The bosonic string measure at two and three loops and symplectic transformations of the volume form

Abstract

Symplectic modular invariance of the string integral has been verified at genus 2 and 3 using the period matrix coordinatization of moduli space. A calculation of the transformation of the product of holomorphic coordinates Π i⩽j dτ ij shows that an extra phase arises together with the factor associated with a specific modular weight; the phase is cancelled in the transformation of the entire volume element including the complex conjugate. An argument is given for modular invariance of the reggeon measure at genus 12.