The Bosonic String Measure at Two and Three Loops and Symplectic Transformations of the Volume Form

TL;DR

This paper investigates the symplectic invariance of the bosonic string measure at genus 2 and 3, analyzing transformation properties of the volume form and proposing invariance at higher genus.

Contribution

It provides a detailed calculation of the transformation of the holomorphic volume element and discusses modular invariance of the reggeon measure at genus twelve.

Findings

Confirmed symplectic invariance at genus 2 and 3

Identified an extra phase in volume element transformation

Argued for invariance of the reggeon measure at genus twelve

Abstract

Symplectic modular invariance of the bosonic string partition function has been verified at genus 2 and 3 using the period matrix coordinatization of moduli space. A calculation of the transformation of the holomorphic part of the differential volume element 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 twelve.