## June 12, 2007

### More Mathematical Blogging

#### Posted by David Corfield

More mathematics blogging at the Secret Blogging Seminar. This time a group blog manned by five recent and future Berkeley mathematics Ph.D.’s, at least 4 of whom have contributed comments to the Café, and who promise

Commentary on our own research, other mathematics pursuits and what ever seems like writing about on any given day. Sort of like a seminar, but with more rude commentary from the audience.

I see one of their number, A J Tolland, gave a talk Categorified TQFT & Quantum Groups (notes):

There’s some emerging folk wisdom which says that there’s a connection between the “n” in n-category and the “n” in n-dimensional. In this talk, we’ll explore one of the simplest examples of this phenomenon, showing how one can recover the axioms of a (finite) quantum group from path integral rules of a “categorically enriched” version of the topological QFT known as Chern-Simons theory. The finiteness restriction is necessary in this case to make the path integral into a well-defined mathematical object.

Sounds like there may be one or two common interests between blogs.

Posted at June 12, 2007 8:03 AM UTC

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### Re: More Mathematical Blogging

More mathematics blogging at the Secret Blogging Seminar.

That’s great! We are getting slowly to the point which I was hoping for already yeras ago: all discussion needs – and much more – can be satisfied online.

But is it just me being blind, or don’t they provide any feeds for their blog (atom, rss, etc.)?

I see one of their number, A J Tolland, gave a talk […]

Now, that’s interesting! Unfortunately at the moment the slides break off right at the point where it becomes really interesting… :-)

Posted by: urs on June 12, 2007 10:40 AM | Permalink | Reply to this

### Re: More Mathematical Blogging

As per WordPress defaults: content and comments.

Posted by: John Armstrong on June 12, 2007 3:51 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

[…] defaults […]

Ah, I see. Thanks!

Posted by: urs on June 12, 2007 4:31 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

Don’t worry Urs. The talk was purely expository; nothing you haven’t seen before. (I’m somewhat embarassed to admit: I put the notes up before the 2nd half of the talk because I’d been a little rough on the audience during the first half.)

Posted by: A.J. on June 12, 2007 4:28 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

nothing you haven’t seen before

But I am not completely sure what you have in mind when you

recover the axioms of a (finite) quantum group from path integral rules of a “categorically enriched” version of the topological QFT known as Chern-Simons theory

Probably a “finite quantum group” is nothing but a finite group with a cocycle on it? (We are talking about Dijkgraaf-Witten theory, I suppose?)

But still, from which starting point do you mean to derive that?

Posted by: urs on June 12, 2007 4:36 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

Yep, it’s just Dijkgraaf-Witten theory. The talk was just an exposition of Dan Freed’s ideas.

So, by “derive”, we mean that the path-integral gluing and sewing relations tell us that the category associated to $S^1$ is the category of representations of the quantum double of functions on a finite group, in the same way that the path integral gluing and sewing in the case of 2d topological field theory tells us that the algebra associated to $S^1$ is a Frobenius algebra.

This is all in Dan Freed’s notes. I’ve added, at most, errors in my own notes.

Posted by: A.J. on June 12, 2007 6:01 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

Urs wrote:

Probably a “finite quantum group” is nothing but a finite group with a cocycle on it? (We are talking about Dijkgraaf–Witten theory, I suppose?)

A “finite quantum group” is the name some people use for a finite-dimensional Hopf algebra. An example of this is the group algebra $\mathbb{C}[G]$ of a finite group. Another easy example is the algebra $Fun(G)$ of complex-valued functions on $G$, with pointwise multiplication.

As you probably know, $\mathbb{C}[G]$ and $Fun(G)$ are dual Hopf algebras. The category of representations of $\mathbb{C}[G]$ is the category of representations of $G$. The category of representations of $Fun(G)$ is the category of $G$-graded vector spaces — or if you prefer, vector bundles over $G$.

So, $G$-representation and $G$-grading are dual concepts in a certain sense.

Indeed, if we have an finite abelian group $A$, it has a ‘Pontryagin dual’ — the finite abelian group

$A^* = hom(A,\mathrm{U}(1))$

which is equipped with natural isomorphisms

$Fun(A^*) \cong \mathbb{C}[A]$ $\mathbb{C}[A] \cong Fun(A^*)$

as Hopf algebras. We’ve been talking about this recently elsewhere, so I couldn’t resist pointing out how it fits in here.

More interesting finite quantum groups arise when we take a 3-cocycle

$\alpha: G \times G \times G \to \mathbb{C}^*$

and use this to ‘twist’ the associator in the category of $G$-graded vector spaces. The resulting monoidal category turns out to be the category of representations of some more interesting finite-dimensional Hopf algebra.

Even more interesting stuff happens when we form the ‘quantum double’ of any of the Hopf algebras mentioned so far!

The best way to understand the quantum double goes via its category of representations. Just as any monoid has a ‘center’ which is a commutative monoid, any monoidal category $M$ has a ‘center’ $Z M$ which is a braided monoidal category. This is part of the general philosophy of centers!

If we take the center of

$Rep(\mathbb{C}[G]) \simeq [vector bundles over G]$

we get the category of equivariant vector bundles over $G$ — that is, vector bundles equipped with a lift of the adjoint action of $G$ on itself. This is the category of representations of a certain Hopf algebra, the ‘quantum double’ of $\mathbb{C}[G]$. Amusingly, this is also the quantum double of $Fun(G)$.

Finally, the most fun occurs when we take the quantum double of a ‘twisted’ group algebra.

But still, from which starting point do you mean to derive that?

As you probably know, we can derive all these Hopf algebras starting from an extended TQFT, the Dijkgraaf–Witten model, and considering its vector spaces and 2-vector spaces of states on various manifolds.

Jeffrey Morton is handing in his thesis tomorrow! It shows how to define the (untwisted) Dijkgraaf–Witten model as an extended TQFT in all dimensions.

It sounds like Jeffrey Morton, A. J. Tolland and Bruce Bartlett have a common interest in this Dijkgraaf–Witten stuff. Maybe they should collaborate!

I love it too, as a simple example of lots of fun ideas.

Posted by: John Baez on June 14, 2007 4:55 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

The best way to understand the quantum double goes via its category of representations.

For me, the best way to understand quantum doubles goes via Simon Willerton!

The Baby Version of Freed-Hopkins-Teleman

Posted by: urs on June 14, 2007 5:17 PM | Permalink | Reply to this

### Re: More Mathematical Blogging

Jeffrey Morton is handing in his thesis tomorrow!

Cool! When can we expect a version of it on the archive or on your webpage, Jeffrey?

I noted with interest the recent talk by A.J. Tolland; the notes were interesting and even if they were just an exposition of Freed’s ideas, they seemed to be a slightly different angle on it which I hadn’t seen before.

The best way to understand the quantum double goes via its category of representations.

My personal preference on these things is to use as much geometrical language as possible and try to prevent algebra from gatecrashing on the party :-).

I’m not sure if Urs is as radical as I am, but when it comes to the Drinfeld double I like to think completely in the groupoid picture… and never actually think in terms of the actual algebra “Drinfeld double” , at all. I think that Jeffrey is also quite geometric in his approach. I’m certainly not sure that Simon is as radical as I am, though it’s true he likes the groupoid picture - he wrote a paper on it!(which is where I learnt it from.)

This is where I guess I differ with the philosophy that seems to come out the Khovanov homology school. It seems to me that the philosophy there is to try to categorify the algebraic structures which gave rise to invariants of tangles. So at some point one imagines “categorifying a Hopf algebra” and things like that.

Whereas I would prefer to categorify the geometric structures (or at least categorify the way they are ‘represented’)… not the algebraic ones. Just a personal preference.

Let’s recall here how to think of the braided monoidal category $Rep(DG)$ of reps of the Drinfeld double in purely geometric terms.

You think of it as the category of equivariant vector bundles over $G$. Which is the same thing as to say, as the category of representations of the groupoid $\Lambda G$ (which depicts $G$ acting on itself by conjugation).

How about the monoidal structure, and the braiding? (This was the whole reason one used the language of Hopf algebras, etc.) Well, you also treat that geometrically. If

(1)$\sigma, \rho \in Rep(\Lambda G),$

then you define

(2)$(\sigma \otimes \rho)_g = \Gamma_{m^{-1}(g)} (m^*(\sigma) \otimes m^*(\rho)).$

In words : the fiber of $\sigma \otimes \rho$ at $g \in G$ is the space of sections of the pullback bundle over $G \times G$ induced by the multiplication map

(3)$m : G \times G \rightarrow G.$

And the braiding is written down in a similar way. Of course, these formulas are just fancy geometric spin for the convolution product, i.e.

(4)$(\sigma \otimes \rho)_g = \oplus_{a b = g} \sigma_a \otimes \rho_b$

The point is that everything is geometric, and we avoided things like algebras, coalgebras, antipode maps, and all those things.

It’s just a personal preference, and maybe we’re simply lucky to have the luxury of a geometric viewpoint here because the Drinfeld double is such a simple quantum group. Does there exist a purely geometric framework for interpreting say, $Rep(U_q (sl_2))$? Isn’t that what quantum field theory (e.g. Witten’s work on the Jones polynomial) is supposed to give us?

Posted by: Bruce Bartlett on June 15, 2007 12:39 AM | Permalink | Reply to this

### Re: More Mathematical Blogging

I’m not sure if Urs is as radical as I am

I am a wannabe radical: I want to be as radical as you are. If I am not, it is only because you have thought about this more than I have!

As I keep telling everybody who wants to listen (like I did to the attendants of the latest Oberwolfach Arbeitsgemeinshaft) the best way to understand the Drinfeld double is not to consider it at all – but instead read Simon Willerton’s paper and Bruce Bartlett’s mind.

Because that geometric picture actually explains what is really going on. And it shows how lots of supposedly difficult deep statements in the non-finite case, like the Freed-Hopkins-Teleman result, have simple at least heuristic interpretations (though, as you know, I have an idea how to also reduce the full FHT result to willertonesc tautologies, by using the String group, as described here.)

How about the monoidal structure, and the braiding? […] Well, you also treat that geometrically

And not only that, I think: one even gets it from “canonical quantum theory”. As we discussed in Fusion and String Field Star Product, one way to understand this is to take serious the interaction of strings in this picture, via the pair of pants.

As we discussed there, this too is an old observation by Dan Freed, but I’d think the formulation which I mention at the above link is the one that ties in directly with the $\Lambda G$-philosophy – which I would regard as a special case of the idea of “quantizing on a category” (using Chris Isham’s terminology), following the framework of the charged $n$-particle.

That “quantizing on a category” business is, I think, what really underlies Simon Willerton’s observation, that it is good to think of the circle in terms of its fundamental groupoid:

in general, given an $n$-particle with parameter space $\mathrm{par}$, we want to think of $\mathrm{par}$ as a suitable $n$-category.

I tried to discuss this and the relation to your (Bruce’s) world of ideas in String on $B G$.

I deeply regret having had to stop following these trains of thought a while ago. I am hoping that once Bruce’s thesis is out, I’ll find new time and energy to look into this seriously again.

Posted by: urs on June 15, 2007 9:39 AM | Permalink | Reply to this

### Re: More Mathematical Blogging

Urs, you have a great grand scheme of this stuff which is really very impressive. I had forgotten that the “best” way to understand the monoidal structure and braiding on the reps of the Drinfeld double is as coming from the quantum interaction of strings on the pair of pants! That’s one of the things Freed taught us, and what you have stressed and built a bigger picture from.

I guess that’s the attractive thing about TQFT’s, the reason why many of us here are working on some or other aspect of them. Because it’s all about the interaction of classical and “quantum” geometry; some of the geometry is classical, you put it in by hand, but then some cool geometric structures magically emerge from the quantum side of things.

That seems to me to be the magic of things like Witten’s work on Chern-Simons theory, and things like the A and the B models, and Khovanov homology, and mirror symmetry, and “geometric Langlands”, and so on. These things are way too advanced for me but at least I can see the pattern : geometric structures emerging from quantum theory. And many of these structures have an n-categorical feel to them, which is what John taught us, making them very attractive for us here at the n-category cafe.

And it’s exciting because I get the feeling that it’s still a big mystery; I don’t think anyone has come up with a coherent account of it all.

Posted by: Bruce Bartlett on June 15, 2007 10:43 AM | Permalink | Reply to this

### Re: More Mathematical Blogging

By the way, it just occurs to me that the same spell which trivializes the Drinfeld double and the FHT theorem might also help the wizard create the philosopher’s group $U(1)$ by transmutation of lesser materials:

Let’s think of a space over $U(1)$

$X \to U(1)$

as the realization of the nerve of a morphism of categories. If there is a groupoid $\mathrm{Gr}$ such that

$X \simeq |\mathrm{Gr}|$

then this ought to be

$\mathrm{Gr} \to \Sigma \mathbb{Z} \,.$

Et voilà

Posted by: urs on June 15, 2007 11:08 AM | Permalink | Reply to this

### Re: More Mathematical Blogging

It sounds like Jeffrey Morton, A. J. Tolland and Bruce Bartlett have a common interest in this Dijkgraaf-Witten stuff. Maybe they should collaborate!

Maybe we should. I’m a bit tied up at the moment, but I’d enjoy talking about this stuff later this summer.

Posted by: A.J. on June 19, 2007 4:36 PM | Permalink | Reply to this

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