The n-Category Café

Urs wrote:

I am looking forward to meeting Bruce Bartlett and Simon Willerton in Sheffield.

And don’t forget Eugenia Cheng!

Back in my hotel in Sheffield. Some time to kill before meeting Bruce and Simon on some Bluegrass festival tonight. Should probably rather be taking a nap, exhausted as I am, but anyway.

Was a very good conference. Here some random impressions, from memory (i.e. without any notes in front of me).

I had my personal déjà vu when André Henriques announced his talk on connections on String bundles.

In fact, André reported on some ideas that Jacob Lurie had on this topic – inspired by having heard Johan Baez talk about it in oslo recently – , and on a conversation he had with him about it.

They are coming from a homotopy-theorist’s perspective as far as the total space of the 2-bundle is concerned, and use a simplicial version of Breen-Messing’s version of Anders Kock’s version of Lawvere’s version of synthetic differential geoemtry as far as the connection is concerned.

It was noteworthy to see how they encountered the fake-flatness constraint using the direct application of their notion of 2-connection. They did’t address it as such, but noticed that the 2-connection on the String bundle forces the underlying $\mathbb{Z\G$-bundle it is lifted from to be flat. But of course that’s precisely what fake flatness implies for the structure 2-group $\mathrm{String}_k(G) = (\hat \Omega_k G \to P G)$.

To circumvent this, they thought of some workaround which they addressed as a one-and-a-half connection.

I very much enjoyed discussing these things with André. I was glad to finally have met him.

Concernign a discussion we had last night:

Question: what is the relation between $k$-tangles and $k$-spheres?

If I remember correctly what John was telling me in Vienna a few months ago, the $k$-sphere can be thought of as being modelled by the weak $\omega$-ctageory which is freely weakly generated on a single weakly invertible generator in degree $k$.

That’s supposed to be one way to see why the reason why sphere spectra are so intricate while Eilenberg-MacLane spaces are so easy, homtopy-theoretically: the latter are strictly generated on a single invertible generator in some degree.

I was thinking: try to write down what a weakly generated thing on a dualizable generator looks like and pass from globular to the Poincaré-dual string diagrams. Doesn’t the result look precisely like a generators-and-relations description of the $k$-tuply monoidal category of (framed, maybe) $k$-tangles in sufficiently high codimension?

Or something like that.

Anyway, if this is anywhere close to being right, it would seem to imply some interesting general nonsense about topological field theories and sphere spectra.

Ulrich Bunke shocked me with telling me that it is well known that the Witten genus of $X$ is the genus of the Dirac operator on $X$ which is twisted by a bundle of vertex operators, with the latter being essentially nothing but the bundle associated to a $\hat Omega_k G$-bundle.

He pointed me to some book from the 90s where this is described, but I don’t have my notes here right now.

While this sounds most plausiby, I was shocked to hear that this is already known, since my impression had been all along that it is precisely a statement of this sort which we would like to prove in order to finally make the Witten genus computation in terms of Dirac operators precise. (Just a few days ago I had hear Wurzbacher talk about this issue and seem to have understood that he said that a good answer to this is not available.)

Anyway, this is fun: the Witten genus should really be the index of a Dirac operator twisted by a String 2-bundle, whatever that means in detail – but we know, using the fact that $\mathrm{String} = |(\hat Omega_k G \to P G)|$ and using the canonical 2-rep, that such a string 2-bundle has pretty much precisely the desired vertex operator algebras as fibers.

So we are pretty close here to making precise a cool statement:

The Witten genus is the index of a Dirac operator twisted by a String 2-bundle.

To make this come out real smoothly we may need to spend a thought or two on categorified Clifford algebra. But it looks to me like this should actually nicely match with the stuff we were talking about concerning this point recently. But we’ll see. (Sorry, no links, I am working on some stupid slot-machine terminal.)

Lots of talks in this conference of NPQ-manifolds, Lie $n$-algebroids, groupoids and the like, of course.

The following question seems to stare in our faces. I am not aware that anyone has tried to answer it:

Isn’t it striking that all the information in a weak $\omega$ Lie groupoid, after we pass to its Lie algebra, is entirely encoded in a simple homological statement saying that there is dome $D$ such that $$D^2 = 0 \,.$$

Shouldn’t that make us wonder? Since the concept of $\omega$-groupoid itself didn’t have quite concice definition.

So, let’s see. An $\omega$-groupoid is a Kan complex. Which is a simplicial complex.

And with a cell complex, we have$\partial^2 = 0$.

On the other hand, the $D^2 = 0$ controlling the world of Lie $\omega$-algebras is not quite saying “the boundary of a boundary vanishes”.

It is rather saying something like:

the filler of a filler vanishes.

In some sense. Does that ring any bell with anyone?

We also heard a talk on weak reps of Lie $n$-algebras “up to homotopy”.

I am wondering: take a qDGCA which has the Chevalley-Eileneberg algebra of a Lie algbra in degree +1, as usual, and apart from that only stuff in negative degree, say from -1 to -$n$.

A quick inspection shows that any degree +1 nilpotent differential on such a best looks a lot like a weak action of the Lie algebra on the $n$-vector space which is given by the $n$-term chain complex that we effectively have in the negative degrees.

That would make good sense. But I should sit down and check the formulas. Or did anyone already?