Comments on the Spectrum of CHL Dyons

TL;DR

This paper examines the degeneracy formulas for dyons in N=4 heterotic string compactifications, addressing S-duality invariance, contributions from higher genus surfaces, and the nature of negative discriminant states.

Contribution

It clarifies the S-duality invariance of dyon degeneracy formulas and introduces a criterion for genus-two contributions, highlighting the role of higher genus surfaces and multi-centered states.

Findings

S-duality invariance can be made manifest with a specific prescription.

Higher genus Riemann surfaces contribute to degeneracies beyond genus two.

Negative discriminant states correspond to multi-centered configurations, not single black holes.

Abstract

We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with $N =4$ supersymmetry. The partition function for these dyons is given in terms of Siegel modular forms associated with genus-two Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance of the resulting degeneracies and give a prescription that makes the invariance manifest. We show, using M-theory lift of string webs, that the genus-two contribution captures the degeneracy only if a specific irreducibility criterion is satisfied by the charges. Otherwise, in general there can be additional contributions from higher genus Riemann surfaces. We analyze the negative discriminant states predicted by the formula. We show that even though there are no big black holes in supergravity corresponding to these states, there are multi-centered particle-like configurations with subleading entropy in agreement with the microscopic prediction and our prescription for S-duality invariance. The existence of the states is moduli dependent and we exhibit the curves of marginal stability and comment on its relation to S-duality invariance.