Modular Groups for Twisted Narain Models

TL;DR

This paper explores the modular symmetry groups of twisted Narain models, specifically analyzing $Z_N$ orbifolds like the $Z_7$ case, and investigates how discrete Wilson lines influence these groups.

Contribution

It provides a detailed method to determine modular groups for $Z_N$ orbifolds, including the non-trivial $Z_7$ case, and shows how Wilson lines can alter these groups.

Findings

The modular group for the $Z_7$ orbifold does not contain $ ext{SL}(2, extbf{Z})^3$ as previously speculated.

Discrete Wilson lines can modify the modular symmetry group, sometimes breaking $ ext{SL}(2, extbf{Z})$.

The paper offers explicit generators for the modular group in the $Z_7$ orbifold case.

Abstract

We demonstrate how to find modular discrete symmetry groups for $Z_N$ orbifolds. The $Z_7$ orbifold is treated in detail as a non-trivial example of a $(2,2)$ orbifold model. We give the generators of the modular group for this case which, surprisingly, does not contain $\sltz^3$ as had been speculated. The treatment models with discrete Wilson lines is also discussed. We consider examples which demonstrate that discrete Wilson lines affect the modular group in a non-trivial manner. In particular, we show that it is possible for a Wilson line to break $SL(2,{\bf Z})$.