gives the following spectrum for the monodromy : $$\label{hvee}
\Lambda  =  {\mu + \rho \over k+ h^\vee} \, \ \ \ \ {\rm in\  the\
sector}\ \ {\cal H}_\mu\otimes\bar{\cal H}_\mu\ .$$ Here ${\cal H}_\mu$ and $\bar{\cal H}_\mu$ are integrable highest-weight representations of the left and right current algebras, respectively; they are labelled by a highest weight vector $\mu$ of the corresponding representation of $g$. Furthermore, $\rho$ is the Weyl vector of $g$, [*i.e.*]{} one half of the sum of all positive roots of $g$. Semiclassical reasoning does not actually determine the finite shift of $k$ by the dual Coxeter number $h^\vee$ — a detailed quantum calculation is needed for this. Using (\[hvee\]) one finds the following spectrum for the topological defect operator, $$\label{topW}
{\cal O}_{\rm chir}(\lambda^\ast; R)   = {\rm Tr}_R \left( 
e^{2\pi i \Lambda} \right) = {S_{R\mu} \over S_{0\mu}} \, \ \ \ \ \ 
{\rm in\ \  }  {\cal H}_\mu\otimes\bar{\cal H}_\mu\ .$$ Here $S$ is the modular transformation matrix of the chiral characters, and $0$ (and $R$) denotes the Kac-Moody representation built on the trivial (or the $R$) representation of $g$. Note that the trace of $M$ in the above expression can be computed for an arbitrary representation $R$, 