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June 8, 2009

Strings, Fields, Topology in Oberwolfach

Posted by Urs Schreiber

This week’s workshop at MFO is on Strings, Fields, Topology. We started collecting notes and other material at

Oberwolfach Workshop, June 2009 – Strings, Fields Topology .

This includes today

- Christoph Schweigert and Ingo Runkel on [[CFT]] and algebra in modular tensor categories;

- Dan Freed on [[differential cohomology]] of [[string theory]] [[background fields]];

- Kevin Costello on quantum field theory in terms of [[factorization algebras]].

Posted at June 8, 2009 11:30 PM UTC

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Tuesday

I have added to [[Oberwolfach Workshop, June 2009 – Strings, Fields, Topology]] today’s talk notes (Tuesday, June 9) on

- Ulrich Bunke and Thomas Schick lecturing on their models for [[differential cohomology]] and in particular differential K-theory

- after that we heard part II and III of Kevin Costello’s lectures on his work on quantum field theory, in terms of [[BV-theory]] and [[factorization algebras]]

This is impressive stuff. The notes on the nnLab can hardly convey the full picture here. You are supposed to switch to reading his book on renormalization, too:

Kevin Costello, Renormalization and the Batalin-Vilkovisky formalism (web)

This is important stuff. Read it.

Posted by: Urs Schreiber on June 9, 2009 7:44 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

Can someone who is there clarify how Costello’s B(M) is a set of colors?
is it the big disk that is most relevant?

Posted by: jim stasheff on June 9, 2009 9:36 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

Hi Jim,

you wrote:

Can someone who is there clarify how Costello’s B(M)B(M) is a set of colors?

So the point is that B(M)B(M) contains not just the abstract disk D nD^n, but maps ϕ:D n→M\phi : D^n \to M (probably taken to be embeddings) into MM. So the operad does not just have the single color given by the abstract disk, but one color per map ϕ:D n→M\phi : D^n \to M.

In other words, there is (precisely )one kk-ary operation in the operad per (k+1)(k+1)-tuple of maps ϕ i:D n→M\phi_i : D^n \to M with the ϕ 1≤i≤n\phi_{1 \leq i \leq n} factoring through the ϕ k+1\phi_{k+1} and non-intersecting.

This kk-ary operation parameterized by all the disks in MM is a close cousin of the familiar operator product in vertex operator algebras.

In fact, the idea is that as we let the disks shrink to points, and make everything depend holomorphically on some complex structure on a 2-dimensional MM, this does becomes precisely the operad whose algebras are vertex operator algebras, with its characteristic dependency of the binary operation on a complex parameter. That parameter is the remnant of the different “colors given by disk embeddings” of Costello’s factorization algebras.

(Though in the talk there was some discussion of this point, it didn’t quite become clear if the intended precise statement here has been formulated already).

Posted by: Urs Schreiber on June 10, 2009 12:17 AM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

Urs responded (and I interleave)

So the point is that B(M) contains not just the abstract disk D n, but

ALL (appropriate)

maps $D^n \to M$ (probably taken to be embeddings) into M. So the operad does not just have the single color given by the abstract disk, but one color per map .

This k-ary operation parameterized by all the disks in M is a close cousin of the familiar operator product in vertex operator algebras.


In fact, the idea is that as we let the disks shrink to points, and make everything depend holomorphically on some complex structure on a 2-dimensional M, this does becomes precisely the operad whose algebras are vertex operator algebras, with its characteristic dependency of the binary operation on a complex parameter. That parameter is the remnant of the different colors given by disk embeddings.


Jim: the classical little disks operad was geometric - parameterized by the center of the little disk and the radius

a holomorphic analog is `obvious’ but has it been written somewhere?

Posted by: jim stasheff on June 10, 2009 2:21 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

a holomorphic analog is ‘obvious’ but has it been written somewhere?

Apparently it hasn’t been written out yet. In fact, in the talk there was quite a bit of discussion of this point. The punchline seems to be that while it looks very obvious, it may require still a little care.

But see also David Ben-Zvi’s comment here.

Posted by: Urs Schreiber on June 10, 2009 3:02 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

Firstly – Urs, thanks for your kind comments on my talks, and for posting notes.

I think the main problem with the discussion of holomorphic factorization algebras in my talk on Monday was that I got the definition wrong (which is a little embarassing). However, I think it’s not hard to give a reasonable definition (using, for instance, parametrized holomorphic discs embedded in a Riemann surface).

The link between vertex algebras and holomorphic factorization algebras is little more than an analogy write now: I think it would be hard to prove a precise theorem (the best results I know along these lines are those of Huang, who shows that some version of Segal’s CFT axioms are equivalent to vertex algebras).

All of these definitions of factorization algebra could be regarded as a little tentative. For us, the main aim was to come up with a definition which encodes all the salient properties of the examples we construct using perturbation theory. However, there are many ways to modify the technical details of our definitions in such a way that they still encompass our examples.

Kevin

Posted by: Kevin Costello on June 10, 2009 9:39 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

There is an equivalence of categories between vertex algebras (in the original definition) and G_a equivariant factorization algebras on A^1 (using ordinary differential operators), and it’s not too hard to write down, given the existing literature. Factorization for higher genera is trickier, and any equivalence with vertex algebras seems to require additional conditions like O_X-flatness.

Posted by: Scott Carnahan on June 17, 2009 8:34 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

For those of us who can’t be there,
these real time notes are a wonderful gift. Thanks

Posted by: jim stasheff on June 9, 2009 9:37 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

For those of us who can’t be there,

these real time notes are a wonderful gift. Thanks

Thanks for the feedback!

Posted by: Urs Schreiber on June 10, 2009 12:20 AM | Permalink | Reply to this

Wednesday

This morning we had

- Alexander Kahle on superconnections and index theory (I have uploaded reaonably readable notes on that)

- Gabriel Drummond-Cole about the bigger ∞\infty-operadic story behind the Baranikov-Kontsevich passage betweem BV and hypercommutative operads. This was a pretty cool talk, but unorthodox enough to prevent me from taking any coherent typed notes on it.

Maybe Bruce Bartlett or somebody else will be so kind to upload his or her handwritten notes.

- Scott Wilson on categorical algebra, and generalized Hochschild cohomology

as far as the speaker got, this has large overlap with the material recalled at a past blog entry Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

This afternoon is of course the traditional hike (and a “Hopkins event” soccer match, as well as the “Hopkins event” “explain the Hopkins Kervaire invariant 1 proof” ).

So no more notes today. In particular since I need to prepare for my unexpected talk tomorrow…

Posted by: Urs Schreiber on June 10, 2009 11:34 AM | Permalink | Reply to this

Re: Wednesday

Here are some notes I wrote to remind myself what to say tomorrow. Needs trimming and polishing, but it’s a start:

Background fields in twisted differential nonabelian cohomology

Posted by: Urs Schreiber on June 11, 2009 12:07 AM | Permalink | Reply to this

Thursday

There are now notes uploaded for André Henriques’ talk todday on his study of a 3-category of conformal nets with Chris Douglas and Michael Hill.

I just quote Mike Hopkins’ comment after the talk

This stuff is terrific.

If you are interested you should have a look at the notes in preparation that André provides on his webpage:

Chris Douglas, Andr´ Henriques, Michael Hill, Geometric String structures – notes on ℤ 2\mathbb{Z}_2-graded conformal nets (ps)

Posted by: Urs Schreiber on June 11, 2009 10:11 PM | Permalink | Reply to this

Re: Thursday

Though Andre and I did work with Mike Hill on string structures, the above project on conformal nets is rather work with Arthur Bartels.

Thanks for linking to the workshop notes and draft.

Posted by: Chris on September 15, 2009 6:58 AM | Permalink | Reply to this

wrapup

Bruce Bartlett has now uploaded lots of pdfs with scans of his handwritten notes on the talks, and has typed a list of abstracts into the the entry.

Based on that I have in turn started equipping these abstracts with cross-links. Many of them to currently non-existing entries, indicated by the gray shading. I think this is a good opportunity and motivation to start creating the corresponding entries. For many of them the conference notes themselves provide a first bit of content.

For instance I am going to move large parts of the notes for the lecture by Thomas Schick on differential cohomology to the corresponding entry differential cohomology, etc.

There are lots of other gray-ish links now which I was and am planning to create an entry for, but likely won’t find the time soon.

Have a look at the grayish links and see if any of them inspire you to click on the green question mark and start providing a bit of content for these keywords!

Ideally some of the speakers will feel sufficiently appalled by the insufficiency of the material at the links provided with their talk notes to give it a go themselves.

Just imagine this ideal world where every online abstract and set of notes on a talk comes equipped with its linked list of keywords to nnLab entries explaining this stuff. There is nothing to stop us from going to that world…

Posted by: Urs Schreiber on June 13, 2009 4:37 PM | Permalink | Reply to this

Re: wrapup

For those of us (hopefully not just me) who like to print things out, it would be nice to have [[abstracts]] as a separate `page’

jim

Posted by: jim stasheff on June 14, 2009 3:19 PM | Permalink | Reply to this

Re: wrapup

Posted by: Eric on June 14, 2009 5:39 PM | Permalink | Reply to this

Re: wrapup

Thanks, Eric.

If you have a minute, could you add cross links between the list of abstracts and the rest of the material. And add the conference identification information to the list of abstracts, such that if there is another Oberwolfach conference this year whose abstract we nnLabify it all makes sense to the reader.

Generally, we should add a new wiki-“category” “conference” or the like.

I am in a haste and busy with something else. Otherwise I’d do it myself.

Posted by: Urs Schreiber on June 14, 2009 5:56 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

I have done a big editing upgrade of the notes from Oberwolfach page on the nLab. I have a big debt to pay to all the other note-takers around the internet whose notes have been a big help to me! (A certain Herr Ben-Zvi stands out here, as does a certain Herr Schommer-Pries).

The notes are now arranged in a single page, with clickable titles on top which take you down to the talk nsummary at the bottom.

I don’t want to call them ‘abstracts’, because they are ( mostly :=) ) not the speaker’s actual abstract but rather my and other nLab editors personal impression of the talk, equipped with hyperlinks to other nLab pages on those topics. That may seem like a silly or pretentious distinction, but it gives the whole thing a reason to be on the nLab, as opposed to just a webpage of the conference of some kind. The info is meant to be integrated into the nLab.

A highlight of the conference was of course Mike Hopkins’ talk on the history behind the Kervaire invariant, although he has spoken about this elsewhere. It was exciting to see him even speculate about the existence of those exotic framed manifolds to exceptional Lie groups like E8! Or at least to some kind of ‘complex structure’, see the last page of these notes. I think John will appreciate that :-).

Posted by: Bruce Bartlett on June 15, 2009 2:02 PM | Permalink | Reply to this

Re: Strings, Fields, Topology in Oberwolfach

Thanks, Bruce, great.

We just talked about this in private, but I want to say it here in public, too:

one good thing about having this stuff on the nnLab is that eventually it helps us weave that web of links that’s gonna become our joint online super-brain (to be distinguished from the super-brane that John is looking into), if you allow me that bit of pathos.

With useful talks Labified, we can link from them to nnLab entries explaining the stuff there, link from entries to talks as furether references, incorporate pieces of talks entirely into nnLab entries and so forth.

I notice that researchers start to add references to nnLab entries concerning their work, when they see these are missing. Ideally eventually they’ll also feel motivation for and gain by adding material itself. That’d be the win-win situation to strive for.

Posted by: Urs Schreiber on June 15, 2009 2:25 PM | Permalink | Reply to this

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