We study Siegel modular forms associated with the theta lift of twisted elliptic genera of $K3$ orbifolded with $g^{\prime }$ corresponding to the conjugacy classes of the Mathieu group $M_{24}$. For this purpose we rederive the explicit expressions for all the twisted elliptic genera for all the classes which belong to $M_{23}\subset M_{24}$. We show that the Siegel modular forms satisfy the required properties for them to be generating functions of $1/4$ BPS dyons of type II string theories compactified on $K3\times T^2$ and orbifolded by $g^{\prime }$ which acts as a ${\mathbb{Z}}_N$ automorphism on $K3$ together with a $1/N$ shift on a circle of $T^2$. In particular the inverse of these Siegel modular forms admit a Fourier expansion with integer coefficients together with the right sign as predicted from black hole physics. This observation is in accordance with the conjecture by Sen and extends it to the partition function for dyons for all the 7 CHL compactifications. We construct Siegel modular forms corresponding to twisted elliptic genera whose twining character coincides with the class $2B$ and $3B$ of $M_{24}$ and show that they also satisfy similar properties. Apart from the orbifolds corresponding to the 7 CHL compactifications, the rest of the constructions are purely formal.

• Received 20 June 2017

DOI:https://doi.org/10.1103/PhysRevD.96.086020