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

TL;DR

This paper predicts the signs of certain Fourier coefficients of Siegel modular forms based on black hole properties and tests these predictions in specific string theory models, linking black hole physics with modular form coefficients.

Contribution

It introduces a novel prediction for the signs of Fourier coefficients of Siegel modular forms derived from black hole characteristics and verifies these in string theory models.

Findings

Predicted signs match computed Fourier coefficients in tested models.

Black hole properties impose constraints on modular form coefficients.

Results support the connection between black hole physics and modular form structures.

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.