On the positivity of black hole degeneracies in string theory
Kathrin Bringmann, Sameer Murthy
TL;DR
The paper proves a positivity conjecture for microscopic BPS degeneracies in ${ m N}=4$ and ${ m N}=8$ string theories by exploiting the (mock) modular structure of generating functions for black hole states. It analyzes four- and five-dimensional black holes arising from Type II compactifications on $T^{6}$, $T^{6}/T^{5}$, and $K3 imes T^{2}$, showing that their degeneracies are Fourier coefficients of (mock) Jacobi forms and Siegel modular forms, and that these coefficients are positive when the discriminant satisfies $4mn-r^2>0$. For $m=1$ and $m=2$, the authors provide two distinct proofs: a circle-Method-based analytic approach and an explicit algebraic method using theta decompositions, Hurwitz–Kronecker class numbers, and modular forms. The results reinforce the link between black hole entropy and exact BPS state counting, and they introduce robust techniques for proving coefficient positivity in mock Jacobi and related automorphic objects, with potential applications to broader modular-form contexts in quantum gravity.
Abstract
Certain helicity trace indices of charged states in N=4 and N=8 superstring theory have been computed exactly using their explicit weakly coupled microscopic description. These indices are expected to count the exact quantum degeneracies of black holes carrying the same charges. In order for this interpretation to be consistent, these indices should be positive integers. We prove this positivity property for a class of four/five dimensional black holes in type II string theory compactified on T^6/T^5 and on K3 \times T^2/S^1. The proof relies on the mock modular properties of the corresponding generating functions.