BORDERED RIEMANN SURFACES, SCHOTTKY-GROUPS AND OFF-SHELL STRING AMPLITUDES

We propose an off-shell extension of the closed bosonic string scattering amplitudes as functional integrals over bordered Riemann surfaces. The tree level off-shell N-scalars amplitude is handled with the help of Schottky groups and the Burnside theta-series. Using these tools and the asymptotic behaviour of the partition function at the boundary of the moduli space, we show that the off-shell amplitudes exhibit the tachyon mass-shell poles in the external momenta. In addition, their residues are shown to be the semi-off-shell amplitudes and the well-known Koba-Nielsen amplitudes.