Product Representation of Dyon Partition Function in CHL Models

TL;DR

The authors derive product representations for Siegel modular forms that govern dyon degeneracies in CHL $\mathbb{Z}_N$ orbifolds, generalizing the Borcherds–Gritsenko–Nikulin framework to weight $k=\tfrac{24}{N+1}-2$ forms. Starting from an invariant integral over the CHL orbifold CFT, they extract holomorphic data through a Poisson-resummed orbit analysis, yielding explicit product formulas for $\widetilde{\Phi}_k$ (and, via a modular transform, for $\Phi_k$) built from twisted elliptic genera of $K3$. The $N=2$ case is worked out in detail, with explicit $F^{(r,s)}$ and $c^{(r,s)}(n)$, and the resulting $V\to0$ factorization into eta-quotients provides a strong consistency check against known limits. The framework is then extended to general $N=2,3,5,7$, producing coherent product representations and normalization choices that ensure duality invariance and correct asymptotics for dyon degeneracies in CHL models.

Abstract

A formula for the exact partition function of 1/4 BPS dyons in a class of CHL models has been proposed earlier. The formula involves inverse of Siegel modular forms of subgroups of Sp(2,Z). In this paper we propose product formulae for these modular forms. This generalizes the result of Borcherds and Gritsenko and Nikulin for the weight 10 cusp form of the full Sp(2,Z) group.