ee bosons in the semi-classical limit, as anticipated. At fixed $kf^2$ and for each interaction vertex, the power of the coupling constant $f$ is equal to the number of structure constants that appear. Since we are interested in computing OPEs involving the currents and the primary fields, let us write these fields in terms of the bosons $X^a$: $$\begin{aligned}
\label{current(X)}
  \frac{j^a_{L,z}}{c_+} &=& (\partial g g^{-1})^a =  i (f \partial X^a + f^2 \frac{{f^{a}}_{bc}}{2} X^c \partial X^b + 
\frac{f^3}{6} {f^a}_{bc} {f^c}_{de}  \partial X^e X^d X^b +...)
  \nonumber \\
  \frac{j^a_{L,\bar z}}{c_-} &=& (\bar{\partial} g g^{-1})^a = i (f \bar{\partial} X^a + f^2 \frac{{f^{a}}_{cb}}{2} X^b \bar{\partial} X^c + 
\frac{f^3}{6} {f^a}_{bc} {f^c}_{de} \bar \partial X^e X^d X^b +...),\end{aligned}$$ \[phi(X)\] = e\^[i f X\_a t\^a]{} = i f X\_a t\^a - f\^2 X\_a t\^a X\_b t\^b +... where in the last line the generators $t^a$ are taken in the representation associated to the primary field $\phi$.

The semi-classical behavior of the current-current OPE {#the-semi-classical-behavior-of-the-current-current-ope .unnumbered}
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We study the semi-classical behavior of the OPE between two $z$-components of the left-current. The discussion generalizes straightforwardly to other current-current OPEs. We assume that the only operators that appear in the result of this OPE are composites of (derivatives of) left currents. This is true at the WZW point, and can presumably be proven at any point using conformal perturbation theory. Let us isolate one term in this OPE : \[jjOneTerm\] j\^a\_[L,z]{}(z) j\^b\_[L,z]{}(w) = ... + [A\^[ab]{}]{}\_[a\_p a\_[p-1]{}...a\_[2]{} a\_1]{}(z-w, |z - |w) :j\^[a\_1]{}\_[L,z]{}:j\^[a\_2]{}\_[L,z]{}...:j\^[a\_[p-1]{}]{}\_[L,z]{}j\^[a\_p]{}\_[L,z]{}:...::(w) +...Our goal is to evaluate the behavior of the tensor ${A^{ab}}_{a_p...a_1}(z-w, \bar z - \bar w)$ when the parameter $f$ is small. The reasoning will not depend on the particular current compo