is related to the integer level $k$ through $$\kappa = \psi^2 k/2\ ,$$ with $\psi$ the length of long roots of the Lie algebra $g$ of $G$. The currents generate the independent left and right symmetry transformations $$g \to u(x^+)^{-1}\; g\; \bar u(x^-)\ ,$$ under which they themselves transform as : $$\label{trans}
J \to u^{-1} J u + i\kappa\; u^{-1} \partial_+  u \qquad
\ {\rm and} \qquad
\bar J \to \bar u^{-1}  \bar J  \bar u + i\kappa\;
\bar u^{-1} \partial_-  \bar u\ .$$ The Poisson bracket algebra of the Fourier moments, $J_n^a$, of these currents is the classical counterpart of the standard left- and right-moving affine Kac-Moody algebras $\hat{g}$ with central extension $k$. This algebra implies the transformation rules (\[trans\]), and vice versa.

We will be interested in bulk defects which interact linearly with the currents of the model, and preserve a global $G$ symmetry. The classical observable that corresponds to the quantum transfer matrix of such defects reads[^6] $$\label{basic}
{\cal O}_{\omega}(\lambda,\bar\lambda; R ) =  {\rm Tr}_R\; {\rm P \; exp}
\;\left( i  \int_0^{2\pi}  d\sigma\;  ( \lambda\; J^a   \; - \;
\bar \lambda\;  \omega(\bar J^a) ) \; t^a \right)\ .$$ Here $t^a$ are the generators of $g$ in an 