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 self-maps of a circle: 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?