Dyon degeneracies from Mathieu moonshine

Abstract

We construct the 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 $M_{24}$. We complete the construction for all the classes which belong to $M_{23}\subset M_{24}$ and two other classes outside the subgroup $M_{23}$. For this purpose we provide the explicit expressions for all the twisted elliptic genera in all the sectors of these classes. 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. Our analysis completes the construction of the partition function for dyons as well as the twisted elliptic genera for all the $7$ CHL compactifications.

arXiv commentSection on comparison with earlier literature added, References added

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Cited by in corpus (9)

• Killing Spinors for Finite Temperature Euclidean Solutions at the BPS Bound
• Gravitational couplings in $\mathcal{N}=2$ string compactifications and Mathieu Moonshine
• On Mathieu moonshine and Gromov-Witten invariants
• Horizon states and the sign of their index in $\mathcal{N}=4$ dyons
• Positivity of discrete information for CHL black holes
• Heterotic strings on $(K3\times T^2)/{\mathbb{Z}}_3$ and their dual Calabi-Yau threefolds
• Gravitational couplings in $\mathcal{N}=2$ heterotic compactifications with Wilson lines
• Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the ${\mathbb{Z}}_2$ Model
• Mock Modularity Of Twisted Index In CHL Models