Negative discriminant states in N=4 supersymmetric string theories

Abstract

Single centered BPS black hole solutions exist only when the charge carried by the black hole has positive discriminant. On the other hand the exact dyon spectrum in heterotic string theory compactified on T^6 is known to contain states with negative discriminant. We show that all of these negative discriminant states can be accounted for as two centered black holes. Thus after the contribution to the index from the two centered black holes is subtracted from the total microscopic index, the index for states with negative discriminant vanishes even for finite values of charges, in agreement with the results from the black hole side. Bound state metamorphosis -- which requires us to identify certain apparently different two centered configurations according to a specific set of rules -- plays a crucial role in this analysis. We also generalize these results to a class of CHL string theories.

LaTeX file, 32 pages; v2: reference added; v3: added new section 3.3

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• Dyonic black hole degeneracies in $\mathcal{N}=4$ string theory from Dabholkar-Harvey degeneracies
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• Rademacher expansion of a Siegel modular form for $\mathcal{N}=4$ counting
• Siegel Paramodular Forms from Exponential Lifts: Slow versus Fast Growth
• Negative index Jacobi forms and quantum modular forms