Free field realizations

[Daniel Moskovich]

I have a really silly question which I've been scrounging around for an
answer for for too long so I'll just ask it here:
What, conceptually, is a free field realization of an affine Lie
algebra, and why do we want one? And what IS a free field anyway?
Checking literature, I can get all kinds of technical definitions, but
not something conceptual... OK, you can write nice explicit formulas
for Verma modules etc., but why?

[Darth Sidious]

Daniel Moskovich wrote:
> I have a really silly question which I've been scrounging around for an
> answer for for too long so I'll just ask it here:
> What, conceptually, is a free field realization of an affine Lie
> algebra, and why do we want one? And what IS a free field anyway?

A free field is a field whose dynamics is generated by a
gaussian action (a quadratic form in the said field). For free fields
you can easily compute correlation functions and OPE's. This
allows you to easily compute the central charge, for example.

> Checking literature, I can get all kinds of technical definitions, but
> not something conceptual... OK, you can write nice explicit formulas
> for Verma modules etc., but why?

You're talking about 2 dimensional CFT's right? I guess that it's
a bit conforting to see that local QFT's are behind these conformal
field theories. Of course, the thing is that we can say a lot about
these theories only from their symmetry. There's no need to start
with an action, and no need to introduce free fields. But since
some of the results are universal after all it's a good idea to work
with some explicit model.