2d gravity stress tensor and the problem of the calculation of the multi-loop amplitudes in the string theory

Abstract

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 surface 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.