glossing over a technicality here) $p$-functor

that assigns $n$-morphisms in the category of $p$-torsors of the structure $p$-group $G_p$ to n-dimensional hypermvolumes inside $M$ in the $p$-path $p$-groupoid $\mathcal{P}_p(M)$.

This is the most banal thing that one could say about $p$-holonomy, and yet it says everything there is to say. This is the content of the notes mentioned above.

I should mention that John Baez was telling me all along that this must be true. Still, it is kind of interesting to see how it works out and for instance gives rise to the cocycle description of a (fake-flat) nonabelian gerbe with connection and curving.

Anyway, my point is that $\mathcal{P}_p(M)$ (the $p$-category whose objects are points in $M$, whose morphisms are paths in $M$, whose 2-morphisms are surfaces in $M$, and so on) as well as $G_p-p\mathrm{Tor}$ are both $p$-groupoids and hence a principal $p$-bundle with $p$-connection is nothing but a $p$-groupoid morphism.

This says nothing but that when you compute $p$-holonomy along some $p$-dimensional surface which retraces itself, you have done nothing - up to equivalence. This, again, is a most banal statement, really.

Locally, as I show, all this is described by simplicial maps of the sort shown in the third figure of the previous entry. This says that locally this $p$-groupoid morphism is given by $p$-functors from $p$-paths to $G_p$ itself, which are glued on $(n+1)$-fold overlaps by $p$-functor $n$-morphisms.

I find it interesting that this description in terms of local $p$-functors ,glued together on overlaps, easily generalizes away from groupoids. Without much ado I can write down the same simplicial map for much more general $p$-functors that do not need to take values in a $p$-group.

For instance they might take values in a $p$-groupoid. (Recall that I am now talking about the local functors which previously took values in a $p$-group only.) I show in section 13 of my thesis that the differential version of such a gadget is related to the algebroid Yang-Mills theory that have been considered by Thomas Strobl.

I don’t know if this situation still gives rise to a nonabelian gerbe. The Breen & Messing definition of nonabelian gerbes is very general.

But the target of these local functors does not have to be a $p$-groupoid either and still the whole thing makes sense and is certainly no longer described by any gerbe. I wonder if there are any interesting examples of such more general ‘$p$-bundles with $p$-connection and $p$-holonomy’.

One thing that might go wrong, though, is the independence of the local description of the choice of good covering. For the $p$-group case this independence follows easily from the equivalent intrinsic description in terms of $\mathrm{hol} : \mathcal{P}_p(M) \to G_p-p\mathrm{Tor}$. But this no longer works when we do away with weak invertibily.

All right, I am rambling on here and should stop at some point.