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Stream: deprecated: mathematics

Topic: vertex (operator) algebras

Tim Hosgood (Nov 30 2022 at 13:24):

An algebra is a vector space $V$ along with a morphism $V\to\operatorname{End}(V)$ (satisfying some properties). The morphism gives a multiplication on $V$.

A vertex algebra is a vector space $V$ along with a morphism $V\to\operatorname{End}(V[t,t^{-1}])$ (satisfying some properties). The morphism gives $\mathbb{Z}$-many "compatible" multiplications on $V$.

Is there a nice categorical way of understanding how the second generalises the first? I know this is a vague question, but I guess I'm expecting to hear "power series monad" or "operad" or something like this. All the references I know on V(O)As are very much not from a categorical/combinatorial perspective, and it would be nice to see if category theory offers a nice succinct way of packaging up all the information hidden away in the "satisfying some properties" part of the definition.

Tim Hosgood (Nov 30 2022 at 13:25):

(apparently vertex operator algebras are equivalent to holomorphic algebras over the holomorphic punctured sphere operad, but this isn't quite the type of answer I was hoping for (unless somebody has some nice explanation for that))

John Baez (Nov 30 2022 at 16:09):

To me the punctured sphere idea is the best way to make the definition of vertex operator algebra succinct and categorical. Basically we just take the usual string diagram for multiplication in a commutative algebra, namely a cobordism where two circles merge to one:

.

and take it very seriously, putting a conformal structure on the surface (and treating the circles as 'punctures').

John Baez (Nov 30 2022 at 16:10):

This simultaneously explains why VOAs are important - namely, the capture a certain portion of conformal field theory - and replaces the usual complicated algebraic axioms for a VOA with pretty pictures.

John Baez (Nov 30 2022 at 16:15):

I guess you've seen this stuff on the nLab:

The traditional definition of vertex operator algebra (VOA) is long and tends to be somewhat unenlightening. But due to Huang it is now known that vertex operator algebras have equivalently a characterization which manifestly captures the relation to conformal field theory:

There is a [[monoidal category]] or [[operad]] whose morphisms are conformal spheres with $n$-punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along punctures. This can be regarded as a category $2\mathsf{Cob}_{\textrm{conf}}^0$ of 2-dimensional genus-0 conformal [[cobordisms]].

As shown by theorems by [[Yi-Zhi Huang]] and [[Liang Kong]], a vertex operator algebra is precisely a [[holomorphic representation]] of this category, or an algebra for this [[operad]], i.e. a [[monoidal functor]]

$V : 2\mathsf{Cob}_{\mathrm{conf}}^0 \to \mathsf{Vect}$

such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.

Tim Hosgood (Nov 30 2022 at 18:07):

John Baez said:

To me the punctured sphere idea is the best way to make the definition of vertex operator algebra succinct and categorical. Basically we just take the usual string diagram for multiplication in a commutative algebra, namely a cobordism where two circles merge to one and take it very seriously, putting a conformal structure on the surface (and treating the circles as 'punctures').

this is nice! but raises more questions for me...

1. are you saying that if we used string diagrams instead of these puffed up things with conformal structure, then we'd recover exactly usual algebras? so VOAs are exactly what you get by taking algebras and making the strings into tubes and adding conformal structure?
2. if the answer to the above is "yes" (or, even if not, I guess), how does passing from strings to conformal punctured spheres correspond to replacing $V$ by $V[t,t^{-1}]$? maybe this has something to do with the fact that lots of people seem to call the process of adjoining $[t,t^{-1}]$ something like "taking the loop space"

John Baez (Nov 30 2022 at 18:13):

1. Yes, algebras with extra properties and/or structure: probably commutative algebras, maybe with some extra structure.

John Baez (Nov 30 2022 at 18:20):

2. Throughout string theory there's a constant called the "string tension", and as it goes to infinity string theory should approach a particle theory, because the expected radii of the little loops of string goes to zero. So, if I were really good at string theory I could take a particular VOA, see the string tension as a parameter in the definition of this VOA (which evil mathematicians may have set equal 1, the way they do with parameters they don't like), and then take the limit as the string tension goes to infinity and get a commutative algebra.

John Baez (Nov 30 2022 at 18:21):

Since I'm not very good I would do better to just look up the answer somewhere, or ask around: "how can we see a VOA as a deformation of a commutative algebra?"

John Baez (Nov 30 2022 at 18:24):

Or I could just guess. I think your guess is not bad. The Laurent polynomials $\mathbb{C}[t,t^{-1}]$ are spanned by elements $t^n$, which are an orthonormal basis of functions on the unit circle $\{|t| = 1\}$ in the complex plane - that's how those Laurent polynomials are related to the "loops of string" that I'm talking about, namely the circles you get by slicing this picture horizontally:

.

So, we could try to insert a parameter into the definition of a specific VOA that lets us replace $\{|t| = 1\}$ with $\{|t| = r\}$ and then take the limit $r \to 0$.

John Baez (Nov 30 2022 at 18:29):

I'm talking about a specific VOA to play it safe, but there could easily be a systematic "functorial" way to do this for all VOAs.

John Baez (Nov 30 2022 at 18:32):

Okay, I think I've gone as far as I want to go now... so I will hand you over to this capable-sounding paper:

• Bong H. Lian and Andrew R. Linshaw, Vertex algebras and commutative algebras.

whose abstract says:

The paper then proceeds to describe progress made in the field of VOAs in the last 15 years which is based on fruitful analogies and connections between VOAs and commutative algebras. First, there are several functors from VOAs to commutative algebras that allow methods from commutative algebra to be used to solve VOA problems.

Tim Hosgood (Dec 01 2022 at 00:43):

John Baez said:

I'm talking about a specific VOA to play it safe, but there could easily be a systematic "functorial" way to do this for all VOAs.

I would be very interested in this, for sure!

Tim Hosgood (Dec 01 2022 at 00:43):

but thanks for all your answers — there's a lot for me to digest :-)

John Baez (Dec 01 2022 at 21:16):

Sure! By the way, that paper discusses 'abelian vertex algebras' and says these are equivalent to differential graded supercommutative algebras; since these are commutative monoid objects in a certain symmetric monoidal category I suspect these are the 'particle limits' we were talking about, where the 2d cobordisms I drew reduce to the usual string diagrams you might draw for operations on a commutative monoid.

Simon Burton (Dec 01 2022 at 22:50):

John Baez said:

the usual string diagram for multiplication in a commutative algebra

This looks like a 2-morphism in a monoidal 2-category. Can you say a bit more about how you get a commutative algebra here?

John Baez (Dec 04 2022 at 00:02):

Monoidal 2-categories are overkill here, though very interesting.

John Baez (Dec 04 2022 at 00:03):

The category 2Cob, where objects are collections of circles and morphisms are 2-dimensional cobordisms, is a symmetric monoidal category.

John Baez (Dec 04 2022 at 00:03):

It's a famous fact that 2Cob is the free symmetric monoidal category on a commutative Frobenius monoid.

John Baez (Dec 04 2022 at 00:04):

A 2d TQFT is by definition a symmetric monoidal functor $F: 2\mathsf{Cob} \to (\mathsf{Vect}, \otimes)$.

John Baez (Dec 04 2022 at 00:05):

By the stuff I just said, there's a correspondence between 2d TQFTs and commutative Frobenius monoids in $(\mathsf{Vect}, \otimes)$.

John Baez (Dec 04 2022 at 00:07):

But these are just commutative Frobenius algebras.

John Baez (Dec 04 2022 at 00:12):

So in a 2d TQFT, this picture just indicates multiplication in a commutative Frobenius algebra:

John Baez (Dec 04 2022 at 00:14):

.

John Baez (Dec 04 2022 at 00:16):

We can get into monoidal 2-categories if we're interested in extended TQFTs - that's what that 'cobordism hypothesis' stuff is about - but we were simply talking about how a vertex operator algebra is a generalization of a commutative algebra.

John Baez (Dec 04 2022 at 00:17):

If you read the 'genus zero' stuff here:

The traditional definition of vertex operator algebra (VOA) is long and tends to be somewhat unenlightening. But due to Huang it is now known that vertex operator algebras have equivalently a characterization which manifestly captures the relation to conformal field theory:

There is a [[monoidal category]] or [[operad]] whose morphisms are conformal spheres with $n$ punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along punctures. This can be regarded as a category $2\mathsf{Cob}_{\textrm{conf}}^0$ of 2-dimensional genus-0 conformal [[cobordisms]].

As shown by theorems by [[Yi-Zhi Huang]] and [[Liang Kong]], a vertex operator algebra is precisely a [[holomorphic representation]] of this category, or an algebra for this [[operad]], i.e. a [[monoidal functor]]

$V : 2\mathsf{Cob}_{\mathrm{conf}}^0 \to \mathsf{Vect}$

such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.

John Baez (Dec 04 2022 at 00:18):

and think about it a while, you'll see it cuts us down from commutative Frobenius algebra to mere commutative algebras.

John Baez (Dec 04 2022 at 00:18):

(The conformal structure, on the other hand, makes things a lot more complicated.)