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.

Section on comparison with earlier literature added, References added

References in corpus (13)

• Notes on the K3 Surface and the Mathieu group M_24
• CHL Dyons and Statistical Entropy Function from D1-D5 System
• Note on Twisted Elliptic Genus of K3 Surface
• Lectures on Quantum Black Holes
• Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories
• Comments on the Spectrum of CHL Dyons
• Product Representation of Dyon Partition Function in CHL Models
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• Dyon Spectrum in Generic N=4 Supersymmetric Z_N Orbifolds
• How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?
• No More Walls! A Tale of Modularity, Symmetry, and Wall Crossing for 1/4 BPS Dyons
• Siegel Modular Forms and Black Hole Entropy
• $\mathcal{N}=2$ heterotic string compactifications on orbifolds of $K3\times T^2$

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