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S-matrix is Path Integral with Vertex Operators inserted. I know how to compute Shapiro-Virasoro amplitude. So I don't have problems with calculations but with understanding.

In this calculations formalism of 2-dimensional CFT is used. But there is no S-matrix in CFT, only correlators (N-point functions).

I can treat embedding of world sheet into Minkowski space-time like scalar conformal fields with color indices. In this sense it is pure CFT where again no S-matrix is available. In QFT we have assymptoticaly free particles, but due to conformal invariance we can't build such states in CFT.

What I actually compute when I compute Polyakov's path integral with vertex operators?

-Anatoly Korybut