In complex analysis, the Schwarz-Christoffel map is known for taking a polygon in the complex plain to the upper half of the complex plain, with the vertices getting mapped to the real line.

In string theory, this is used to obtain a canonical representation of worldsheets (which can be considered to be limits of polygons).

For the worldsheet of a string splitting into two strings, the differential equation of the Schwarz-Christoffel map $w(z)$ is given by

\(\frac{dw}{dz} = A\frac{(z-x^{*})}{(z+1)(z-1)}\)

where $x^{3}$ is the point of interaction. I naively read the right hand side as being proportional to the pole on the real axis corresponding to the incoming string, devided by the poles corresponding to the two outgoing strings. If my interpretation is right, does this hold for maps of worldsheets for more complicated interactions too, such that the poles corresponding to the incoming (on shell) strings always appear in the numerator and the poles of the outgoing (on shell) strings in the denominator of a similar differential equation for the corresponding Schwarz-Christoffel map? Are these differential equations somehow related to the S-matrix?