A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order $N=2$ in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

中文翻译：

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CHL Calabi-Yau 三重形式：曲线计数、Mathieu Moonshine 和 Siegel 模形式

CHL 模型是 $\mathrm{K3} \times E$ 与 $N$ 自同构的商，该自同构辛作用于 K3 表面，并通过椭圆曲线 $E$ 上的 $N$-扭转点移动. 我们推测椭圆CHL模型的原始Donaldson-Thomas配分函数是Siegel模形式，即出现在Mathieu Moonshine中的相应扭曲缠绕椭圆属的Borcherds提升。该猜想与 David、Jatkar 和 Sen 对弦理论的预测相匹配。我们使用拓扑顶点来证明该猜想的几个基本情况。通过退化为 $\mathrm{K3} \times \mathbb{P}^1$，我们还将 DT 分区函数表示为 Fock 空间上一个算子的扭曲迹。这产生了进一步的计算证据。讨论了该猜想对非几何 CHL 模型的扩展。我们详细考虑 $N=2$ 阶的 CHL 模型。我们推测了所有曲线类中所有阶二 CHL 模型的 Donaldson-Thomas 不变量的公式。该猜想是根据两种 Siegel 模形式制定的。其中之一，Iwahori 子群的 Siegel 形式，据我们所知尚未出现在物理学中。这种差异在附录中与 Sheldon Katz 进行了讨论。