Heterotic string on the CHL orbifold of K3

TL;DR

This paper explores heterotic string compactifications on CHL orbifolds of K3, revealing connections to Mathieu moonshine and modular forms, and analyzing gauge coupling corrections in these models.

Contribution

It extends heterotic string compactifications to CHL orbifolds, linking their indices to Mathieu group symmetries and modular forms, and studying gauge threshold corrections.

Findings

Supersymmetric index expressed via McKay-Thompson series.

Moduli dependence of gauge couplings captured by Siegel modular forms.

Connections established between string compactifications, moonshine, and dyon partition functions.

Abstract

We study ${\cal N}=2$ compactifications of heterotic string theory on the CHL orbifold $(K3\times T^2)/\mathbb{Z}_N$ with $N= 2, 3, 5, 7$. $\mathbb{Z}_N$ acts as an involution on $K3$ together with a shift of $1/N$ along one of the circles of $T^2$. These compactifications generalize the example of the heterotic string on $K3\times T^2$ studied in the context of dualities in ${\cal N}=2$ string theories. We evaluate the new supersymmetric index for these theories and show that their expansion can written in terms of the McKay-Thompson series associated with the $\mathbb{Z}_N$ involution embedded in the Mathieu group $M_{24}$. We then evaluate the difference in one-loop threshold corrections to the non-Abelian gauge couplings with Wilson lines and show that their moduli dependence is captured by Siegel modular forms related to dyon partition functions of ${\cal N}=4$ string theories.