generators of the Kac-Moody algebra[^10] $$[J^a_n, J^b_m] = i f^{abc}\;  J^c_{n+m} + {\kappa}\, n\; \delta^{ab}
\delta_{n+m,0}\ .$$ A similar expression defines the regularised right-moving currents. Note that this regulator preserves the classical algebra of analytic symmetry transformations, [*i.e.*]{} transformations generated by positive-frequency modes of the currents.

Let us concentrate first on chiral defects, and expand the loop observables (\[bas\]) in a power series of the coupling constant, $$\label{path}
{\cal O}_{\rm chir}(\lambda; R) =  \sum_{N=0}^\infty
{(i\lambda)^N }
\; {\cal O}^{(N)}(R) \ ,$$ where $$\label{path1}
{\cal O}^{(N)}(R) =
{\rm Tr}_R\; (t^{a_1}\cdots t^{a_N}) \left( \prod_{i=1}^{N}
\int_0^{2\pi} 
d\sigma_i \right) \;\theta_{\sigma_1>\cdots >\sigma_N}\;
J^{a_1}(\sigma_1)\cdots J^{a_N}(\sigma_N)\;$$ with $\theta_{\sigma_1>\cdots >\sigma_N}=1 $ if $\sigma_1>\sigma_2 \cdots >\sigma_N$, and $\theta_{\sigma_1>\cdots >\sigma_N}=0$ otherwise. Classically, the order of the currents in the above expression is irrelevant, but in the quantum theory a precise order must be specified. To guide our choice we will insist that the following two symmetries of the classical observables be preserved: (i) the path can start at any point $\sigma_0$ on the circle, and (ii) the result is invariant if we reverse the orientation of the loop, and change $R$ with its conjugate representation.[^11] Neither of these symmetries would be preserved if we kept the order of the currents 