Higher genus Siegel forms and multi-center black holes in N=4 supersymmetric string theory

TL;DR

This paper proposes a conjecture linking Fourier coefficients of a degree three Siegel form to the degeneracy of three-center BPS bound states in type II string theory on K3×T^2, supported by physical consistency checks.

Contribution

It introduces a novel conjecture connecting Siegel modular forms to multi-center black hole degeneracies in N=4 string theory.

Findings

Evidence from wall-crossing and holographic bounds supports the conjecture.

Counting functions involving modular forms appear in degeneracy limits.

Consistency with single- and two-center degeneracy counts is demonstrated.

Abstract

We conjecture that the Fourier coefficients of a degree three Siegel form, $1/\sqrt{\chi_{18}}$, count the degeneracy of three-center BPS bound states in type II string theory compactified on $K3 \times T^2$. We provide evidence for our conjecture in the form of consistency with physical considerations of wall-crossing, holographic bounds, and the appearance of suitable counting functions (involving the inverse of the modular discriminant $\Delta$ and the inverse of the Igusa cusp form $\Phi_{10}$) in limits where the count degenerates to involve single-center or two-center objects.