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October 21, 2008

John McKay Visits Kent

Posted by David Corfield

John McKay, of McKay corrrespondence fame, came to speak to us at Kent yesterday. In a hour we were given his views on the past, present and future of the study of finite simple groups. The past was accessible enough, back to Plato and Empedocles, and beyond them to the Scottish stones. We were told to pester the Ashmolean Museum if they are reluctant to show them, since they are obliged to do so.

Naturally, the present and future were more difficult. Before briefly giving you a chain of terms I managed to jot down, a question. What biographical detail connects McKay with Robert Moody?

So the main take home message was that the body of work stretching from 1964, when Janko discovered the first of his sporadic groups, up until the writing up of the proof of the classification of finite simple groups has employed a very limited outlook, largely group theory and finite geometry. Much greater contact with the rest of mathematics is needed. Three promising avenues are:

1. Integrable systems: replicable functions, Schwarzian, Grunsky coefficients, Bergman kernel, energy stress tensor, and the DKP equation have something to do with Witten’s 24 dimensional compact, orientable, spin manifold with the action of monster group on the free loop space.

2. Symplectic geometry: There’s a correspondence between (E 8,E 7,E 6)(E_8, E_7, E_6) and (Monster, Baby monster, Fischer group Fi 24Fi_24) and with

(120tritangentsonsexticofgenus4,28bitangentsonquartic,27linesoncubic). (120 tritangents on sextic of genus 4, 28 bitangents on quartic, 27 lines on cubic).

And 360 cusps link to the 120 tritangents.

3. Hirzebruch - Manifolds and Modular Forms: Chern classes, symmetric function theory, replicable functions (again). A Hopf algebra which commutes a Hecke algebra. A Witt ring. Nine fields with class number 1, which have something to do with the Monster as the symmetries of a conformal field theory.

So, as you can see, it’s largely come over to me as an incomprehensible stream of consciousness, but it was fun to listen to. And we got to hear McKay’s opinions on the vertex operator algebra approach to moonshine: why explain something complicated with something even more complicated?

Posted at October 21, 2008 9:06 AM UTC

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Re: John McKay Visits Kent

David wrote:

And we got to hear McKay’s opinions on the vertex operator algebra approach to moonshine: why explain something complicated with something even more complicated?

Vertex operator algebras are just an algebraist’s way of talking about conformal field theory. A conformal field theory is just a 2d quantum field theory that has all angle-preserving transformations as symmetries. These ‘conformal transformations’ are fundamental in complex analysis — as is the jj function, whose coefficients are mysteriously linked to representations of the Monster group.

This all seems quite promising. So, while I’m no expert, I don’t regard this line of thought as ‘explaining something complicated by something more complicated’. True, it takes a while to learn quantum field theory — but even elementary particles have mastered it, so we can too.

I do, however, sympathize with McKay’s remark in the following way: the axioms for a vertex operator algebra are a very sneaky distillation of the ideas of quantum field theory — not what I’d consider a clear perspective on the subject. Worse, they’re often presented in a hideously unmotivated way — presumably to protect algebraists from the pain of learning some physics. In fact, I almost think nobody should be allowed to study vertex operator algebras without first taking a course on quantum field theory. It’s like learning Lie algebras without knowing about Lie groups: it cuts you off from the key source of inspiration.

Posted by: John Baez on October 21, 2008 5:23 PM | Permalink | Reply to this

Re: John McKay Visits Kent

Even more naively, a vertex operator is just a differential operator - an (exponential) function of x and p satisfying [x, p] = i.

Posted by: Thomas Larsson on October 22, 2008 9:52 AM | Permalink | Reply to this

Re: John McKay Visits Kent

Even more naively, a vertex operator is just a differential operator - an (exponential) function of x and p satisfying [x,p]=i[x,p] = i.

This would be (just) the degenerate case of vertex operator algebras in (0+1)-dimensional QFT aka quantum mechanics, where a vertex operator is just an operator in the Heisenberg picture “localized” at some fixed time.

The standard term “vertex operator algebra” is commonly used for 2-dimensional QFT, where everything crucially depends on one more parameter #.

Posted by: Urs Schreiber on October 22, 2008 2:36 PM | Permalink | Reply to this

Re: John McKay Visits Kent

The simplest geometric way to think of a vertex operator algebra is as a representation of the category whose morphisms are punctured conformal spheres and whose composition is gluing at punctures.

Algebraically this is easily thought of as an associative algebra, where the product operation depends on a complex parameter: there is not just the product, but a product operation for all zz \in \mathbb{C}.

The two pictures relate in the obvious way: the product is the image of the Riemann sphere with two incoming and one outgoing punctures, one incoming at z=0z=0, the outgoing at z=z=\infty and the remaining incoming one at that parameter zz.

(This is by theorems by Huang and Kong, as we mentioned recently.) Put this way, nothing here should be mysterious. Of course the standard notation for this simple idea takes a little getting used to. That product operation is usually denoted

Y(,):V××VV, Y(-,-)- : V \times \mathbb{C} \times V \to V \,, i.e. if you specify one vector vv and a complex number zz you get a linear map Y(v,z):VV Y(v,z) : V \to V called the “operator product with vv” but to be thought of simply as “multiplication with vv at a distance zz”.

The general simple idea here of an algebra whose product depends on a parameter is really not even intrinsic to the conformal setup. Just recently Steve Hollands suggested in QFT in terms of consistency conditions to consider this formalization of “operator product expansion” more generally for general QFTs. The basic idea is really pretty elementary and non-mysterious. Have a look at Hollands’ article.

We talked about that here.

Posted by: Urs Schreiber on October 22, 2008 2:22 PM | Permalink | Reply to this

Re: John McKay Visits Kent

Indeed, Huang and Kong’s operadic approach to vertex operator algebras is the only one I’d recommend to mathematicians wishing to study VOAs while remaining ignorant of quantum field theory. It’s geometrical, beautiful, and much more conceptual than the agonizing axioms one usually sees.

However, I still think that learning VOAs without first learning quantum field theory is like trying to fly a fighter jet without having piloted a Cessna. You may succeed in doing some damage… but probably mostly to yourself.

Posted by: John Baez on October 27, 2008 4:34 PM | Permalink | Reply to this

Re: John McKay Visits Kent

This all sounds seriously fascinating and a ray of light giving hope to those of us who study the Finite Simple Groups. (One often gets the feeling that the rest of mathematics views them as somehow old news barely worth the time of day and with no future at all.)

The trouble is that a list of words without context is not very helpful. Presumably there is evidence to suggest the connection between the Finite Simple Groups and each of 1, 2 and 3, but where can one go to find out about it?

Any chance of some book/paper/arXiv references?

Posted by: Ben Fairbairn on October 21, 2008 8:24 PM | Permalink | Reply to this

Re: John McKay Visits Kent

It was extraordinarily condensed. It looks like this conference covered much of what he spoke about, and it has videos.

Ah, from the program, I see I should have put ‘dKP’ rather than ‘DKP’ – dispersionless Kadomtsev Petviashvily (dKP) hierarchy.

Posted by: David Corfield on October 21, 2008 9:55 PM | Permalink | Reply to this

Re: John McKay Visits Kent

I assume Ben has read Terry Gannon’s book Moonshine Beyond the Monster — but for anyone just starting, that’s a great place to start.

Posted by: John Baez on October 27, 2008 4:37 PM | Permalink | Reply to this

Re: John McKay Visits Kent

A talk on Arnold’s trinities! Oh, I wish I had been there. Does anyone have notes?

Posted by: Kea on October 21, 2008 8:57 PM | Permalink | Reply to this

Re: John McKay Visits Kent

One reference is:

The Virasoro Algebra and Some Exceptional Lie and Finite Groups by Michael P. Tuite

This has been published in SIGMA 3 (2007) and is available at math.QA/0610322

Posted by: Bruce Westbury on October 21, 2008 9:23 PM | Permalink | Reply to this

Re: John McKay Visits Kent

I find talking to John an extremely heady experience, kind of like the scene in the movie π\pi when our hero meets the young Kabbalist in the diner.

I remember him taking some sequence of three exceptional objects, producing the numbers 1, 2, and 3, and saying “of course, these are the Schur multipliers of the Monster, the Baby Monster, and the” some other group, Higman-Sims maybe? At this point I protested. So he produced a ream of other coincidences to buttress this, some involving representation theory of the Virasoro algebra in characteristic p, and well…

I’m surprised that 2-representation theory wasn’t more explicitly looked to as a savior. The impression I get from Andre Henriques is that some groups want to act on vector spaces, and for some you should look at a higher categorical level for their representations; the String group and the Monster group being the most definitive examples.

Posted by: Allen Knutson on October 21, 2008 11:11 PM | Permalink | Reply to this

Re: John McKay Visits Kent

I’m surprised that 2-representation theory wasn’t more explicitly looked to as a savior. The impression I get from Andre Henriques is that some groups want to act on vector spaces, and for some you should look at a higher categorical level for their representations; the String group and the Monster group being the most definitive examples.

I once attempted to think about this. The Monster group MM is the group of symmetries of the Moonshine vertex operator algebra VV. So it acts on the category of representations of VV. Since Rep(V)Rep(V) is a semisimple category (in fact it only has one simple object), I thought this would be a nice example of a 2-representation. But then someone told me that there are no non-trivial U(1)U(1)-valued 2-cocycles on the Monster (this is the ingredient you would need to ensure that MM acting on Rep(V)Rep (V) has a chance of being ‘interesting’). Then I found found the talk Orbifold Confrmal Field Theory and the Cohomology of the Monster by Geoffrey Mason on Google. This talk made me realize that indeed there is interesting stuff that can be said about the Monster and 2-representations, but the relevant 2-vector space is not the category Rep(V)Rep(V) of ‘ordinary’ representations of the monster, but rather the category of ‘twisted’ representations, which you think of as forming a 2-representation of the loop groupoid of the Monster. Happily, he even mentions at one point that

This fact is the basis for understanding Monstrous Moonshine (see below).

All the stuff he talks about (transgression, cocycles, Hochschild cohomology, loop groups) is classic higher-categorical extended TQFT mumbo-jumbo.

Sadly though I don’t understand vertex operator algebras :-) Not by a long shot. And that is where I inevitably meet my shipwreck.

Posted by: Bruce Bartlett on October 22, 2008 1:03 AM | Permalink | Reply to this

Re: John McKay Visits Kent

I get from André Henriques is that some groups want to act on vector spaces, and for some you should look at a higher categorical level for their representations; the String group and the Monster group being the most definitive examples.

Using the strict version of the String 2-group of GG, which involves the Kac-Moody extension of the loop group, there is a nice 2-representation on Bimodules2VectorSpacesBimodules \hookrightarrow 2VectorSpaces in the vonNeumann algebraic context, which essentially reproduces the kind of construction that Stolz and Teichner originally considered for String-bundles in Wiaeo? and leads to associated String 2-bundles whose fibers are vonNeumann algebras A xA_x, but thought of as placeholders for their module categories Mod A xMod_{A_x}.

Konrad and I discuss this in example 4.19, p. 73 in the context of String 2-bundles with connections. It arises there as a special case of a very general type of 2-representations of strict 2-groups.

There will be 50 ways to look at String and its representation theory, but to me this looks like one good reason to consider the strict version of String(G)String(G): it basically gives the theory of affine groups a 2-categorically refined perspective.

Posted by: Urs Schreiber on October 22, 2008 7:14 AM | Permalink | Reply to this

Re: John McKay Visits Kent

…for some you should look at a higher categorical level for their representations; the String group and the Monster group being the most definitive examples.

Does this mean that in some sense the Monster is really a 2-group?

Posted by: David Corfield on October 22, 2008 9:16 AM | Permalink | Reply to this

Re: John McKay Visits Kent

Well, perhaps there is an argument to be made that the Monster is really a 2-group, or at least a 2-groupoid. I know some people will laugh and think that this is some kind of hyper-deranged n-category madness, but anyhow, consider the Stolz-Teichner programme. One of their projects is to understand a conformal field theory very roughly as a 2-functor

(1)Z:2CobSuperVect. Z : 2Cob \rightarrow SuperVect.

where 2Cob is very roughly the 2-category of points, lines and surfaces equipped with conformal/euclidean metrics.

The Moonshine module is a vertex operator algebra, so it is a conformal field theory, so it should be formulatable as such a 2-functor.

The Monster is the group of symmetries of a conformal field theory. And the group of symmetries of a 2-functor really forms a 2-groupoid.

Posted by: Bruce Bartlett on October 22, 2008 2:23 PM | Permalink | Reply to this

Re: John McKay Visits Kent

There is some good experimental evidence that higher structure should be present in generalized moonshine. I’ve heard some suggestions involving 2-groups and monster-equivariant elliptic cohomology, which apparently looks a lot like a geometrically rich variation on 2-vector spaces. The Hopkins-Kuhn-Ravenel generalized characters of such a cocycle seem to be connected to genus 1 orbifold correlators.

If the monster acted strictly on the orbifold conformal field theory, you might expect an action of centralizers of elements on the corresponding twisted modules, but for many conjugacy classes, you get a projective action, arising from McKay-Thompson series having unusually high level as modular functions, and this seems to come from nontrivial elements of H^3(Monster,C*), which has not been calculated. I don’t know how to encode this sort of data into a 2-group, though.

Nora Ganter has some ArXiv preprints concerning homotopy-theoretic aspects of moonshine, and Morava also wrote a short note.

Posted by: Scott Carnahan on October 23, 2008 2:14 AM | Permalink | Reply to this

Re: John McKay Visits Kent

It would be fun if n-thinking led to a rethink of the sporadics. Something like, there’s an infinitely family of 2-groups whose shadows at the 1-level are sometimes sporadic simple groups and sometimes not groups at all. We discussed this kind of phenomenon a while ago.

Posted by: David Corfield on October 23, 2008 1:34 PM | Permalink | Reply to this

Re: John McKay Visits Kent

David wrote:

It would be fun if nn-thinking led to a rethink of the sporadics.

‘Fun’? It would be AWESOME! Just as awesome as if the Monster was the symmetry group of quantum gravity in 2+1 dimensions, as Witten has suggested.

I guess I should start thinking about this a bit. I don’t have the technical expertise to prove anything about the Monster, but I might be one of the first to make some wild guesses.

Posted by: John Baez on October 24, 2008 4:34 AM | Permalink | Reply to this

Re: John McKay Visits Kent

The Monster is not a semidirect product, but to get down to detail, how do you think about your Poincaré 2-group? Is it that anything ‘good’ about the 1-group is understood by the 2-group, and understood better in many cases? When you’ve worked out the 2-represention theory of the 2-group will it subsume the 1-representation theory?

Posted by: David Corfield on October 24, 2008 9:06 AM | Permalink | Reply to this

Re: John McKay Visits Kent

David wrote:

The Monster is not a semidirect product, but to get down to detail, how do you think about your Poincaré 2-group? Is it that anything ‘good’ about the 1-group is understood by the 2-group, and understood better in many cases?

I don’t know. Frankly I’ve spent very little time thinking about what this 2-group really ‘means’. Usually I think a mathematical object XX' provides a more detailed view into a subject than some other object XX if you can decategorify XX' and get XX. But if we take the Poincaré 2-group, and form the group of isomorphism classes of objects, we don’t get the Poincaré group — we get the Lorentz group.

The only reason I called it the ‘Poincaré 2-group’ is that, just like the Poincaré group, it’s built using the action of the Lorentz group on the translation group. We can build a ‘Poincaré nn-group’ for any nn following the same pattern. But what do they mean? I don’t know.

When you’ve worked out the 2-represention theory of the 2-group will it subsume the 1-representation theory?

Not in any obvious way. By the way, the 2-representations of this 2-group were worked out quite a while ago by Crane and Sheppeard. What we’re doing now is to make their work a bit more rigorous and a lot more general.

After that, Laurent Freidel and Aristide Baratin want to continue Crane and Sheppeard’s work by studying some 4d state sum models built using 2-representations of the Poincaré 2-group. They’ve made some interesting progress. But since I have little conceptual understanding of where this 2-group fits in the grand scheme of things, I’ve always been a bit reluctant to work on that. And, I’m way too busy.

All this probably has rather little to do with the Monster.

Posted by: John Baez on October 24, 2008 6:10 PM | Permalink | Reply to this

Re: John McKay Visits Kent

Allen said

I find talking to John an extremely heady experience.

Evidently Morava too,

I could never have approached this subject but for the untiring interest of John McKay, who has often seemed to me an emissary from some advanced Galactic civilization, sent here to speed up our evolution.

Posted by: David Corfield on October 27, 2008 5:05 PM | Permalink | Reply to this

Re: John McKay Visits Kent

The biographical detail linking McKay and Moody is that they both grew up in the small village of Otford. McKay gave the impression that only a couple of hundred people lived there. Perhaps the Wikipedia population figure of 3,528 takes in a larger area, or maybe the population has grown.

Even so, were that success rate to be replicated throughout the UK we might expect around 30,000 mathematicians of their stature.

Posted by: David Corfield on October 23, 2008 1:08 PM | Permalink | Reply to this

Re: John McKay Visits Kent

But if you divide the UK into population centres of about 3500 people then you would only need 170 UK mathematicians for there to be an even chance of being able to find two from the same place.

Posted by: PhilG on November 4, 2008 4:50 PM | Permalink | Reply to this

Re: John McKay Visits Kent

As a context today, I assume the classification complete. We need
to think of extending Chevalley’s
Tohoku J (1955?) construction to
include sporadics (at least M and
may be more - all of them if we are lucky.)

Posted by: John McKay on November 4, 2008 1:13 PM | Permalink | Reply to this

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