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September 11, 2008

Planar Algebras, TFTs with Defects

Posted by Urs Schreiber

I am in Vienna at the ESI attending a few days of the program Operator algebras and CFT.

This morning we had a nice talk by Dietmar Bisch on

Dietmar Bisch, Paramita Das, Shamindra Kumar Ghosh
The planar algebra of group-type subfactors
(arXiv)

I had not really looked into planar algebras before. The Wikipedia entry gives some information but leaves out the crucial pictures. Noah Snyder and his guest blogger Emily Peters on Secret Blogging Seminar have a useful series of posts on the topic – with the pictures.

The generic picture is this one:

which I have stolen from Noah Snyder’s web site.

This picture shows a representative of a generic morphism of the planar tangle operad, whose objects are unions of circles with marked points and whose morphisms are isotopy classes of such disks as above, with disks in the interior taken out and non-intersecting lines drawn in the remainder, which may end on any of the boundaries. In addition, one generally considers this colored in some index set, i.e have a color assigned to each face.

Composition in the operad is the obvious one obtained from gluing in disks into the holes of another disk, such that all the boundary labels match. A planar algebra is a representation of this planar tangle operad, i.e. an algebra for the operad.

If you’d ask me how I would summarize this in few words i’d say:

A planar algebra is a 2d genus-0 TFT with defect lines.

When I mentioned this description to people here who know about defect lines they certainly agreed. But it seems that this description is not something used in the literature at this point (or is it somewhere??).

But it might be useful…

Of particular interest is the special case where the set of colors has precisely two elements {a,b}\{a,b\} and where one considers only defect lines aba \to b and bab \to a.

A combination of results of Jones and collaborators says the following:

a) Every finite index type II 1II_1 vonNeumann algebra subfactor NMN \subset M comes with a sequence M iM_i of subfactors called its standard invariant.

b) By assigning M kM_k to the circle with 2k2k marked points (and, by assumption, the intervals in between necessarily labeled alternatingly by aa and bb) these subfactors M kM_k induce the strudcture of a planar algebra.

c) Every planar algebra of subfactors arises this way as a genus 0 2d TFT with {a,b}\{a,b\}-defects.


There is some understanding of the relation between subfactors and boundary conditions for 2-dimensional conformal field theory, with a crucial puzzle piece being

Roberto Longo, Karl-Henning Rehren
Local fields in boundary conformal QFT
(arXiv)

My current best understanding of the general story is the consideration given in the examples starting on page 29 of AQFT from nn-functorial QFT. As indicated in the figure on page 32, one can start with a lattice model of a 2-d QFT which consists just of the worldsheet of a single “string bit”, i.e. a discrete piece of string stretching from one boundary condition to another, such that its space of states along this interval, in the sense of extended QFT, is given by the bimodule NM M_{N} M_M given by te subfactor inclusion NMN \subset M.

If there is similarly a “string bit” for a dual module, stretching from the MM-boundary to the NN-boundary, then one can glue such strips alternatingly ababa \to b \to a \to b \to \cdots.

As described in section 7.8 and chatted about in AQFT from lattice models(?), the lattice QFT obtained this way is the Hopf spin chain model considered in

Florian Nill, Kornél Szalachányi
Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry
(arXiv).

Now, this consideration is set on 2-dimensional Minkowski space. But notice that it features precisely the phenomenon of a QFT built out of alternating defect and dual defect which one sees in the planar algebras/defect 2d TFTs arising from subfactors.

It would surprise me if this is a coincidence…

Posted at September 11, 2008 2:13 PM UTC

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53 Comments & 1 Trackback

Re: Planar Algebras, TFTs with Defects

Very good observation, Urs! I’d tried to come up with an elegant higher-category-flavored definition of ‘planar algebra’, but I got stuck. Unfortunately, the people I was talking to (on various blogs) seemed uninterested in finding such a definition.

Of particular interest is the special case where the set of colors has precisely two elements {a,b}\{a,b\} and where one considers only defect lines aba \to b and bab \to a.

In everything I’ve read, it seemed like the only case people were interested in — and that’s why I got stuck when trying to get an elegant definition. I see now that dropping this is the key to attaining elegance.

It’s a bit like trying to find an elegant definition of ‘groups with 16 elements’. Clearly the right approach is to first define groups, and then add the condition of having 16 elements when you need it. But equally clearly, the theory of groups with 16 elements is just a portion of the very interesting theory of general groups.

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? (Maybe 2-categories with duals, or double categories, or cyclic operads?)

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.

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. (Maybe you’re the only one who does?)

Posted by: John Baez on September 11, 2008 6:39 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Posted by: jim stasheff on September 11, 2008 6:59 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Yes, it’s Vaughan Jones.

Posted by: Urs Schreiber on September 11, 2008 9:23 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Yes, it’s Vaughan Jones.

By the way, for anyone interested there is various useful course material explaining in detail the stuff we are talking about here on Jones’ website.

He provides lecture notes on

vonNeumann algebras (pdf),

subfactors, standard invariants etc. (pdf),

annular structure of subfactors (pdf),

and last not least:

planar algebras (pdf).

with all the cool pictures.

Posted by: Urs Schreiber on September 11, 2008 10:04 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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 2Cob Def2\mathrm{Cob}_{Def} of 2-dimensional cobordisms with defect lines drawn on them to VectVect?

With that in hand, take any refinement of 2Cob2Cob 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 Def2Cob_{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 conf2Cob_{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.

Posted by: Urs Schreiber on September 11, 2008 8:41 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Posted by: Bruce Westbury on September 11, 2008 8:56 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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

Then maybe the 2-category I’m talking about is the ‘free 2-category with duals on CC’, in a certain interesting sense. The \dagger operation in CC should get promoted to an honest duality in this 2-category: that is, f :abf^\dagger : a \to b becomes the ambidextrous adjoint of f:baf : 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 Def2Cob_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!)

Posted by: John Baez on September 12, 2008 1:11 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

My guesses above are still mixed up. Those descriptions might apply if we were just taking disk-shaped regions with tangles in them and gluing them together along a portion of their boundary. But picture Urs drew shows a disk with some holes cut out.

Posted by: John Baez on September 12, 2008 2:32 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Genera

Posted by: Scott Carter on September 12, 2008 4:09 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

which can equivalently be rephrased as: a monoidal functor from the category 2Cob Def2Cob_{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.

Yes, I should have said: 2Cob def 02Cob^0_{def} the monoidal category of genus-0 2d cobordisms with defect lines on them.

I said it that way a few minutes after the reply to your comment in the reply to Bruce’s comment.

So, a planar algebra is a monoidal functor

pa:2Cob def 0Vect pa : 2Cob_{def}^0 \to Vect

and it corresponds to those full 2d TFTs with defects and without restriction on the genus which arise as extensions of this functor

2Cob def 0 pa Vect TFT 2Cob def \array{ 2Cob_{def}^0 &\stackrel{pa}{\to}& Vect \\ \downarrow & \nearrow_{TFT} \\ 2Cob_{def} }

through the canonical inclusion 2Cob def 02Cob def 2Cob_{def}^0 \hookrightarrow 2Cob_{def} .

Posted by: Urs Schreiber on September 12, 2008 6:56 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Has someone come up with an elegant operadic approach to describing genus-0 conformal field theories? I bet someone has. If so, we could water that down by neglecting the conformal structure, and get a definition of genus-0 2d topological quantum field theories.

But genus-0 2d TQFTs are quite boring unless we allow a bit of extra spice, like defect lines.

So: does there already exist an elegant operadic approach to describing genus-0 conformal field theories with defect lines?

Posted by: John Baez on September 11, 2008 6:45 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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?

Posted by: Urs Schreiber on September 11, 2008 9:01 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

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 — and maybe even a bit disreputable for some reason.

Posted by: John Baez on September 11, 2008 6:55 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Posted by: Urs Schreiber on September 11, 2008 9:18 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Wow this is very interesting because just yesterday I was looking at Noah Snyder’s nice website, because I have had my head swirling with fusion categories for the last few months. John wrote:

Very good observation, Urs! I’d tried to come up with an elegant higher-category-flavored definition of ‘planar algebra’, but I got stuck.

I’m pretty hazy on what a planar algebra is defined to be, but on Noah Snyder’s website he says:

A planar algebra is a combinatorial model for a pivotal fusion category

Isn’t this in some sense already an elegant higher categorical formulation of a planar algebra?

Posted by: Bruce Bartlett on September 11, 2008 7:04 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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 def 02Cob^0_{def}, the monoidal category of 2d-cobordisms of genus 0 with defect lines drawn onto them, to VectVect.

Posted by: Urs Schreiber on September 11, 2008 9:32 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Yes you’re right, sorry I am just confused about some things. For instance, Bruce Westbury wrote:

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.

And Noah Snyder wrote on his webpage:

A planar algebra is a combinatorial model for a pivotal fusion category…The relationship between planar algebras and fusion categories is that the planar algebra describes the Hom spaces between tensor powers of a chosen fundamental object in the fusion category.

I’m having some trouble combining these two pictures together, and I’m also confused about another problem: if planar algebras are like fusion categories, how do we see the associators (6j symbols) in the planar algebra framework?

Posted by: Bruce Bartlett on September 11, 2008 10:23 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

The expert on string bits is Manin.

Posted by: Kea on September 11, 2008 10:27 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

John: maybe we should forget cyclic operads and work with 2-categories with duals

No. Use both. That’s what we’re trying to do. One won’t get a nice description for the Fourier transform (2d path integral) otherwise.

Posted by: Kea on September 12, 2008 1:24 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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!

Posted by: Noah Snyder on September 12, 2008 5:45 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

This point of view is what you’re describing here, right?

Posted by: David Corfield on September 12, 2008 11:32 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

David asked Noah:

This point of view

[that higher categories with extra duality structure are usefully obtained as algebras for certain operads]

is what you’re describing here, right?

I find the general attitude of FQFT useful here to organize the picture, the way it connects geometry with algebra:

a notion of cobordisms category CobCob potentially with extra bells and whistles is a “geometric entity” in a general sense, in that most usually its morphisms are concretely represented by certain spaces.

The FQFT itself, being a linear representation of CobCob, yields, with the image of Cob in Vect, an algebraic structure. The general question one asks in FQFT theory is hence:

Which category of algebraic structures is equivalent to FQFTs on the given Cob?

The famous archetypical simple example is: reps of 2Cob are commutative Frobenius algebras.

One notices that Frobenius algebras are very special one-object linear categories and that they are a special case of a more general notion, “Frobenius algebroids” and “Calabi-Yau-categories”. This is then the first instance of how certain categories with extra duality structure are characterized as FQFTs, as cobordism representations.

Now, instead of talking about monoidal functors from a monoidal category Cob to Vect, it can be useful to rephrase this in terms of operads. Trivially so in simple examples such as 2Cob, but with the potential for much more sophisticated applications.

The operad plays the role of the “geometric” entity.

And then, this is the point of view I think in the operadic periodic table, one finds that large classes of useful higher categories with extra duality structure arise as representations aka algebras for these operads.

Which from the FQFT point of view says: large classes of interesting higher categories with duals are equivalent to certain FQFTs.

It’s a biased point of view. But I find it helpful when thinking about this stuff and in particular when organizing mentally the literature on this business.

Posted by: Urs Schreiber on September 12, 2008 5:08 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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

Second some more substantial comments. There are a few things I’d been meaning to get to in my SF and PA series of posts (but I’ve been really distracted trying to finish papers in time for application deadlines). 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. In particular, suppose you’re studying “intermediate subfactors” so you have three (II_1, finite index in each other, etc.) factors A contained in B contained in C. Now your bimodules come in more flavors, and so the corresponding planar algebras have 3 shadings. The analogue of Temperley-Lieb for planar algebras with 3 shadings is called the Fuss-Catalan planar algebra, and is used in work of (I think) Jones and Bisch on intermediate subfactors. So there is at least some interest among subfactor people in allowing more kinds of defect lines, as you put it. Similarly, for studying ordinary spherical tensor categories, you want to allow oriented defect lines.

Basically here’s the general setup. Take any spherical 2-category and take your favorite chosen collection of tensor generating morphisms in the 2-category (this choice is important and the rest will depend on the choice!) Then you get an example of a version of a planar algebra where for the operad you allow spaghetti and meatball diagrams with the regions labelled by objects in the category, and the spaghetti labelled by your chosen tensor generators.

Thus planar algebras are well-tuned for studying problems where you care about a fixed tensor generator. 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! Similarly, for subfactors where the inclusion A subset B gives you a particular chosen A-B bimodule B, then you might as well exploit your chosen tensor generator.

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.

Finally, on John’s comment on whether you get full-fledged 2-d TQFTs, you can actually start with the same pictures, but just interpret them differently! That is, the picture in the post can be viewed as a pair of pants (for a 3-legged alien, naturally). You end up with a slightly different version of the axioms of a planar algebra (for example, whether you’re allowed to isotope the holes in the picture around each other), but basically there’s a similar picture to planar algebras that you might call “surface algebras” where you allow yourself to glue together those diagrams in more complicated ways. There hasn’t been a whole lot done, but there has been a little (sadly, I can’t seem to locate the reference).

Posted by: Noah Snyder on September 12, 2008 5:31 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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,

(1)V= id iX i V = \oplus_i d_i X_i

where the d id_i are the Frobenius-Perron dimensions?

Posted by: Bruce Bartlett on September 12, 2008 10:20 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

I don’t understand any of this stuff, but I bet that canonical generating object will usually have Perron–Frobenius dimension >2\gt 2. So, if I could understand the first remark you quoted from Noah, maybe I could understand the second remark. Maybe.

Posted by: John Baez on September 12, 2008 4:23 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

I’m on an iPhone at the airport, hopefully these comments won’t come out too garbled.

I mentioned the FP dim smaller than 2 question not because it’s necessary for the formalism, but because it’s a question that has a nice answer! Namely there’s a quantum McKay correspondence that gives an ADE classification. I wanted to mention an example of an actual question where planar algebra are useful for finding an answer.

Posted by: Noah Snyder on September 12, 2008 5:55 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

John wrote: “So, if I could understand the first remark you quoted from Noah, maybe I could understand the second remark. Maybe.”

Basically here’s what you do. I’ll do the pivotal tensor category case, but everything should work for the right sort of 2-category.

Take your tensor category and choose an object X. Now consider the full subcategory whose objects are strings of tensor products of X and X*.

Next apply the usual method of drawing pictures in pivotal categories, but to the full subcategory. Since the only objects in this full subcategory are tensor products of X and X* you might as well drop the labels on the strings in your diagram and just use an upward arrow to denote X and a downward arrow to denote X*.

What you have now is a version of planar algebras where you’ve broken the symmetry. You have rectangles instead of circles, and spaghetti must connect to the top or bottom of rectangles. But since the category is pivotal, you can move strings from the top to the bottom, so you might as well just use circles.

To recover the category from a planar algebra, first break symmetry and look at rectangle pictures. That gives you a tensor category (via stacking and disjoint union). Then take Karoubi envelope (formally add the image of every idempotent) to recover your original category (or at least the subcategory tensor generated by X and X*).

To illustrate the point further, there’s a version of non-pivotal monoidal categories tensor generated by some object X, where you’d just define them as algebras over an operad of pictures where the holes are rectangles and the spaghetti can never double back.

Posted by: Noah Snyder on September 13, 2008 6:01 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Thanks! I don’t completely understand this yet, but this seems like the kind of thing I can understand.

I’m relieved that the ‘Perron–Frobenius dimension <\lt 2’ stuff doesn’t play a role in these abstract considerations. I understand things best when there’s a clean separation between the general abstract nonsense and the specific concrete nonsense. I like ‘em both, but right now I’m struggling with the former!

Posted by: John Baez on September 13, 2008 6:49 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

You might want categorical dimension here instead of FP dimension (they agree in the unitary case, see ENO for details). But yeah, that should be a tensor generator.

On the other hand, most existing technology for planar algebras works best when the tensor generator is simple.

But recall that for groups a faithful representation is always a tensor generator, so it’s not such a bad restriction.

Posted by: Noah Snyder on September 12, 2008 6:07 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Posted by: Urs Schreiber on September 12, 2008 12:37 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

I should have said “pivotal” everywhere I said “spherical.” Adding spherical makes the theory nicer and seems pretty harmless.

If you wanted a full 2d Tqft with defect lines, then looking at the action of a sphere with one meatball and some spaghetti would imply sphericality.

Posted by: Noah Snyder on September 12, 2008 6:19 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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?

Posted by: Jens on September 13, 2008 11:33 AM | Permalink | Reply to this

Spherical cats

Even over at Shores of the Dirac sea they are now talking about spherical cats.

Posted by: Urs Schreiber on September 12, 2008 1:03 PM | Permalink | Reply to this

no coincidence

I ended the above entry with

It would surprise me if this is a coincidence…

Reading a bit more and talking to a couple of people (am grateful to Pasquale Zito once again) I see that it is indeed not a coincidence.

The standard invariant of a subfactor NM N \hookrightarrow M given by chain of inclusions of finite algebras, the relative commutants P 0P 1P 2 P_0 \hookrightarrow P_1 \hookrightarrow P_2 \hookrightarrow \cdots has different equivalent defintions (page 6 of Jones’ lecture Intermediate subfactors) one of which being proposition 2.1.10 on the same page, which says that this algebra is the algebra of endomorphisms in the 2-category Bimod of kk powers of MM regarded as an NN-NN-bimodule:

P k=End Bimod( NM N N NM N N NM N N k) P_k = End_{Bimod}( \underbrace{ {}_N M_N \otimes_N {}_N M_N \otimes_N {}_N M_N \otimes_N \cdots }_{k} )

Now, denoting by ρ:NM\rho : N \hookrightarrow M the subfactor’s inclusion of algebras as well as the corresponding MM-NN bimodule ρ:(MM ρN)Bimod \rho : (M \stackrel{M_\rho}{\to} N) \in Bimod we get the dual amplimorphism ρ¯:MN\bar \rho : M \to N and the corresponding dual NN-MM bimodule bimodule ρ¯:(NN ρ¯M)Bimod \bar \rho : (N \stackrel{N_{\bar \rho}}{\to} M) \in Bimod and ρ\rho and ρ¯\bar \rho form an ambijunction. (Some details for instance in Pasquale Zito’s thesis).

In particular we should have (Nρ¯MρN)(NN ρ¯ρN)(N NM NN). (N \stackrel{\bar \rho}{\to} M \stackrel{\rho}{\to} N) \simeq (N \stackrel{N_{\bar \rho \circ \rho}}{\to} N) \simeq (N \stackrel{{}_N M_N}{\to} N) \,.

This would mean that indeed the tower of endomorphism algebras coming from End Bimod(ρ¯ρρ¯ρ) End_{Bimod}( \bar \rho \circ \rho \circ \bar \rho \circ \rho \circ \cdots ) appearing in Ocneanu’s work, discussed in Pasquale’s thesis and mentioned in the above entry is the same as the standard invariant of the subfactors.

Which is all just plausible because it should just be another way to say indeed that the planar algebra obtained from the subfactor NMN \hookrightarrow M corresponds to the spherical bicategory generated from the two objects NN and MM with the ambijoint morphisms NρMN \stackrel{\rho}{\to } M and Mρ¯NM \stackrel{\bar \rho}{\to } N between them.

Somehow my impression is that all this is very obvious to the experts, or at least it should be to those such as Noah Snyder I guess, who can translate back and forth between the spherical 2-category picture and that of planar algebra.

In Planar Algebras, I it takes Jones a bit of work to construct the planar algebra from the subfactor. Does this become just a consequence of abstract nonsense with the right 2-categorical technology available?

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

Posted by: Urs Schreiber on September 12, 2008 5:42 PM | Permalink | Reply to this

Re: no coincidence

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.

Posted by: Noah Snyder on September 12, 2008 6:31 PM | Permalink | Reply to this

Re: no coincidence

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.

Posted by: Bruce Westbury on September 13, 2008 4:54 AM | Permalink | Reply to this

Re: no coincidence

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.

Please do!

Posted by: Urs Schreiber on September 13, 2008 7:39 AM | Permalink | Reply to this

Re: no coincidence

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

Posted by: John Baez on September 13, 2008 6:19 AM | Permalink | Reply to this

Re: no coincidence

I recommend this for anyone…[link to Michael Müger’s paper].

Indeed. This paper has helped me a lot many times.

Posted by: Bruce Bartlett on September 13, 2008 10:58 AM | Permalink | Reply to this

How does one twist a fusion category by the Galois group?

Could I take this opportunity to ask the following (elementary) question about fusion categories to Those Who KnowTM^TM? The paper

Etingof, Nikshych and Ostrik, On fusion catgories (2002), arXiv link

is a fundamental one and I have spent a lot of time trying to understand it (and related papers, like this one). A fundamental tool which is used over and over again is that one can apparantly “twist” a fusion category by an element of the “Galois group” or something to get a new one, cf. page 5:

…since one can twist a fusion category by an automorphism of \mathbb{C} and get a new fusion category.

and in the proof of Proposition 8.24:

…Applying [something] to the the category g i(C)g_i(C)… [g ig_i is an element of a Galois group]

No doubt I’m being really dumb, but can someone explain to me what is meant here? I know I’m missing out on a powerful tool by not understanding this.

Posted by: Bruce Bartlett on September 13, 2008 12:07 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

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 π\pi’s or ee’s, except in harmless combinations like e 2πim/ne^{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(¯|)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(|)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(¯|)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…

Posted by: John Baez on September 13, 2008 6:00 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

Hi John,

Thanks a lot, yes you are definitely right that is the spirit of what is going on. I’d like to understand it a bit more explicitly though. You see, my understanding of the whole field of fusion categories is that one of the central motivations is:

Find and understand the precise equations which tell us what we need to build a (pivotal, spherical, …) fusion category.

That is my understanding for instance of reading Calaque, Etingof, Ostrik and Nikshych’s work. For instance, on page 8 of Lectures on tensor categories they say:

One of the basic questions of the theory of fusion categories is: Given a fusion ring AA, can it be realized as the Grothendieck ring of a fusion category? If yes, in how many ways?

You’re 100% right that all the data (associators, duality, …?) always tends to boil down to some algebraic equations. But my understanding is that outside of the simple examples, it’s not always easy to actually write down those equations explicitly. I mean, you yourself have some lecture notes demonstrating how to get from the associator to the 6j symbols/matrices:

John Baez, From the Associator to the 6j symbols, Quantum Gravity Seminar winter 2005.

But the equations satisfied by the 6j symbols (the pentagon equation) are quite awkward and cumbersome to write down.

Mmm, so perhaps “twisting a fusion category by an element of the Galois group” means the following: “We don’t know what the equations are explicitly, but we know at least that there are some equations, and thus we can twist everything by the Galois group of these equations”. Thinking it over, I guess that’s still a powerful insight. Bit weird though.

Posted by: Bruce Bartlett on September 13, 2008 7:20 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

A point that has not been mentioned is that fusion categories are rigid. You cannot have a moduli space of fusion categories. This also means that you can choose bases for the Hom vector spaces so that all the 6j-symbols lie in some number field. If you then apply a field automorphism it is clear that you get a fusion category. If you are working over the complex numbers this may not be the same as the one you started with.

An easy example of something similar is that if you have a finite group G of exponent e then given a complex irreducible representation you can choose a basis so that all matrices have entries in the number field generated by a primitive e-th root of unity. Given one such representation you can apply a Galois automorphism and this may give you a different such representation.

Another example of rigidity is that compact hyperbolic 3-manifolds are rigid. This means that the SL(2) character variety of the fundamental group is a finite set. Given a hyperbolic structure there is a number field such that every element of the fundamental group is represented by a matrix with entries in this number field. You can then apply a Galois automorphism and get a possibly different homomorphism and so a different hyperbolic manifold. I dont know what the geometrical significance of this is.

Posted by: Bruce Westbury on September 13, 2008 8:48 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

Bruce wrote:

John Baez, From the Associator to the 6j symbols, Quantum Gravity Seminar winter 2005.

There’s no need to restrict to SU(2) - just use irreps instead of reps

But the equations satisfied by the 6j symbols (the pentagon equation) are quite awkward and cumbersome to write down.

Curious! It was seeing those equations (I assume you mean Biedenharn-Elliot) as my favorite pentagon that led me BACK to realizing the 6j symbols were for associativity. One hang up I had was realizing the physicists (well, some) thought that vector spaces had preferred basis, assumed bu not mentioned.

Posted by: jim stasheff on September 14, 2008 12:55 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

Thanks for reminding me about the Biedenharn-Elliot equations; I have to admit I have never actually read a paper where these equations are explicitly discussed though a Google search reminded me that the Wizard talked about them ages ago. For some reason these equations have never been mentioned in the literature I’ve been reading about fusion categories, even though it seems clear that the central motivation of the ENO paper,

One of the basic questions of the theory of fusion categories is: Given a fusion ring A, can it be realized as the Grothendieck ring of a fusion category? If yes, in how many ways?

seems to be roughly about calculating the solutions to these equations (which are rigid, as Bruce Westbury pointed out). Mmm.

Posted by: Bruce Bartlett on September 14, 2008 10:18 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

At the back of vol II of Biedenharn-Louck are some marvelous graphics, beyond the pentagon.

Posted by: jim stasheff on September 15, 2008 2:20 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

Are you referring to Biedenharn, Louck, Racah-Wigner Algebra in Quantum Theory? Is there a volume II? Have to admit I’ve never heard about these books.

Posted by: Bruce Bartlett on September 15, 2008 3:17 PM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

Biedenharn and Louck wrote two enormous books in Rota’s red Encyclopedia of Mathematics series: Angular Momentum in Quantum Physics: Theory and Application and Racah-Wigner Algebra in Quantum Theory.

Anyone who enjoys modern terminology, category theory, mathematical elegance and other new-fangled stuff is bound to find these books a bit appalling. But there’s a lot of good stuff buried in them! They’re packed with obscure and tantalizing facts about SU(2)SU(2), the 6j and higher ‘nn-j’ symbols for this group, the Biedenharn–Elliot identities (now called pentagon identities), and so on.

They’re sort of like your grandparents’ attic, full of quaint and curious souvenirs and perhaps even some precious forgotten jewels mixed in with lots of dusty junk.

Posted by: John Baez on September 23, 2008 5:15 AM | Permalink | Reply to this

Re: How does one twist a fusion category by the Galois group?

You find us talking about the automorphisms of \mathbb{C} in this post. Paul Yale wrote an article about it.

Posted by: David Corfield on September 13, 2008 10:20 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Ingo Runkel and Rafal R. Suszek have a new preprint: arXiv:0808.1419, “Gerbe-holonomy for surfaces with defect networks”. On page 44: they point out the relation you mentioned and quote Jones’ paper.

My personal view is that CFTs with topological defects are related to certain “categorification” of CFTs.

Actually, it is Huang who proved that the category of certain algebras over a partial operad consisting of spheres with punctures and local coordinates is isomorphic to the category of vertex operator algebras. The full field algebra was first introduced by Huang and myself in: arXiv:math/0511328. For 2-d open-closed CFT, maybe the following paper is helpful: Cardy-Condition for Open-Closed Field Algebras . Also, if anyone is interested in the note Urs mentioned, he or she can ask me for it.

John,

I am currently visiting Caltech. I will stay here for 6 months. I believe that it is not so far away from Riverside. If you are interested in CFT from VOA point of view, I can come to meet you.

Posted by: Liang Kong on September 20, 2008 10:16 PM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

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.

Posted by: John Baez on September 23, 2008 6:44 AM | Permalink | Reply to this

Re: Planar Algebras, TFTs with Defects

Liang tells me that he is currently experiencing problems with accessing the nnCafé. If you haven’t yet, maybe send him an email.

Posted by: Urs Schreiber on September 23, 2008 9:16 AM | Permalink | Reply to this
Read the post Back and Catching Up
Weblog: The n-Category Café
Excerpt: A list of things to read.
Tracked: September 21, 2008 2:55 PM

Re: Planar Algebras, TFTs with Defects

Dear John:

I am sorry for late reply. I did not check the blog so often.
I will be very happy to give a talk.
I am available except Wednesday and Oct. 7th.

Best regards,
Liang

Posted by: Liang Kong on September 27, 2008 12:28 AM | Permalink | Reply to this

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