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There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. I am not even sure whether I gradually do understand more or not.

In this question I decided to single out one of the simplest "building blocks" or "starting points" of the whole stuff, since I could not really find much about it.

My question is: how exactly shall I understand mathematically one separately taken vertex operator? What kind of object is it?

Very imprecisely and vaguely, I developed this kind of intuition: a vertex operator is an operator-valued distribution concentrated at a single point, i. e. an operator-valued delta-function. Is this correct?

Trying to be more specific - maybe we are given some kind of geometric substrate, like some manifold, say; a vertex operator is like a delta-function on that manifold, with an operator $T$ at a single point $x$ of this manifold and nothing else elsewhere. Now, by analogy with a "usual" delta-function, which can be viewed as a functional assigning to functions $f$ the values $f(x)$, we might think that a vertex operator is a functional assigning to... here begins something that I am not sure about:

  • assigning to sections $s$ of some vector bundle the results of action $T(s(x))$;
  • or assigning to vector bundles $E$ operators acting on the fibres $E_x$;
  • or...?

Does the above viewpoint make sense at all? Can it be made more rigorous?

The only place that I could find where singled out vertex operators are discussed at length is chapter 13 in Pressley & Segal's "Loop groups", where these operators are also called blips and are described in terms of $L\mathbb T$, the group of (smooth? analytic?) self-maps of the group $\mathbb T=\mathbb R/\mathbb Z$ with pointwise group structure: if $L\mathbb T$ acts on some Hilbert space $\mathscr H$, they define the operator $\psi_x$ on $\mathscr H$, for $x\in\mathbb T$, as the one that corresponds to the limit of actions of those self-maps which make one complete roundtrip in a neighborhood of $x$ and are identity elsewhere.

Vaguely this indeed resembles delta-functions. Can this be reformulated in terms of functional view of delta-functions as above?

If my intuition is just not relevant - what is the correct one?

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  • $\begingroup$ It's not clear to me upon reading your question whether you already understand what vertex [operator] algebras are and are trying to understand what vertex operators are, or whether you're trying to understand vertex algebras in the first place. It's possible to understand (and rigorously define) the algebras in a purely formal / algebraic way without trying to care about what the operators actually "are". $\endgroup$
    – Gro-Tsen
    Jan 25, 2019 at 11:17
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    $\begingroup$ Have you looked at: encyclopediaofmath.org/index.php/Vertex_operator ? $\endgroup$
    – S. Carnahan
    Jan 25, 2019 at 12:23
  • $\begingroup$ @Gro-Tsen Well I cannot say I understand V[O]As but I agree that it is possible to do this without ever mentioning vertex operators. But this question is precisely about the latter - I hope to grasp the algebraic side better if I will have some tangible images in mind like analogy with delta distributions $\endgroup$ Jan 25, 2019 at 12:38
  • $\begingroup$ @S.Carnahan No I have not seen this before, thank you! The entry indeed contains lots of densely packed information and several very important references. However it is, I think, packed a bit too densely. As for the references, if I understand correctly, the key ones are Corrigan & Fairlie, Date-Kashiwara-Miwa, and Fubini-Veneziano, is that correct? $\endgroup$ Jan 25, 2019 at 12:41
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    $\begingroup$ In any case it would be great if you (or anybody else) would provide an answer based on your link but shifted more towards the vertex operators per se... $\endgroup$ Jan 25, 2019 at 12:42

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