The n-Category Café

Really for Urs, but I donb’t know how to comment on an original post.

In this situation, those who know will know which Jones you refer to, but in a mixture of operads and subfactors it would be good to identify: V Jones not JDS Jones I presume.

Hi John,

thanks for your comment!

So, do you think there’s an elegant abstract algebraic definition of a ‘2d genus-0 TQFT with defect lines’, perhaps using a bit of higher category theory?

Now I am not sure what you mean. Isn’t that what I said in the entry, that a 2d genus-0 TQFT with defect lines is an algebra for the colored planar tangle operad? Which can equivalently be rephrased as: a monoidal functor from the category $2\mathrm{Cob}_{Def}$ of 2-dimensional cobordisms with defect lines drawn on them to $Vect$?

With that in hand, take any refinement of $2Cob$ to a 2-category and I guess we get a straightforward refinement of that to a 2-category of 2d cobordisms with defect lines on them.

And then of course we’d want to eventually drop the genus-0 condition and consider all of $2Cob_{Def}$. As you say next:

Of course there’s another possibility: that all the really interesting examples of planar algebras come from full-fledged 2d TQFTs, and the genus-0 condition is just a distraction.

Yes. Experience with 2d CFT shows that treating only the genus-0 sector is a useful first restriction to then tackle afterwards the representation theory of the full setup.

Namely in that work by Huang and Kong which I mentioned recently it is shown that vertex operator algebras are precisely the reps of the (holomorphic and) genus-0 part of $2Cob_{conf}$, something which is certainly a first crucial building block for any attempt to go further to higher genus.

I believe the theory of von Neumann algebras will soon be seen as part of a bigger theory of ‘higher Hilbert spaces’. I think you know what I mean.

Yes, I think so.

(Maybe you’re the only one who does?)

Today I had lots of nice discussion with André Henriques again. André knows many things. He is all into operator algebras now, for good reason.

If you take a planar algebra with just one colour you get a spherical category. So planar algebras are to spherical categories as 2-categories are to monoidal categories.

John wrote:

So, do you think there’s an elegant abstract algebraic definition of a ‘2d genus-0 TQFT with defect lines’, perhaps using a bit of higher category theory?

Urs replied:

Now I am not sure what you mean. Isn’t that what I said in the entry, that a 2d genus-0 TQFT with defect lines is an algebra for the colored planar tangle operad?

I didn’t see that part — sorry!

Now, can we make this very nice and purely algebraic by saying something like

THE COLORED PLANAR TANGLE OPERAD IS THE FREE ???? OPERAD ON ????

?

You see, all these really important structures tend to have nice universal properties — and that’s what I’m really looking for.

I can’t help but think it’s important that the operad you’re talking about is a cyclic operad. It could be the free cyclic operad on a few operations.

But since planar tangles form a monoidal category with duals, and planar tangles with colored regions form a 2-category with duals, maybe we should forget cyclic operads and work with 2-categories with duals. I like those! And so do Bruce Bartlett and Jamie Vicary.

For example, say we start with some fixed category $C$ of colors and defect lines going between them. You gave an example where $C$ had two objects and two nontrivial morphisms. I presume $C$ should be category with duals, i.e. a $\dagger$-category, so every defect line $f : a \to b$ has defect line $f^\dagger : b \to a$ going back?

Then maybe the 2-category I’m talking about is the ‘free 2-category with duals on $C$’, in a certain interesting sense. The $\dagger$ operation in $C$ should get promoted to an honest duality in this 2-category: that is, $f^\dagger : a \to b$ becomes the ambidextrous adjoint of $f : b \to a$.

(All this jargon is equivalent to some very simple pictures, but it could still be very useful in the long run, to fit things into the big picture.)

Which can equivalently be rephrased as: a monoidal functor from the category $2Cob_Def$ of 2-dimensional cobordisms with defect lines drawn on them to Vect?

That doesn’t sound right. This category will have morphisms including a bunch of higher-genus surfaces with defect lines drawn on them. So, I doubt (symmetric) monoidal functors from this category to Vect will be exactly the same as genus-zero 2d TQFTs with defect lines, unless some marvelous theorem says ‘once you’ve got genus zero, you get all the higher genuses for free’.

But, I think we agree that the genus-zero restriction is likely to be ‘technical’ rather than ‘fundamental’ in nature.

(Genuses? Genii? They both sound stupid. And ‘genii’ reminds me of an old Bugs Bunny cartoon version of The Thousand and One Nights, where Bugs, playing Aladdin, suddenly starts singing ‘I dream of genie with the light brown hair’. But never mind!)

Has someone come up with an elegant operadic approach to describing genus-0 conformal field theories?

Yes, this is precisely what Huang does, and now his student Liang Kong continues to due. See the references I mentioned here.

Liang has given me more notes which he has written up for talks which gives a good summary. But apparently he hasn’t made that public yet.

Huang and Kong slightly vary Segal’s original setup for functoruial 2dCFT in that they do not consider conformal cobordisms with boundary circles, but conformal surfaces with parameterized point insertions. But there is a way to pass between these two pictures.

In any case, conformal spheres with parameterized insertion points form an operad under the more or less obvious gluing at insertions poits using the parameterization, and they show that holomorphic representations of this operator (i.e. those that depend holomorphically on the insertion points) are precisely vertex operator algebras, i.e. the “chiral part” of a 2D CFT.

Then more recently Liang Kong carried this much further and conisdered what he calls “full field algebras.” See for instance his Full field algebras, operads and tensor categories.

A “full field algebra” is obtained by removing the holomorphicity constraint in the above and considering arbitrary reps of that operad. These full field algebras consist of an extension of the thing obtained by tensoring a holomorphic and an anti-holomorphic chiral VOA.

This is interesting, because it is precisely analogous to the way non-chiral bulk CFTs work in the language of conformal nets (operator algebras, algebraic quantum field theory): there the chiral algebra is a local net on the real axis thought of as a light-ray and an extension of the bulk net obtained by assigning to any causal diamond the tensor product of the two algebras given by feeding the two projections of the causal diamond onto the left- and the right moving light-ray by two local nets on the real axis is the bulk net.

Sorry for the long sentence.

If so, we could water that down by neglecting the conformal structure, and get a definition of genus-0 2d topological quantum field theories.

Yes, but I’d say the result is what we already know: the planar tangle operad. No?

It’s interesting that you mention ‘string bits’. They’re clearly necessary for a full-fledged higher-categorical approach to string theory, since for that you need to be able to chop strings into polygons, and the edges of these are string bits.

Sometimes when I try to describe to string theorists the idea of parallel 2-transport I use the term “string bits” to convey the idea of an extended QFT picture where data is assigned to pieces of worldsheet. But in that context it is a bit loosely speaking.

But, I’ve only seen a few people use the term ‘string bits’, and I got the impression that this idea was a bit outside the mainstream

There was a time when string bits were the latest hype. They are still very important, because they correspond to AdS/CFT duals of certain N=4 SYM observables. As such they survived in what is currently maybe the most active part of string theory, namely spin chain model computations in AdS/CFT. How those “string bits” are related to the ones I talk about here would have to be investigated in detail. But there ought to be a relation.

I’m pretty hazy on what a planar algebra is defined to be,

Hm, I thought I indicated the definition in the first few lines of the entry. It’s not esoteric:

a planar algebra is an “algebra for”, i.e. a representation of the “planar tangle operad”. See the comments around the picture right at the beginning to see how this is defined.

So to each circl with colored marked points the planar algebra assigns a vector space, and to all such disks as in the picture above it assigns a linear map from the tensor product of the vector spaces assigned to the interior circles to the vector space assigned to the outer circle. And this assignment of linear maps is supposed to respect the composition of disks obtained by gluing one disk into one of the interior circles of another.

So, in other words, a planar algebra is simply a monoidal functor from $2Cob^0_{def}$, the monoidal category of 2d-cobordisms of genus 0 with defect lines drawn onto them, to $Vect$.

John, I think you should turn your argument on its head here. The point is that in any spherical 2-category you get (for free!) an action of a cyclic operad (in fact the whole “annular Temperley-Lieb” operad). What Jones’s Annular Tangles shows (building on earlier work by Graham and Lehrer) is that this annular structure tells you important facts that would otherwise be hard to extract from your 2-category!

In their usual axiomatization a spherical 2-category has a choice of directions (vertical and horizontal compositions). But why let the axiomatization get in the way of seeing all the structure! Why not picka “polar” axis instead, and look at how your 2-category behaves under the action of the annular category? This is “annular composition” which is no better or worse than “vertical composition” or “horizontal compositions.” The spherical 2-categories have more symmetry than their axioms do!

Suppose you want to classify fusion categories with a self-dual tensor generator of Frobenius-Perron dimension less than 2. Then planar algebras are exactly the right formalism for attacking this problem!

I’m almost seeing the light… can you explain why you need the FP dimension to be less than 2?

Another way to say this is that planar algebras aren’t something for studying tensor categories (or 2-categories), they’re something for studying a particular object (resp. morphism) in that tensor (resp. 2-) category.

Okay… the light is dawning… but wait. In the spherical category situation, by focusing on that object and its tensor powers, don’t you also gather quite a lot of/all information about the spherical category anyway? Isn’t there always the canonical generating object,

where the $d_i$ are the Frobenius-Perron dimensions?

Hi Noah,

thanks for your comments!

First a nitpicky comment. I think you need some more conditions like “finite index” in a).

Thanks, I’ll fix that.

Second some more substantial comments.

[…]

One of them is that the shaded Planar Algebras are just a version of spherical 2-categories with two objects. You can have more objects if you want.

Yes, I think this I said clearly. This is what matches so nicely with the point of view of “defect lines” in 2d QFT, which are a topic apparently gaining momentum.

Basically here’s the general setup. Take any spherical 2-category

By the way: how does one quickly see the need for the sphericality condition? I think I can easily derive it if I allow my planar algebra to have morphisms with no outer circle.

Because then I can form a sphere by starting with the disk with no inner circle taken out but one defect line forming a circle and composing that with the disk with no outer circle but a single inner circle.

(In FQFT language: I am composing the cap with the cup, and the cap carries a closed defect line.)

The result is a sphere with a circle drawn on it. Which means that I cannot intrinsically distinguish inside from outside of the circle. Mapping that to the algebraic side this is the sphericality.

Thus planar algebras are well-tuned for studying problems where you care about a fixed tensor generator.

Which, it seems, can be thought of from the TQFT point of view nicely: the fixed tensor generator is the “space of states” of the string stretching once between the given boundary conditions. I think we see a variant of Baez-Dolan here, in that a TFT functor is fixed already by low dimensional data with duality properties on it.

In the absence of defect lines this data is that assigned to a point. Here with defect lines present, the data is that assigned to the defects. Which is only natural.

There hasn’t been a whole lot done, but there has been a little (sadly, I can’t seem to locate the reference).

Perhaps Subfactors and 1+1-dimensional TQFTs by Kodiyalam, Pati, and Sunder is what you have in mind?

Urs, I’m pretty sure you’re right here that the approach you outline would simplify the proof in Planar Algebras 1. Certainly it would clarify things for people such as you and I if the argument were broken into two steps: subfactor to category to planar algebra. The first step uses facts about subfactors and conditional expectation, while the second step is mostly 2d topology.

What’s the reference I should look at for the relation between spherical 2-categories and planar algebras?

I wrote something up but did not publish because I did not feel there was enough interest. Perhaps I should reconsider.

I recommend this for anyone trying to connect subfactors and category theory:

• Michael Müger, From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories

I don’t really know anything about this stuff, but I imagine all the formulas that define your fusion category — formulas for associators and braidings and the like — involve numbers that are algebraic. Right? No $\pi$’s or $e$’s, except in harmless combinations like $e^{2\pi i m /n }$ which are algebraic numbers?

If so, then you could really set up your fusion category using the field of algebraic numbers $\overline{\mathbb{Q}}$ instead of the field of complex numbers.

Then, any automorphism of $\overline{\mathbb{Q}}$ would act to turn your fusion category into a new one! Just take all the numbers in your formulas and hit them with this automorphism; you get a bunch of new numbers that still work, since all the identities they need to satisfy are algebraic.

These automorphisms form what’s called the Galois group $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$.

Now, you say those guys actually talk about automorphisms of $\mathbb{C}$ instead of automorphisms of $\overline{\mathbb{Q}}$. As an algebra over $\mathbb{R}$, $\mathbb{C}$ just has two automorphisms: the identity and complex conjugation. That’s sort of dull, so I can’t imagine them making a big deal about that.

But if we’re just doing abstract algebra, rather than analsis, we can also think of $\mathbb{C}$ as an abstract field, and then its automorphism group is enormous, thanks to the uncountable collection of transcendental numbers that can be permuted in all sorts of goofy ways. This group would be called $Gal(\mathbb{C}|\mathbb{Q})$. I bet the main reason most people don’t talk about it much is that it’s so huge — even $Gal(\overline{\mathbb{Q}}|\mathbb{Q})$ has been known to drive men insane.

(In case you’re wondering: I don’t know of cases where women were driven insane by this group.)

Anyway, the main point is that whenever you’re doing linear algebra over a given field, the group of automorphisms of that field acts on everything in sight. People have gotten interested in this for topological quantum field theory, so these guys may be thinking about something like that.

Just wild guesses…

The results you mention sound very interesting!

Maybe you can come and give a seminar at UCR — we have a number of people interested in vertex operator algebras and related topics.