The Immortal Dyon Index Is Positive

Abstract

The microstates of supersymmetric black holes in asymptotically flat four-dimensional spacetime are expected to be bosonic due to the spherical symmetry of their horizons. This implies that the index counting the difference between bosonic and fermionic black hole microstates must be positive, as conjectured by Sen in arXiv:1008.4209. We show that the conjecture is satisfied by the index which counts 1/4-BPS states in N = 4 string theory forming single centered black holes, called immortal dyons. The proof relies on the relation between the indexed degeneracies and the coefficients of a family of mock modular forms, thus showing that black holes predict the sign of the Fourier coefficients of mock modular forms.

Figures (1)

• Figure 1: Walls of marginal stability in the upper half plane parametrized by $\tau =-\frac{v_2}{\sigma_2}+i \frac{\sqrt{\rho_2\sigma_2-v_2^2}}{\sigma_2}$, with $\rho_2\sigma_2-v_2^2\gg1$, which for the attractor values becomes $\tau = \frac{\ell}{2m}+i\frac{\sqrt{\Delta}}{2m}$. The $\mathcal{R}$ chamber is the one delimited by the walls connecting $0$, $1$ and $i\infty$. The delimited area in grey is a fundamental domain for $GL(2,\mathbb{Z})$, which lies inside the $\mathcal{R}$ chamber and contains the attractor values allowed by our choice of representatives \ref{['eq:representatives']}.

Theorems & Definitions (25)

• Theorem 1
• Corollary 1.1
• Proposition 1
• Proposition 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3