properties of their correlation functions. According to it, starting with a regular unflowed correlator, a spectral flow operator is inserted for each unit of spectral flow carried by each vertex. After Mellin-transforming the amplitude thus obtained to the basis in which the Cartan generator of SL($2,\mathbb R$) is diagonal, also called the $m$-basis for short, the dependence on the “unphysical” insertion points is removed and the worldsheet dependence, properly adjusted. The computation concludes after transforming back to space-time picture or $x$-basis. No constraint on the value of $\omega$ is imposed.

Transforming correlation functions in the $m$-basis back to the space-time picture is not a simple task. In some cases in which the affine symmetry dictates the functional dependence of a correlator on the space-time coordinates, as for the regular term of the propagator or some three-point functions, no inverse Mellin transformation is needed and the FZZ recipe can be easily carried out. However, knowing the dependence of a correlator on the space-time coordinates may not be enough for the FZZ procedure to be fully completed.

In this paper, we compute the contact term of the propagator in a general setting. Such term is expected to exist in a spectral flowed 