How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?

TL;DR

The paper addresses how the sign of the helicity trace index for single-centered 4d BPS black holes, $B_6$, is determined by horizon degeneracy and fermion zero modes, and predicts a negative value in the attractor chamber. It derives an explicit integral representation of $B_6$ in CHL models in terms of a Siegel modular form $\widetilde{\Phi}$ and analyzes chamber dependence through contour choices, enabling a test of the negativity against microscopic results from heterotic on $T^6$ and CHL theories. The author provides concrete kinematic charge constraints for which the attractor point lies in the relevant chamber and performs extensive finite-charge tests showing that $-B_6$ is positive for charges satisfying these constraints, in agreement with the horizon-based prediction. The work highlights the consistency between macroscopic black hole entropy data and the Fourier coefficients of Siegel modular forms, while illustrating the role of wall-crossing and duality in determining the contributions of single-centered versus multi-centered configurations.

Abstract

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,Z) or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T^6, as well as in Z_N CHL models for N=2,3,5,7.

Figures (1)

• Figure 1: A schematic diagram representing the chamber ${\cal R}$ in the upper half $\tau$ plane, bounded by the walls of marginal stability, for ${\hbox{Z Z}}_N$ orbifolds of heterotic string theory on $T^6$ for $N=1,2,3$. The shapes of the circles and the slopes of the straight lines bordering the chamber depend on the charges and other asymptotic moduli, but the vertices are universal.