# Free Field Realization of Current Algebras and its Hilbert space

I have some conceptual confusion regarding the interplay between current algebras, their free field representation and the Hilbert space generated from it.

Let's sketch a simple example, $$\mathfrak{so}(n)_1$$. As is readily seen, taking $$n$$ free fermions $$\psi^i(z)$$ with OPE $$\psi^i(z)\psi^j(w) \sim \frac{\delta^{ij}}{z-w}~,$$ one can write down the generators of the $$\mathfrak{so}(n)_1$$ current algebra,

$$J^a(z) \propto t^a_{ij}:\psi^i(z)\psi^j(z):~,$$

where $$t^a_{ij}$$ are $$\mathfrak{so}(n)$$ representation matrices of the vector representation in which the fermions transform. With the right prefactor this yields the desired current algebra. Now one usually goes on and constructs the Sugawara tensor and verifies that these currents are primary fields of conformal weight 1, by construction. The Hilbert space now contains states

$$|0\rangle~, \quad J^a_{-1}|0\rangle~, \quad J^a_{-1}J^b_{-1}|0\rangle~, \quad J^a_{-2}|0\rangle~, \quad ...$$

of conformal weight 0, 1, 2, 2, respectively.

For the $$n$$ free fermions we could write down an action as a non-linear sigma model which in conformal gauge looks something like

$$S = \frac{1}{2\pi} \int d^2z\, \psi^i\bar\partial\psi^i$$

with energy-momentum tensor

$$T(z) = \frac{1}{2}:\psi^i(z) \partial \psi^i(z):$$

For this free theory the textbook treatment consists of regarding the fermionic modes as raising operators filling the Hilbert space with states like

$$|0\rangle~,\quad \psi^i_{-1/2}|0\rangle~, \quad \psi^i_{-1/2}\psi^j_{-1/2}|0\rangle ~, \quad \psi^i_{-3/2}|0\rangle ~, \quad ...$$

and then going on about fermionic zero modes etc.

Now the first question is how to compare those two stories. In the yellow CFT book, the quantum equivalence of the above energy-momentum tensors is derived, hence I would assume them to lead to the same Hilbert space. Do they? Furthermore, what happens to the fermionic oscillators if they are only used to build the current algebra?

Some other thoughts are on the decomposition of representations. Is it wrong to think of the construction of the currents which transform in the adjoint representation as picking out the adjoint in the product of $$\mathbf{n} \otimes \mathbf{n}$$ (twice the vector representation)? At least in this way one would interpret the spectrum for the free theory, which usually happens by decomposing them into representations of the little group.

The confusion arose in the context of defining a string theory with a Lie group manifold as target space, using a WZW model. For example for doing string theory on $$S^3$$ one can study the $$\mathfrak{su}(2)_k$$ WZW model which for $$k=1$$ has a free field realization through 2 complex fermions (which actually seems to extend to $$\mathfrak{su}(2) \oplus \mathfrak{su}(2) = \mathfrak{so}(4)$$ bringing us back to the situation described at the beginning). Anyway, in this case it seems that in order to describe a theory in three spacetime dimensions we write down a free theory in four and proceed using just the currents that came from it to compute the spectrum. Is this viewpoint correct?

I believe it likely that at some point my understanding trailed off and I lost the big picture. I would find it very helpful if somebody could point that out and comment on the usefulness of such free field realizations as they appear also for supersymmetric generalizations of WZW models and such.