### 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 $(E_8, E_7, E_6)$ and (Monster, Baby monster, Fischer group $Fi_24$) and with

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

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?

## Re: John McKay Visits Kent

David wrote:

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 $j$ 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 beallowedto 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.