October 23, 2007

On String- and Chern-Simons n-Transport

Posted by Urs Schreiber

I am making the last preparations for a little journey to Great Britain.

Tomorrow starts the Conference: Lie Algebroids and Lie Groupoids in Differential Geometry in Sheffield. Next Monday then I am invited to speak at the Oxford geometry seminar.

On both occasions I’ll talk about selected topics from

String- and Chern-Simons $n$-Transport
(pdf slides)

in Sheffield with an emphasis on the Baez-Crans type/String-like Lie $n$-algebras and related matters, in Oxford with an emphasis on bundle gerbes.

These slides currently serve for me the purpose of a substitute for our cool-but-non-existing-higher-Wiki and are supposed to be treated as such. They should be comparatively enjoyable to read (on the screen, don’t ever try to to print them) if use is made of the hypertext tools provided by your pdf-reader. (Use the arrow keys to read sequentially, remeber your pdf-reader’s internal back button for convenient hyperlink navigation within the pdf document).

To get going, you might want to surf to section Introduction, subsection Plan and have a look at the menu of links provided there. In sub-subsection Categorfication, local trivialization, differentiation you’ll find an “animated and subtitled” version of the classical transport cube playing the role of a 3-dimensional table of contents.

This classical cube is the one whose first edge is local trivialization, whose second edge is differentiation and whose third edge is categorification. Keeping these three directions in mind should help see the big picture behind the details.

I am looking forward to meeting Bruce Bartlett and Simon Willerton in Sheffield. I had been in Sheffield before only once, about 17 years ago, or so, when I stayed for 2 weeks with a guest family.

Posted at October 23, 2007 7:26 PM UTC

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Re: On String- and Chern-Simons n-Transport

Urs wrote:

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

And don’t forget Eugenia Cheng!

Posted by: John Baez on October 25, 2007 6:08 AM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

But that goes without saying. Next, you’ll be telling him to mind the gap what he steps off the train.

:)

Posted by: James on October 25, 2007 12:38 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Quite. Who could forget Eugenia?

Posted by: Tom Leinster on October 25, 2007 1:45 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

don’t forget Eugenia Cheng!

Did say hi to her. But there was little more time, she is not attending the conference and we are actually not located in Sheffield, but in a (beautiful) village called Blakewell, a few kilometers outside Sheffield.

As time permits, I’ll try to say a bit here about the talks we had. Ulrich Bunke gave a great one on topological T-duality, in which the statement

(Topological) T-duality is 2-group Pontryagin duality.

was made precise. Great stuff. One crucial insight used is that a $G$-torsor is the same thing as an extension of $\mathbb{Z}$ by $G$. This way, in my words, a line 2-bundle (gerbe) overa torus bundle (spacetime) is, locally, a 3-group

$\Sigma^2 \mathbb{Z} \times \Sigma \mathbb{Z} \times Z \,.$

T-duality acting on this line 2-bundle over a torus-fibered space is then nothing but appliying $\mathrm{Hom}_{3Grp}(-- , \Sigma^2 \mathbb{Z})$ to that, which turns it into

Posted by: Urs Schreiber on October 25, 2007 2:27 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Urs wrote:

we are actually not located in Sheffield, but in a (beautiful) village called Blakewell

Are you sure you don’t mean Bakewell? It’s famous for its tarts.

Posted by: Tom Leinster on October 25, 2007 4:03 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

With careful attention to the intended meaning of ‘tart’, mind you…

Posted by: Todd Trimble on October 25, 2007 4:26 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Whilst it might be more famous for its tarts, it is in fact Bakewell Pudding which is the local speciality. As well as the wikipedia article cited above, you can see a picture on the website of the hotel where we’re having the conference, indeed they claim to be responsible for its invention (and give us the option of eating it every night – yummy!)

Posted by: Simon Willerton on October 25, 2007 5:14 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Ooh, look! Look over there — one of the locals is doing that charming “whilst” thing again!

By the way, I just saw a great cartoon. A woman lying on a table in a hospital, in the final stages of labor. Her husband with a video camera aimed between her legs, saying “Push! Push! YouTube has a 10 minute maximum on its video clips!”

It reminded me of the Catsters.

Posted by: John Baez on October 25, 2007 6:02 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Uh…thanks, I think…

Well actually I just almost died laughing. Where’s the link? Don’t tell me it was a cartoon on paper?

Posted by: Eugenia Cheng on October 25, 2007 9:10 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Eugenia wrote:

Where’s the link? Don’t tell me it was a cartoon on paper?

Yeah, alas — it was the “Bizarro” cartoon that appeared in (approximately) yesterday’s newspaper. While some Bizarro cartoons can be found online (e.g. here), I haven’t been able to find this particular one — I can only find old ones. Maybe I’d need to subscribe to DailyINK or something.

It was meant as a compliment. Drs. Cheng and Willerton are trying to deliver an idea in 10 minutes, not a baby, but it’s sort of similar.

Posted by: John Baez on October 26, 2007 1:24 AM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

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 $G$-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.

Posted by: Urs Schreiber on October 27, 2007 6:39 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

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.

Posted by: Urs Schreiber on October 27, 2007 6:47 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

I wrote:

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

(Give or take one dimension here or there.)

Bruce, Simon, Eugenia and I talked about that all night in the pub. Luckily Bruce had John&Jim’s HDA&TQFT with him, so that we could remind ourselves of some things.

Personally, I felt like oscillating back and forth betweenn “oh, it’s obvious” and “argh, I’ll never gonna get it”.

One thing seemed clear though: we want to re-degenerate (I would have said instead “to suspend” but was urged not to) the Tangle hypothesis (p. 23) which says

The $n$-category of framed $n$-tangles in $n+k$ dimensions is equivalent to the free weak $k$-tuply monoidal $n$-category with duals on one object.

and turn it from a statement about $k$-tuply $n$-categories to one about $(n+k)$-categories with only trivial cells up to level $k$ (which is really just the definition of $k$-tuply monoidal, of course).

Anywa, this way the tangle hypothesis seems to be about things generated from something with duals in degree, $k$. And the question is:

how is that related to the freely weakly generated infty category on a single generator in some degree, i.e. to that thing which is supposed to be the infty-category model of a sphere?

Posted by: Urs Schreiber on October 28, 2007 12:44 AM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

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.)

Posted by: Urs Schreiber on October 27, 2007 6:58 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

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?

Posted by: Urs Schreiber on October 28, 2007 1:09 AM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

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?

Posted by: Urs Schreiber on October 28, 2007 1:30 AM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Who talked about Lie n-alg reps up to homotopy?
the topic goes back to at least

“Constrained Poisson algebras and strong homotopy representations,” Bull. Amer. Math. Soc.

19 (1988), 287-29?

Posted by: jim stasheff on October 29, 2007 1:17 PM | Permalink | Reply to this

Re: On String- and Chern-Simons n-Transport

Who talked about Lie $n$-alg reps up to homotopy?

Marious Crainic did. He said he’ll be able to provide a write-up of whathe talked about in a few weeks.

Actually, he only talked about repsof ordinary Lie algebras, but on complexes of vector spaces.

I really have the suspicion that this mustbe related to qDGCAs concentrated in degree +1, and then -1, -2, and so on, with the differential on the +1 part simply the Chevalley-Eilengerg differential of degree +1.

Then any way you’d extend the CE differential to the full qDGCA will produce something that looks like a Lie algebra acting on the complex of vector spaces that is sitting in the negative degrees.

But I didn’t follow Crainic’s details closely enough to be able to check if this is what he is essentially doing. My suspicion is that it should at least be related. What do you think?

Posted by: Urs Schreiber on October 29, 2007 1:27 PM | Permalink | Reply to this
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