May 14, 2008

HIM Trimester Geometry and Physics, Week 1

Posted by Urs Schreiber I had thought I could get some work done here, but I hardly find the time to reply to my email.

But there are lots of things that would deserve blogging about.

The main two activities currently are one daily seminar on supergeometry in preparation for the Stolz-Teichner program next week, the other is a daily inernal seminar where we teach each other whatever comes to mind.

Vertex operator algebras from Segal’s axioms

In the internal seminar I very much enjoyed a talk by Liang Kong on how vertex operator algebras “arise from Segal’s CFT axioms” if you massage those a bit. This is due to theorems by Huang and Huang-and-Kong, and gives a rather nice geometric description of the otherwise rather unwieldy definition of a VOA. The following literature information thanks to Liang Kong:

The main statement is in Yi-Zhi Huang’s book:

Two-dimensional conformal geometry and vertex operator algebras Progress in Mathematics, Vol. 148, Birkhauser Boston, 1997.

There is an online paper cover the main result in the book: arXiv:q-alg/9504019.

For the Theorem that proper completion of VOA gives algebra over a true operad, read the following two papers: arXiv:math/9808022, arXiv:math/0010326

For the open-string case: arXiv:math/0308248

For the open-closed case and higher genus: arXiv:math/0610293

Supergeometry

We are following Deligne-Morgan, Quantum fields and strings, a course for mathematicians (#). Today it was my turn with talking about integration over supermanifolds. While I followed it, I was a bit unsatisfied with their presentation, I must say. Here is my attempt:

Integration over Supermanifolds
(pdf, 6 pages)

Afterwards I had some discussion about supergroupoids. I was talking about integrating super $L_\infty$-algebras to super $\infty$-groupoids recently in

On action Lie $\infty$-groupoids and action $L_\infty$-algebroids
(blog, pdf)

but there I use a notion of superspace more general than that of supermanifolds.

The only reference mentioning the obvious idea of looking at groupoids internal to supermanifolds that we could find is

Rajan Mehta
Supergroupoids, double structures, and equivariant cohomology
(arXiv:math/0605356).

That thesis also consider groupoids internal to “Q-manifolds” (which I would call groupoids internal to $L_\infty$-algebroids).

Incidentally, here is a related question that we happened to come across also in discussion with Constantin Teleman over tea:

when you google “supergroupoid” you get precisely two non-identical hits: the definition by Mehta (groupoid internal to supermanifolds) and the definition by John Baez (groupoid with $\mathbb{Z}_2$-grading).

Similarly, when asked what can be said about his theorem when groups are replaced by supergroups, Constantin Teleman assumed a supergroup is a group with a $\mathbb{Z}_2$-grading. Other people would think it is a group internal to supermanifolds.

“Supergroups” in the sense of groups with a $\mathbb{Z}_2$-grading also appear in the nice proof of Doplicher-Roberts.

While it might seem like a clash of terminology, we know, on the other hand, that super-things in the first sense tend to live in supercategories in the second sense! Super D-brane categories, for instance, live in supercategories in the second sense, as I described in Lazaroiu on $G$-flows on categories.

So something interesting is going on. But I am not completely sure yet what.

Posted at May 14, 2008 6:02 PM UTC

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Re: HIM Trimester Geometry and Physics, Week 1

Thanks Urs this is really useful.

Posted by: Bruce Bartlett on May 15, 2008 12:58 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Hi Bruce,

I wish you could be here, we’d have lots of things to talk about. But I try to keep you posted with the interesting stuff.

Yesterday Orit Davidovich gave a very nice first part of an “Introduction to modular tensor categories” where in the list of examples she went through many of the finite group constructions (Simon’s result on the Drinfeld double, in particular) which you know and love.

On the other, I couldn’t make it to Categories, Logic and Physics, II in London yesterday. Did you give your talk? I’d be very interested in a summary of the highligts of the meeting. Fancy a guest post? :-)

Posted by: Urs Schreiber on May 15, 2008 6:58 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Yes thanks the conference went well, it was well attended, and it looks like the next one might be over a weekend at Oxford (hooray!). The talks will hopefully be available on the webpage soon; maybe we can discuss more then.

Posted by: Bruce Bartlett on May 16, 2008 5:18 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

the webpage

Looking at this webpage, I see that you didn’t give a talk about functorial QFT after all?

Concerning Hiley’s talk: how does he get a Clifford algebra from a groupoid?

Posted by: Urs Schreiber on May 16, 2008 6:29 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

I have a question about super things, is this stuff just sitting inside some other structure? it seems like a special ase of something, like some sort of equivariant theory. Also, it seems sort of DG, in the sense that these superalgebras are just simple chain complexes with “full” cohomology, and that while this is apparently the right thing to use, that maybe this is some sort of “trivial” case. maybe this is a nonsense comment, but i would be curious to know what you think.
ps my boss has that book you guys are going through, deligne morgan et al, I am not so great at such stuff, if you have anything else typed up, it would be awesome if you were willing to share it!

thanks so much
sean

Posted by: sean tilson on May 15, 2008 3:12 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

This deserves a more detailed reply than I have time for right this moment, so just quickly one remark on this:

Also, it seems sort of DG

Withouth the D, yes! :-)

Categorification introduces $\mathbb{N}$-grading and $D$s, superification introduces $\mathbb{Z}_2$-grading and no $D$s.

More precisely:

the algebra of functions on a supermanifold is:

$\wedge^\bullet_A V \,,$

where $A = C^\infty(X)$ is an algebra of smooth functions over a smooth manifold $X$, and where $V$ is an $A$-module in odd degree.

People say we have an “N-super manifold” if $V$ here is actually $\mathbb{N}_+$-graded.

They say “NQ-super manifold” when $V$ is $\mathbb{N}$-graded and $\wedge^\bullet_A V$ is equipped with a degree +1 graded derivation of $\mathbb{N}$-graded-commutative algebras over the ground field $d : \wedge^\bullet_A V \to \wedge^\bullet_A V$ with $d^2 = 0$. I prefer to call this latter structure an $L_\infty$-algebroid (here for more details) because this is the $\infty$-categorification and many-object version (“oidification”) of a Lie algebra.

I find the similarity between superification and categorification striking. It must mean something, but I am not exactly sure yet what. Looking at the above, it seems as if superification corresponds to a notion of “category” where the Gray tensor product of two 1-morphisms is not a 2-morphisms but a 0-morphism (an object).

Sorry, got to run now. More later.

Posted by: Urs Schreiber on May 15, 2008 7:10 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

I have expanded the above comment into what is now subsection 2.3. See table 1 on p. 17.

Posted by: Urs Schreiber on May 15, 2008 11:35 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Concerning the relation between supermanifolds and DG manifolds, I should also mention the following:

one observes (apparently Kontsevich did first, later in particular Pavol Ševera kept “popularizing” it) that a DG-structure on a supermanifold $X$ is precisely an action of $\mathrm{Aut}(\mathbb{R}^{0|1})$ on it.

For instance the DG structure on $\Pi T X$, hence the DG structure on the deRham complex (!) arises canonically from the canonical action of $\mathrm{Aut}(\mathbb{R}^{0|1})$ on $\Pi T X \simeq hom(\mathbb{R}^{0|1}, X) \,.$

That’s pretty cute. In a way, it “exlains” DG in terms of super, making super the more primitive concept.

Posted by: Urs Schreiber on May 15, 2008 10:05 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Except for DG you need to go from super to Z-graded

Posted by: jim stasheff on May 16, 2008 1:51 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Except for DG you need to go from super to $\mathbb{Z}$-graded

Yes, and this is part of the data the $\mathrm{Diff}(\mathbb{R}^{0|1})$-action gives:

such an action on $X$ picks two vector fields on $X$, one odd, the other even: the odd one is the differential, the even one is the “Euler vector field”, the one which measures the degree. Together they define a differential and a grading, and this grading is an $\mathbb{N}$-grading, at least when we apply this reasoning to $\Pi T X$. (I am not sure in general if or how it follows that the extra grading measured by the Euler vector field is by $\mathbb{N}$ or by $\mathbb{Z}$.)

See for instance Pavol Ševera’s recollection of this fact, in section 3.2 of his Gorms and Worms or top of p. 4 of $L_\infty$-algebras as 1-jets.

Posted by: Urs Schreiber on May 16, 2008 6:14 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Firstly, I really appreciate all the time you took to go through this stuff. Secondly, why isnt it that you can think of a super object as DG with d=0? and everything repeated? like V_0=V_2k for all k, etc so it is like the super version is some sort of quotient of this chain complex? (like wrapping the real line around the circle) Also, it seems like all this super stuff, while it is a generalization of many concepts geometrically, is just one type of equivariant theory where we forget about the group action, maybe it is there and we know about it, but it doesn’t seem like we are using a Z/2 action (maybe it is the trivial action). Anyway, it seems like a lot of this theory could be done in a G-graded setting, with G a group. What the Algebra of functions you would use is i don’t really know, but maybe it would involve taking some tensor algebra and then replacing the sign map with something related to the quotient of Sym(n) by im(G) under some sort of ‘normal’ embedding (if such an embedding even could exist generally). i dont really know, and i understand that Z/2 is important to physicists in this way, but why not other finite groups? shouldn’t this be a more general thing? it just seems like superification is way to specialized.
sean

Posted by: sean tilson on May 18, 2008 4:45 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

Secondly, why isn’t it that you can think of a super object as DG with $d=0$?

Well, you can, sure.

While terminology differs, I tend to want to say “super” for “internal to $\mathbb{Z}_2$-graded vector spaces with with nontrivial symmetric braiding and say “DG” for “internal to monoids in cochain complexes”. If the latter have trivial differential and if you remember of the $\mathbb{Z}$-grading only the $\mathbb{Z}_2$-parity you get super.

Just beware that the nature of morphisms depends crucially on how you think of your graded/super entity:

morphisms between DG manifolds, respecting the $\mathbb{Z}$-grading (and possibly the differential) are much more restrictive than if you regard them as just supermanifolds and demand morphisms just to respect the $\mathbb{Z}_2$-grading.

Anyway, it seems like a lot of this theory could be done in a $G$-graded setting, with $G$ a group.

i suppose that’s right. We had some discussion of such generalizations of super here some time ago.

For $G$ some abelian group, you’d want to known all the possibe symmetric monoidal structures on the category of $G$-graded vector spaces. For each of these there is a notion of $G$-super geometry.

With similar ease, one can imagine generalizing DG.structure in various directions. For instance: it’s easy to generalize all definitions to the case where we just demand that $d^n = 0$ for soem $n \gt 2$.

While all these generalizations are very straightforward, it is a bit mysterious what they might mean. Maybe their existence indicates that there are vast generalizations of huge part of the algebra that we do. Or maybe nothing really interesting happens beyond $\mathbb{Z}_2$ or $\mathbb{Z}$-grading and $d^2 = 0$.

I once heard a talk where somebody was advocating to look into “complexes” with $d^3 = 0$. But apart from the formal motivation, no motivation for that step was presented. Recently we had speculated here somewhere that $d^n = 0$ might be relevant for extended quantum field theory, where one wants to talk about “boundaries of boundaries” and not assume that these are necessarily trivial.

Posted by: Urs Schreiber on May 19, 2008 10:59 AM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

There’s a lot written on d^n =0
both ancient and modern

and yes, G-grading is also in the litt

Posted by: jim stasheff on May 19, 2008 4:43 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

I was just curious to see if people had thought about such things. It is very interesting that you point out that the morphisms would be different… It seems that we can easily embed the category of Super Vector spaces (faithfully if that isnt already redundant) in the category of Graded vector spaces, or rather Real (or complex) Chain complexes. Geometrically, perhaps not, it seems that there is a lot more built into the definition of super manifolds that one would not guess just from the definition of super vector spaces, maybe there is something hidden that i dont understand yet. It almost seems like the only connection between the odd and even degree parts, at least not until you have a super algebra. anyways, i love this forum of discussing things!

sean

Posted by: Sean Tilson on May 20, 2008 7:24 PM | Permalink | Reply to this

Re: HIM Trimester Geometry and Physics, Week 1

It seems that we can easily embed the category of Super Vector spaces (faithfully if that isnt already redundant) in the category of Graded vector spaces, or rather Real (or complex) Chain complexes.

Well, careful. As far as bare categories go you can. But there is no embedding as monoidal categories. Because that would be asking for an embedding of $\mathbb{Z}_2$ into $\mathbb{Z}$. As sets that’s easily possible. But not as groups!

You do have a canonical monoidal functor the other way round: from $\mathbb{Z}$-graded vector spaces to $\mathbb{Z}_2$-graded ones, since that comes from the group homomorphisms $\mathbb{Z} \to \mathbb{Z}_2$.

Sorry for nitpicking on this. You are probably well aware of this, but I thought I’d amplify it a bit. The point is that we really care about the grading only in as much as it does affect the monoidal structure, because apart from affecting that it isn’t of much interest, so we should really want to respect that monoidal structure.

it seems that there is a lot more built into the definition of super manifolds that one would not guess just from the definition of super vector spaces,

There is some truth to this. Some people in our little seminar had this little surprise when they first learned about $\mathbb{R}^{p|q}$ as a super vector space and then later saw an object with an entirely different-looking definition but addressed by the same symbol, now regarded as a super manifold. At that point, when one starts talking about locally $\mathbb{Z}_2$-graded ringed spaces it is a step somewhat less naive then the general mantra “we work internal to $\mathbb{Z}_2$-graded vector spaces with the nontrivial symmetric braiding”.

On the other hand, I’d have a comment here: I am puzzled by the pupularity of the definition of supermanifolds as locally ringed spaces.

The category of supermanifolds one gets this way is equivalent to that of $\mathbb{Z}_2$-graded commutative algebras of the form $\wedge^\bullet_A N \,,$ where $A = C^\infty(X)$ is the algebra of ordinary smooth functions on an ordinary manifold $X$ and $N$ is a finite rank projective $A$-module, regarded as being in odd degree and I stick to my habit of writing $\wedge^\bullet$ where you might rather want to write $Sym$ (so it means graded-symmetric tensor powers). Morphisms are just morphisms of graded-commutative algebras.

That’s it. The category of supermanifolds. No sheaves, nowhere. No “locally ringed” business. In a way, I think, this way of looking at it is more systematic.

Posted by: Urs Schreiber on May 20, 2008 10:33 PM | Permalink | Reply to this
Read the post HIM Trimester on Geometry and Physics, Week 4
Weblog: The n-Category Café
Excerpt: Talk in Stanford on nonabelian differential cohomology.
Tracked: May 29, 2008 11:03 PM

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