### 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 $\{a,b\}$ and where one considers only defect lines $a \to b$ and $b \to a$.

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

a) Every finite index type $II_1$ vonNeumann algebra subfactor $N \subset M$ comes with a sequence $M_i$ of subfactors
called its *standard invariant*.

b) By assigning $M_k$ to the circle with $2k$ marked points (and, by assumption, the intervals in between necessarily labeled alternatingly by $a$ and $b$) these subfactors $M_k$ induce the strudcture of a planar algebra.

c) Every planar algebra of subfactors arises this way as a genus 0 2d TFT with $\{a,b\}$-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 $n$-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 $_{N} M_M$ given by te subfactor inclusion $N \subset M$.

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

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…

## 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.

In everything I’ve read, it seemed like the

onlycase 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-fledged2d 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?)