  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.
