## May 19, 2009

### TFT at Northwestern

#### Posted by Urs Schreiber

Quite unfortunately I couldn’t make it to this event that started yesterday:

Topological Field Theories at Northwestern University
Workshop: May 18-22, 2009
Conference: May 25-29, 2009
(website, Titles and abstracts)

An impressive concentration of extended TFT expertise.

But with a little luck $n$Café regulars who are there will provide the regrettable rest of us with reports about the highlights and other lights

In fact, Alex Hoffnung already sent me typed notes that he had taken in talks! That’s really nice of him. I am starting to collect this and other material at

Posted at May 19, 2009 1:57 PM UTC

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### Re: TFT at Northwestern

Somebody stop me, I feel an unseemly rhetorical flourish coming on. “This conference will go down in history as the greatest demonstration for topological field theory in the history of at least one nation”.

Posted by: Bruce Bartlett on May 19, 2009 4:30 PM | Permalink | Reply to this

### Re: TFT at Northwestern

I can’t possibly top Bruce’s enthusiasm and I also do not know how to load my notes onto the nLab as Urs did yesterday. Both of these have something to do with a lack of sleep.

Since there may be at least one person who wants to read these notes as they come out, I have posted them to my webpage. I think you can just click on my name below to get there.

I will try to post notes from Toen’s talk in a bit. Christoph Wockel gave a nice talk today but I was not able to take notes worth putting up here since I was stuck in the back and could not see well. So maybe someone else will put those up.

Also Evan Jenkin’s is posting notes at his website. They are handwritten which means they have pictures. I tried to make pictures, but failed/got lazy so take a look at his here.

Posted by: Alex Hoffnung on May 20, 2009 2:51 AM | Permalink | Reply to this

### Re: TFT at Northwestern

David Ben-Zvi and Orit Davidovich pointed me to some videos and notes of lectures by Jacob Lurie (and while you are there Graeme Segal) that will be of interest to people.

Posted by: Alex Hoffnung on May 20, 2009 3:50 AM | Permalink | Reply to this

### Re: TFT at Northwestern

A veritable bombardment of free beautiful geometry. I am so lucky to be living in this time period; ten years ago if you couldn’t make it to a conference you would miss out big time. You still do of course, but now at least a life buoy is thrown out to you and the skipper shouts out valiantly No one left behind on my watch!

Posted by: Bruce Bartlett on May 20, 2009 5:45 PM | Permalink | Reply to this

### Re: TFT at Northwestern

Thanks for your efforts, Alex! Much appreciated around here.

I also do not know how to load my notes onto the nLab

It would be great if you and others could help me with the task of uploading your notes as well as linking to whatever other material deserves to be linked to at [[Northwestern TFT Conference 2009]], because I am running a bit short of time.

How to upload files is described at

you type the symbols

[[NiceNotesByAlex.ps:file]]

into the edit box, then submit the entry. It will appear featuring a greyish box with a clickable question mark at the point where the link is to appear. Then click on that question mark and be directed to a file upload dialog.

Posted by: Urs Schreiber on May 20, 2009 8:20 AM | Permalink | Reply to this

### Re: TFT at Northwestern

Christoph Wockel has been nice enough to send me some notes from the conference so that I do not feel totally left out. I will post them at the nLab with the other notes as he sends them. You can get to the page by the link Urs posted above.

Posted by: Alex Hoffnung on May 28, 2009 4:20 PM | Permalink | Reply to this

### Re: TFT at Northwestern

Thanks Alex, thanks Christoph!

Do you know if there is any further material online available on Kevin Costello’s talk, specifically?

Posted by: Urs Schreiber on May 28, 2009 4:56 PM | Permalink | Reply to this

### Re: TFT at Northwestern

Yes, thanks Alex and Christoph. Urs, it Evan Jenkins’ notes also cover Kevin’s talk. Unfortunately, not enough for me to understand the defintion of what a “factorization algebra” is… though I suspect that would have been the case even if I had been there! Might be important though, a more elegant way to approach chiral algebras, conformal field theory, etc.

Posted by: Bruce Bartlett on May 28, 2009 5:07 PM | Permalink | Reply to this

### Re: TFT at Northwestern

not enough for me to understand the defintion of what a “factorization algebra” is…

By the way, directly relevant is

section 4.1 “topological chiral homology” of Lurie’s TFT thing.

Posted by: Urs Schreiber on May 28, 2009 5:48 PM | Permalink | Reply to this

### factorization algebra

Unfortunately, not enough for me to understand the defintion of what a “factorization algebra” is…

But let me see, is there a secret here? It seems to be a pretty straightforward definition. I may be missing something, let me know:

It seems to me that a factorization algebra is just an algebra (in Vect) over a version of the little disk operad for which all these little disks come with an embedding into a fixed given manifold.

So, for each $n$-disk image $D \subset X$ in your manifold $X$ there is a vector space $V_D$, and for all collections of such disks $D_1, \cdots, D_k \subset D \subset X$ sitting inside a single big little disk $D \subset X$ there is a linear map

$V_{D_1} \otimes \cdots \otimes V_{D_k} \to V_D$

and these satisfy the obvious operadic compatibility condition that says that when you have disks inside disks inside … inside one big disk, it doesn’t matter in which order you map everything finally to the outermost $V_D$.

So, the whole things is like a $n$-fold commutative higher algebroid with one set of elements per disk in $X$.

The best way to think of this probably is as a TFT all by itself on “genus 0” /tree-levl cobordisms embedded in $X$.

In Lurie’s section 4.1 is actually a simpler special case of this, that where these algebroid elements don’t actually depend on the position of the disks inside $X$.

See the displayed equation in the middle of his page 90: he looks at such beasts obtained from just algebras of the little disk operad by taking a collection of disks in $X$, forgetting that they sit in $X$, and just assigning objects to them.

I am guessing (but will have to remind me) that the original Beilinson-Drinfeld definition looked slightly more complicated, as they used (unless I am misremebering) not open disks but points, so that everything is a bit more singular.

Posted by: Urs Schreiber on May 28, 2009 9:36 PM | Permalink | Reply to this

### Re: factorization algebra

That’s a nice description of something, but why is it called a factorization algebra?

as for
as they used (unless I am misremebering) not open disks but points,

you need disks for the operad structure
but the spaces of configs (of disks or points) have the same homotopy type

a remark that’s standard in the old fashioned *topological* operad theory

why might one want to look at *embeddings* of th elittle disks into some target?

Posted by: jim stasheff on May 29, 2009 12:48 AM | Permalink | Reply to this

### Re: factorization algebra

The definition Kevin uses is equivalent to a nonholomorphic/nonalgebraic version of Beilinson-Drinfeld’s.. the passage between discs and points is related to the equivalence between the $E_n$ and Segal machines for describing iterated loop spaces. The main issue is rather the dependence on the positions of points. In the topological setting (where Jacob writes), our spaces depend locally constantly on the insertions, and so we see an $E_n$ algebra structure on local operators in TFT.

In B&D’s setting of 2d CFT, the sheaves of local operators are no longer locally constant, so we get a more refined notion than that of $E_2$ algebra, namely that of vertex algebra. But I think it’s best to try to understand vertex algebras as simply holomorphic refinements of $E_2$ algebras, as this picture suggests.. One of B&D’s nice observations (explained in detail in the book by Frenkel and me) is that the unit axiom in a factorization algebra forces all the sheaves to have flat connections over the configuration spaces of points, ie to be D-modules. Thus the main difference from their POV between TFT and CFT is whether this D-module is actually integrable, ie a local system.. thus a vertex algebra is a nonintegrable analogue of an $E_2$ algebra!

Finally Kevin’s goal is to describe operators in general QFTs, for which he’s advocating the same formalism, just in the $C^\infty$ setting — ie the OPE of operators is captured by a flat but nonintegrable version of an $E_n$ multiplication.. of course the content of his talk is in actually being able to construct all this structure from perturbation theory!

Posted by: David Ben-Zvi on May 31, 2009 9:54 PM | Permalink | Reply to this

### Re: factorization algebra

embeddings into the target locally constant?

and whence the differential operators ,–> D-module?

Posted by: jim stasheff on June 1, 2009 1:54 PM | Permalink | Reply to this

### Re: factorization algebra

Thanks for the details, David.

If I understand the schedule correctly, I’ll hear Kevin Costello talk about this in Oberwolfach tomorrow. If time permits I’ll try to $n$Labify things as far as possible. Your comments will be helpful.

BTW, I notice that what Stefan Hollands and Robert Wald are trying recently (0802.2198) is very similar (if not equivalent) to these factorization algebra approaches to QFT.

Posted by: Urs Schreiber on June 7, 2009 3:58 PM | Permalink | Reply to this

### Toën, elliptic?

I started looking at Alex’s notes on Toën’s lectures.

Does Toën say anything about a relation to elliptic cohomology of his secondary K-theory? The original conjecture (Baas, Dundas, Rognes, etc.) was that whatever 2-K-theory is, it gives a way to think of elliptic cohomology.

Posted by: Urs Schreiber on May 28, 2009 9:23 PM | Permalink | Reply to this

### Re: Toën, elliptic?

Well, Baas et al. have very concrete attempts at finding various variants fo geometrically represented 2K theories wehatver you call them, but the general philosophy that one should look for some loop or 2-dimensional analogues of K-theory classes is I think earlier than any of their work in the field and is due Graeme Segal, as exposed in his Bourbaki seminar with further acknowledgements to Atiyah, Witten etc.

Posted by: Zoran Skoda on June 7, 2009 6:06 PM | Permalink | Reply to this

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