CHL Dyons and Statistical Entropy Function from D1-D5 System

TL;DR

This work proves the CHL dyon degeneracy formula by mapping 4D CHL dyons to a rotating D1-D5 system in Taub-NUT and deriving a duality-invariant statistical entropy function whose extremum gives the degeneracy, $S_{stat} = \log d(Q_e,Q_m)$. It constructs a generating function $f(\hat{\rho}, \hat{\sigma}, \hat{v})$ that equals $- (i\sqrt N)^{-k-2} / \widetilde{\Phi}_k(\hat{\rho}, \hat{\sigma}, \hat{v})$, tying dyon degeneracies to the inverse of a Siegel modular form $\widetilde{\Phi}_k$. The analysis covers twisted-sector twisted D1-D5 configurations with ${1\over 2} Q_e^2 = n/N$, ${1\over 2} Q_m^2 = (Q_1-1) Q_5$, and $Q_e\cdot Q_m = J$, and shows that the asymptotic degeneracy matches the black hole entropy up to first non-leading corrections while maintaining duality under $\Gamma_1(N)$.

Abstract

We give a proof of the recently proposed formula for the dyon spectrum in CHL string theories by mapping it to a configuration of D1 and D5-branes and Kaluza-Klein monopole. We also give a prescription for computing the degeneracy as a systematic expansion in inverse powers of charges. The computation can be formulated as a problem of extremizing a duality invariant statistical entropy function whose value at the extremum gives the logarithm of the degeneracy. During this analysis we also determine the locations of the zeroes and poles of the Siegel modular forms whose inverse give the dyon partition function in the CHL models.