TITLE: What is the motivation for a vertex algebra?
QUESTION [78 upvotes]: The mathematical definition of a vertex algebra can be found here:
http://en.wikipedia.org/wiki/Vertex_operator_algebra
Historically, this object arose as an axiomatization of "vertex operators" in "conformal field theory" from physics; I don't know what these phrases mean.
To date, I haven't been able to gather together any kind of intuition for a vertex algebra, or even a precise justification as to why anyone should care about them a priori (i.e. not "they come from physics" nor "you can prove moonshine with them").
As far as I am aware, theoretical physics is about finding mathematical models to explain observed physical phenomena. My questions therefore are:
What is the basic physical phenomenon/problem/question that vertex operators model?
What is the subsequent story about vertex operators and conformal field theory, and how can we see that this leads naturally to the axioms of a vertex algebra?
Are there accessible physical examples ("consider two particles colliding in an infinite vacuum...", etc.) that illustrate the key ideas?
Also, are there alternative, purely mathematical interpretations of vertex algebras which make them easier to think about intuitively?
Perhaps people who played a role in their discovery could say a bit about the thinking process that led them to define these objects?

REPLY [50 votes]: The original motivation for vertex algebras is explained briefly in the original paper http://www.jstor.org/stable/27441 as follows. For any even lattice one can construct a space $V$ acted on by vertex operators corresponding to lattice vectors. More generally one can write down a vertex operator for every element of $V$. These vertex operators satisfy some complicated relations, which are then used as the definition of a vertex algebra. In other words, the original example of a vertex algebra was the vertex algebra of an even lattice, and the definition of a vertex algebra was an axiomatization of this example. 
This was motivated by my attempt to understand Igor Frenkel's work on the Lie algebra with Dynkin diagram the Leech lattice.  Frenkel had constructed a representation of this Lie algebra using vertex operators acting on the space $V$, and I was trying to use his work to understand its root multiplicities. I did not use any insights from conformal/quantum/topological field theory or operator product expansions when defining vertex algebras (as implied by some of the other answers), for the simple reason that I had barely heard of these concepts and had almost no idea what they were. 
This is not all that helpful for understanding what a vertex algebra really is. A better view is to regard them as something like a commutative ring with an action of the (formal) 1-dimensional additive group. In particular any such ring is canonically a vertex algebra. The difference between rings with group actions and vertex algebras is that the "multiplication" of a vertex algebra from $V\times V$ to $V$ is not defined everywhere: it can be thought of as a "rational map" rather than a "regular map" in some sense. More precisely if we write $u^g$ for the action of a group element $g$ on $u$, then the product $u^gv^h$ is not defined for all group elements $g$ and $h$, but behaves formally like a meromorphic function of $g$ and $h$, which in particular may have poles when $g$ and $h$ are trivial. Making sense of this informal idea produces the definition of a vertex algebra. (For more details see the unpublished paper http://math.berkeley.edu/~reb/papers/bonn/bonn.pdf.) This means that vertex algebras behave like commutative rings: for example, one should define modules over them, tensor products of modules, and so on.