Dyon Spectrum in Generic N=4 Supersymmetric Z_N Orbifolds

TL;DR

We analyze 1/4 BPS dyons in generic ${\cal N}=4$ ${\rm ZZ}_N$ orbifolds and derive an exact degeneracy formula $d(Q_e,Q_m)=(1/N)\int_{\cal C} d\tilde{\rho}\, d\tilde{\sigma}\, d\tilde{v}\; e^{-\pi i( N\tilde{\rho} Q_e^2 + \tilde{\sigma} Q_m^2/N + 2\tilde{v} Q_e\cdot Q_m)}/\tilde{\Phi}(\tilde{\rho},\tilde{\sigma},\tilde{v})$, with $\tilde{\Phi}$ determined by 2d (4,4) SCFT data. The spectrum is organized into KK monopole, D1-D5 center-of-mass, and relative D1-D5 motion sectors, and shown to be invariant under the S-duality subgroup $\Gamma_1(N)$. The paper also computes the asymptotic statistical entropy by extremizing a universal entropy functional involving $g(\tau)$ and a weight $k$, and demonstrates exact agreement with the black hole entropy once the one-loop Gauss-Bonnet correction $\phi(\tau,\bar{\tau})$ is included, matching at the first nonleading order. These results generalize previous prime-$N$ analyses to generic $N$ and reinforce the microscopic/mmacroscopic entropy correspondence in ${\cal N}=4$ theories.

Abstract

We find the exact spectrum of a class of quarter BPS dyons in a generic N=4 supersymmetric Z_N orbifold of type IIA string theory on K3\times T^2 or T^6. We also find the asymptotic expansion of the statistical entropy to first non-leading order in inverse power of charges and show that it agrees with the entropy of a black hole carrying same set of charges after taking into account the effect of the four derivative Gauss-Bonnet term in the effective action of the theory.