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

TL;DR

The paper investigates the singular behavior of the 2D gravity stress tensor in free field theory, revealing obstacles in constructing modular invariant multi-loop string theory amplitudes due to these singularities.

Contribution

It identifies the origin of singularities in the vacuum value of the 2D gravity stress tensor on higher-genus Riemann surfaces, impacting multi-loop amplitude calculations.

Findings

Vacuum value of 2D gravity stress tensor is singular on the fundamental region.

Singularities prevent the construction of modular invariant multi-loop amplitudes.

Singularity arises from the gravity field's vacuum value on genus-n>1 Riemann surfaces.

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.