Black Holes and the Spectrum of Half-BPS States in N=4 Supersymmetric String Theory

TL;DR

The paper investigates whether half-BPS black hole entropy in ${\cal N}=4$ string theory aligns with the degeneracy of elementary string states. It demonstrates that, to all orders in $1/N$, the black hole entropy depends only on the invariant $N=Q^2/2$ and relates this to the degeneracy through a Laplace transform of the black hole partition function ${\cal F}_{BH}(\mu)$, including exponentially suppressed corrections. The analysis extends to CHL compactifications by counting BPS degeneracies in untwisted and twisted sectors via Narain lattices and orbifold data, and it presents an alternative ensemble interpretation based on null-line charge sums. Finally, it discusses the role of the index ${\Omega}_4$ versus absolute degeneracy, noting cases where the index matches the entropy and cases (notably some Type II models) where it does not.

Abstract

The entropy of a half-BPS black hole in N=4 supersymmetric heterotic string compactification is independent of the details of the charge vector and is a function only of the norm of the charge vector calculated using the appropriate Lorenzian metric. Thus in order for this to agree with the degeneracy of the elementary string states, the latter must also be a function of the same invariant norm. We show that this is true for generic CHL compactifications to all orders in a power series expansion in the inverse charges, but there are exponentially suppressed corrections which do depend on the details of the charge vector. This is consistent with the hypothesis that the black hole entropy reproduces the degeneracy of elementary string states to all orders in a power series expansion in the inverse charges, and helps us extend the earlier conjectured relation between black hole entropy and degeneracy of elementary string states to generic half-BPS electrically charged states in generic N=4 supersymmetric heterotic string compactification. Using this result we can also relate the black hole entropy to the statistical entropy calculated using an ensemble of elementary string states that contains all BPS states along a fixed null line in the lattice of electric charges.