Comments on the Spectrum of CHL Dyons

TL;DR

This work analyzes puzzles in the CHL dyon degeneracy formula for ${\cal N}=4$ theories, showing that degeneracies encoded by genus-two Siegel modular forms depend sensitively on moduli and an irreducibility invariant $I=\gcd(Q_e\wedge Q_m)$. A careful S-duality contour prescription is developed to render the spectrum invariant under $SL(2,\mathbb{Z})$, and it is shown that the genus-two count is valid only when $I=1$; for $I>1$ higher-genus contributions arise. The paper then explores states with negative discriminant $\Delta(Q)$, demonstrating that while they lack single-center black holes, they can exist as multi-centered configurations with subleading entropy, with their existence controlled by lines of marginal stability in moduli space. Together, these results delineate the domain of validity for the dyon partition function, relate moduli dependence to duality and web-geometry, and highlight the role of multi-centered solutions in completing the microscopic picture.

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.

Figures (3)

• Figure 1: Charge conservation at a string junction
• Figure 2: A quarter-BPS dyon carrying irreducible charges $Q_e$ and $Q_m$ with gcd($Q_e \wedge Q_m$)= 1
• Figure 3: A quarter-BPS dyon carrying reducible charges $Q_e$ and $Q_m$ with gcd($Q_e \wedge Q_m$)=2