The n-Category Café

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

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 $z \in \mathbb{C}$.

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 $z=0$, the outgoing at $z=\infty$ and the remaining incoming one at that parameter $z$.

(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

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

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.

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.

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.

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.

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 $M$ is the group of symmetries of the Moonshine vertex operator algebra $V$. So it acts on the category of representations of $V$. Since $Rep(V)$ 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 $U(1)$-valued 2-cocycles on the Monster (this is the ingredient you would need to ensure that $M$ acting on $Rep (V)$ 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 $Rep(V)$ 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.

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 $G$, which involves the Kac-Moody extension of the loop group, there is a nice 2-representation on $Bimodules \hookrightarrow 2VectorSpaces$ 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 $A_x$, but thought of as placeholders for their module categories $Mod_{A_x}$.

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 $String(G)$: it basically gives the theory of affine groups a 2-categorically refined perspective.

…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?

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.

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.