Spectrum of CHL Dyons from Genus-Two Partition Function

TL;DR

This paper computes the genus-two partition function of heterotic string theory in a specific orbifold, confirming a proposed dyon counting formula and providing a new derivation via M-theory.

Contribution

It explicitly evaluates the genus-two chiral partition function for a $ ext{Z}_2$ CHL orbifold, linking it to the dyon partition function and extending techniques to $ ext{Z}_N$ orbifolds.

Findings

Partition function is a Siegel modular form of level two.

Confirmed the dyon partition function matches the computed genus-two form.

Provided an M-theory based derivation of the dyon counting formula.

Abstract

We compute the genus-two chiral partition function of the left-moving heterotic string for a $\mathbb{Z}_2$ CHL orbifold. The required twisted determinants can be evaluated explicitly in terms of the untwisted determinants and theta functions using orbifold techniques. The dependence on Prym periods cancels neatly once summation over odd charges is properly taken into account. The resulting partition function is a Siegel modular form of level two and precisely equals recently proposed dyon partition function for this model. This result provides an independent weak coupling derivation of the dyon partition function using the M-theory lift of string webs representing the dyons. We discuss generalization of this technique to general $\mathbb{Z}_N$ orbifolds.