al dimension of such operators as a function of the two parameters $(k,f)$ of the supergroup sigma-model. At the WZW point these operators are descendants in the highest-weight representations of the left affine Lie algebra.

Operators of the form $:j_L \phi :$ {#operators-of-the-form-j_l-phi .unnumbered}
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Let us consider the operator $:j_{L,z}^a \phi:$ defined as the regular term in the OPE between the operators $j_{L,z}^a$ and $\phi$. To compute the holomorphic dimension of this operator we compute its OPE with the stress-tensor, and look at the second order pole. The computation is done following the method described in appendix \[compositeOPEs\]. The fact that the stress-tensor is holomorphic simplifies the calculation. We find: $$\begin{aligned}
 T(z) :j^a_{L,z}\phi:(w)
& = \lim_{:x \to w:} T(z) j^a_{L,z}(x) \phi(w) \cr
%
& = \lim_{:x \to w:} \left \{
\left( \frac{j^a_{L,z}(x)}{(z-x)^2} + \frac{\p j^a_{L,z}(x)}{z-x} \right) \phi(w) %\right. \cr
%& \qquad \left. 
+ j^a_{L,z}(x) \left( \frac{\Delta_{\phi} \phi(w)}{(z-w)^2} + \frac{\p \phi(w)}{z-w} \right)
\right\} \cr
%
&= \lim_{:x \to w:} \left \{
\frac{1}{(z-x)^2}\left( 
-\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{x-w} + :j^a_{L,z} \phi:(w) \right.\right. \cr
&\qquad \qquad  \left. \left. 
+ {A^a}_c \log|x-w|^2 :j^c_{L,z} \phi:(w) + {B^a}_c \frac{\bar x - \bar w}{x-w} :j^c_{L,\bar z} \phi:(w) + ...
\right) \right. \cr
& \quad + \frac{1}{z-x}\left(
\frac{c_+}{c_++c_-} \frac{t^a \phi(w)}{(x-w)^2} + :\p j^a_{L,z} \phi:(w) \right. \cr
&\qquad \qquad  \left. \left. 
+ {A^a}_c \frac{1}{x-w} :j^c_{L,z} \phi:(w) - {B^a}_c \frac{\bar x - \bar w}{(x-w)^2} :j^c_{L,\bar z} \phi:(w) + ... 
\right) \right. \cr
& \quad \left. 
+ \frac{\Delta_{\phi}:j^a_{L,z} \phi:(w)}{(z-w)^2} + \frac{:j^a_{L,z}\p \phi:(w)}{z-w} 
\right\}  \cr
%
&=
-\frac{2}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:j^a_{L,z} \phi:(w)}{(z-w)^2}  \cr
&\qquad + \frac{1}{(z-w)^3} \frac{c_+}{c_++c_-} t^a \phi(w) + \frac{:\p j^a_{L,z} \phi:(w)}{z-w} + {A^a}_c \frac{1}{(z-