A Comment on Duality Transformations and (Discrete) Gauge Symmetries in Four-Dimensional Strings

TL;DR

This paper explores how target space modular invariance relates to discrete gauge symmetries in four-dimensional orbifold string models, revealing that duality transformations correspond to specific gauge transformations but are not part of the residual discrete gauge symmetry.

Contribution

It clarifies the relationship between duality transformations and discrete gauge symmetries in four-dimensional strings, especially in supersymmetric compactifications and Gepner models.

Findings

Duality elements correspond to finite Kähler and gauge transformations.

These gauge transformations are not elements of the residual discrete gauge symmetry.

In Gepner models, duality elements relate to the ${f Z}_{k+2}$ symmetry of minimal models.

Abstract

We discuss the relationship between target space modular invariance and discrete gauge symmetries in four-dimensional orbifold-like strings. First we derive the modular transformation properties of various string vertex operators of the massless string fields. Then we find that for supersymmetric compactifications the action of the duality elements, leaving invariant the multicritical points, corresponds to a combination of finite K\"ahler and gauge transformations. However, those finite gauge transformations are not elements of a remnant discrete gauge symmetry. We suggest that, at least in the case of Gepner models corresponding to tensor products of identical minimal models, the duality element leaving invariant the multicritical point corresponds to the ${\bf Z}_{k+2}$ symmetry of any of the minimal $N=2$ models appearing in the tensor product.