On the Moduli Space of the $T^6/Z_3$ Orbifold and Its Modular Group

TL;DR

This paper investigates the duality group of the $T^6/Z_3$ orbifold's moduli space, describing its structure, symmetries, and potential implications for non-perturbative superpotentials in string theory.

Contribution

It identifies the duality group as $SU(3,3,Z)$, establishes a symplectic embedding linking lattice data to special geometry, and proposes a candidate automorphic function for the superpotential.

Findings

Duality group is $SU(3,3,Z)$ for the orbifold.

Established a symplectic embedding connecting lattice and geometry.

Proposed a formal automorphic function as a superpotential candidate.

Abstract

We describe the duality group $\Gamma=SU(3,3,Z)$ for the Narain lattice of the $T^6/Z_3$ orbifold and its action on the corresponding moduli space. A symplectic embedding of the momenta and winding numbers allows us to connect the orbifold lattice to the special geometry of the moduli space. As an application, a formal expression for an automorphic function, which is a candidate for a non--perturbative superpotential, is given.