Masaki Izumi, Vaughan Jones, Scott Morrison and I recently uploaded to the arXiv the 3rd and final part of the four part series “Subfactors of index less than 5.” This is a project we’ve been working on for a long time (since Emily, Scott and I started running Planar Algebra Programming Camps in spring of ’08), and after three years and a lot of work from many people it’s very exciting to finally have made it there.

In this post I’ll state the main theorem, say a few words about the history, and then explain the main takeaway lesson we learned in this project.

Here’s the main theorem:

Any subfactor of index less than 5 has one of the following standard invariants:

- One of the ADE planar algebras, with index less than 4
- One of the affine ADE planar algebras, with index equal to 4
- An planar algebra, with index greater than 4
- The Haagerup planar algebra and its dual (index )
- The extended Haagerup planar algebra and its dual (index is a certain cubic integer)
- The Asaeda-Haagerup planar algebra and its dual (index )
- The 3311 Goodman-de la Harpe-Jones planar algebra and its dual (index )
- The 2221 Izumi planar algebra and its complex conjugate (index )

The history of this result is that the classification up to 4 was done in the 80’s (with Ocneanu being the main name), the classification up to was begun by Haagerup in 1994 and completed by work of Asaeda-Haagerup, Bisch, Asaeda-Yasuda, and Bigelow-Morrison-Peters-Snyder (which was the first PAPC paper back in 2009). In addition to the main series of papers (part 1 is Scott and I, part 2 is joint with Emily Peters and David Penneys, and part 4 is written by David Penneys and James Tener), the other key paper in extending the classification to index 5 was our paper with Frank Calegari.

The origin of this project was a conversation between Emily, Richard Burstein, and I in the Nashville airport after the Planar Algebras Shanks workshop. We were discussing the possibility of making an “atlas of subfactors” along the lines of Dror Bar-Natan and Scott’s “Knot Atlas” which would automate calculations about small subfactors. My original idea was that if we used global index instead of index, then the classification problem would become finite and thus automatable. As we learned more and more about the existing results, we ended up instead concentrating on two projects: automating and strengthening Haagerup’s approach for searching for small index subfactors (which lead to this project), and automating and improving techniques for constructing small index subfactors (which lead to the paper with Stephen Bigelow). Basically we would sit around in a house at Bodega Bay for a few days (generously lent to us by Vaughan) while I wrote code for the first project, Emily wrote code for the second project, and Scott ran back and forth getting us unstuck.

The most interesting “big picture” lesson to take from this project is the following. Haagerup’s initial classification result made it look like “exotic subfactors” were quite common. Not only did he find a brand new subfactor, to all appearances it looked like there was an infinite family of subfactors all with small index. Asaeda-Yasuda dashed the latter hope, showing that only the first two of this series were possible, but it still seemed likely that exceptional subfactors were quite common. In fact, we long thought that actually classifying all subfactors of index less than 5 was going to be too hard and that we should just search to find new subfactors in that range and construct them. But as it turns out, even though the number of combinatorial possibilities grows rapidly as you go from to 5, the only subfactors in that range were already known. Thus it now appears that exotic subfactors may be relatively rare. In fact, I would not be surprised if the full list of subfactors which are 4-supertransitive (this is a condition analogous to a group action being 4-transitive, but stronger) is just the ADE type graphs, Asaeda-Haagerup, and extended Haagerup.

So, Noah, are you going to explain your numbering of the parts? Why does this paper come before Penneys-Tener?

I wouldn’t be surprised if we find a few more 4-supertransitive subfactors (there’s a reasonable looking candidate at index ), but I’d like to enthusiastically endorse the conjecture that we already have the complete list of 5-supertranstive subfactors, especially since someone proving this would result in Emily owing me champagne.

The real reason is that Part 1 was finished and submitted (with references to the rest of the series) before it was clear that part 4 would be finished before part 3.

It also makes a certain amount of logical sense. The paper with Frank says that any family of a certain form (which we call “vines” and which were the main families in Haagerup’s partial classification) can only give rise to finitely many examples (for arithmetic reasons!). Thus, when going through the argument every time a family of that form comes up you declare victory. But you still have to actually do the calculations applying the theorem. It makes sense to do those calculations all together at the end, because you might have more vines popping up as you kill the “weeds.”

Plus, we don’t want to let Scott Carnahan (whose part 1 refers to parts 2 and 5, but not 3 or 4) have all the strange numbering fun!

Let’s say someone wants to reference this result. Which paper do they cite (part I? all of them?) Is it a theorem of Morrison-Snyder, or a theorem of Izumi-Jones-Morrison-Penneys-Peters-Snyder-Tener?

Or maybe a theorem of Morrison-Snyder-Morrison-Penneys-Peters-Snyder-Izumi-Jones-Morrison-Snyder-Penneys-Tener? This seems like the type of project (multiple parts with different authors proving a single theorem) that might have done with a research announcement in (say) Comptes Rendus Mathematique.

The classification up to index is due to Haagerup/Bisch/Haagerup-Asaeda/Asaeda-Yasuda/Bigelow-Morrison-Peters-Snyder, so it’s not like you have to cite more papers for this.

The Comptes Rendus idea is a good suggestion. The plan we had instead is to write a survey article afterwards, rather than an announcement before. Partly that’s because we were not optimistic that we’d finish the whole project as quickly as we did. When Part 1 was finished, we were still stuck on two key points.

(btw, it’s Tener, not Tenner. I took the liberty of editing the post and comments to fix this)

Oops, sorry James! In my defense, one of my classmates in undergrad is a mathematician whose last name is spelled Tenner.