are invariant under the (vector-like) transformations that have $u(x) = \omega(\bar u (-x))$, as the reader can easily verify. These vector-like transformations are generated by the linear combinations of currents $${\cal J}_n^a = J_n^{a} +  \omega (\bar J_{-n}^a)  \ ,$$ which must therefore have vanishing Poisson brackets with the loop observables, $$\label{dig}
\left\{ {\cal J}_n^a \;,\; 
{\cal O}_\omega \left({\lambda^*\over 2}, {\lambda^*\over 2} ; R
\right) \right\}_{\;_{\rm PB}} = 0\ \ .$$ Note that the ${\cal J}_n^a$ define a current algebra without central extension. Now assuming that (\[dig\]) survives in the quantum theory, and recalling that the Virasoro generators are quadratic in the currents and $\omega$-invariant, we immediately conclude that ${\cal O}_\omega (\lambda^*/2, \lambda^*/2 ; R )$ will obey the defining relations (\[cnds\]) of conformal defects. Note, however, that these defect lines are not topological — they do not, in particular, commute with the $\tau$-evolution. In fact, they correspond, as we shall later see, to unstable fixed points of the RG flow.

The topological observables ${\cal O}_{\rm chir}(\lambda^* ;R)$, together with their right-moving counterparts, measure the invariant monodromies of classical solutions. More explicitly, a general solution of the classical WZW equations factorises into the product of a left- and a right-moving 