Dying Dyons Don't Count

TL;DR

This paper clarifies the counting of 1/4-BPS dyonic states in N=4 string theory by resolving contour ambiguities, ensuring duality invariance, and distinguishing stable 'immortal' dyons from bound states that appear/disappear across moduli space.

Contribution

It introduces a moduli-dependent contour prescription for counting dyons, ensuring duality invariance and correctly capturing bound state phenomena, and proposes a moduli-independent contour for stable dyons.

Findings

The counting formula's ambiguity relates to bound state formation.

A moduli-dependent contour prescription restores duality invariance.

A second, moduli-independent contour counts stable 'immortal' dyons.

Abstract

The dyonic 1/4-BPS states in 4D string theory with N=4 spacetime supersymmetry are counted by a Siegel modular form. The pole structure of the modular form leads to a contour dependence in the counting formula obscuring its duality invariance. We exhibit the relation between this ambiguity and the (dis-)appearance of bound states of 1/2-BPS configurations. Using this insight we propose a precise moduli-dependent contour prescription for the counting formula. We then show that the degeneracies are duality-invariant and are correctly adjusted at the walls of marginal stability to account for the (dis-)appearance of the two-centered bound states. Especially, for large black holes none of these bound states exists at the attractor point and none of these ambiguous poles contributes to the counting formula. Using this fact we also propose a second, moduli-independent contour which counts the "immortal dyons" that are stable everywhere.