terms’ that arise in the process of normal ordering if one uses at least once the $if^{abc}$ piece of the current commutators. These terms have fewer powers of $J\sim \kappa^{1/2}$, and they depend on the precise definition of the normal ordered expression $:{\cal O}_{\rm reg}^{(4)}(R):$. Indeed, different definitions of $:{\cal O}_{\rm reg}^{(4)}(R):$ differ by rearrangements of the positive, or of the negative modes, which only involve the $if^{abc}$ piece of the commutator.

We have only calculated those ‘subleading’ terms that make divergent contributions (in the $s\to 0$ limit) to the quadratic and cubic Casimir operators of $G$.[^12] Put differently, if we write $${\cal O}_{\rm chir}^{\rm reg}(\lambda; R)\, = 
\, \cdots +  A_2(\lambda; R) \,J_0^aJ_0^a\,
+ \, A_3(\lambda; R) \,d^{abc}J_0^aJ_0^bJ_0^c +  \cdots  
  \ ,$$ then the coefficients $A_2$ and $A_3$ read : $$\label{a2}
A_2(\lambda; R)\, = -2\pi^2 I_R \, 
\left[ \lambda^2 +  \lambda^3\xi +
\lambda^4 \left(\kappa\,\xi +  {3\over 4}\, \xi^2 + {\rm finite}\right)
+ O(\lambda^5)\right] $$ and $$\label{a3}
A_3(\lambda; R)\, = \, -{2\pi^3 i\over 3}\; I_R^{(3)}\; \left[
\lambda^3 + \lambda^4\, \left({3\over 2}\, \xi + {\rm finite}\right)
+ O(\lambda^5)\right]\ ,$$ where we have defined $$\label{xi}
\xi = h^\vee\psi^2 \,{\rm log}\, s\, .$$ In calculating these results, we made use of the infinite 