The 2D gravity stress tensor and multi-loop amplitudes in string theory

It is shown that the vacuum value of the 2D gravity stress tensor in the free field theory is singular in the fundamental region on the complex plane where the genus-n > 1 Riemann surfaces are mapped. Because of the above singularity, one can not construct modular invariant multi-loop amplitudes. The discussed singularity is due to the singularity in the vacuum value of the 2D gravity field that turns out to be on the genus-n > 1 Riemann surfaces.