the $x$-basis having flowed primaries and their global descendants as moments were proposed as those being relevant for physical applications.

Note that the eigenvalues of the Cartan generators do not necessarily agree and, therefore, it will also be the case for the global right and left-moving spins, namely, spectral flowed vertex in the $x$-basis are no longer expected to be spinless operators, their space-time planar spin being given by the difference between $J$ and $\bar J$. This number has to be an integer in order for the corresponding correlation functions to be single-valued. On the other hand, since the lowest- and highest-weight states both contribute to the same operator, a flowed vertex in the space-time picture is not labelled by the spectral flow number but, strictly speaking, by its absolute value. We shall denote the flowed vertex operators as $\Phi^{j\omega}_{J}(x|z)$, where $\omega$ is now the (positive) amount of spectral flow and the superscript $j$ was introduced in order to remind the spin of the unflowed state this vertex is built from. These operators should be understood as microscopic states, as for the Euclidean theory, that are not Virasoro primaries and thus not affine primary fields either, although they are quasi-primary 