Dyonic black hole degeneracies in $\mathcal{N} = 4$ string theory from Dabholkar-Harvey degeneracies

Abstract: The degeneracies of single-centered dyonic $\frac14$-BPS black holes (BH) in Type II string theory on K3$\times T^2$ are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form $\Phi_{10}$. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative $\frac12$-BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the $\frac12$-BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of $\frac12$-BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with $U$-duality invariant $\Delta<0$ are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order $\mathbb{Z}[\sqrt{|\Delta|}]$ in the real quadratic field $\mathbb{Q}(\sqrt{|\Delta|})$. We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.