part, $$\label{clal}
g(x^+,x^-) = g_+(x^+)^{-1}  g_-(x^-)\ .$$ Since only $g$ needs to be a single-valued function on the worldsheet, one has $$g_\pm(x^\pm \pm 2\pi) = M g_\pm(x^\pm) \ ,$$ with $M$ an arbitrary but constant group element.[^7] As the reader can easily check, $${\cal O}_{\rm chir}(\lambda^*; R) = {\rm Tr}_R\;  M  \ ,$$ where the trace is evaluated in the $R$ representation. Note that the decomposition (\[clal\]) is not unique — the freedom to redefine $g_\pm =  g_0\;  g_\pm\ $, with $g_0$ an arbitrary constant group element, changes $M$ to $g_0 Mg_0^{-1}$. This is compatible with the fact that our topological observables only measure the conjugacy class of the monodromy.

Using the above freedom, one can bring $g_\pm$ to the canonical form $$g_\pm =   e^{\pm i\Lambda\; x^\pm}\; \tilde g_\pm(x^\pm)\,,$$ where $\tilde g_\pm$ are single-valued functions on the circle, $M = e^{2\pi i\Lambda}$ is the monodromy matrix, and $\Lambda$ can be chosen (by a Weyl reflection) to lie in a positive Cartan alcove of $\hat{g}$. The triplet $(\tilde g_+, \tilde g_-, \Lambda)$ describes the classical phase space of the WZW model. A geometric quantization of this phase space[^8], known as co-adjoint orbit or Kirillov-Konstant quantization [@Alekseev:1990vr; @Gawedzki:1990jc; @Chu:1991pn] (see also [@Pressley:1988qk]), 