Black Holes, Elementary Strings and Holomorphic Anomaly

TL;DR

The paper tests a duality-invariant framework for relating the entropy of two-charge black holes to the degeneracy of elementary BPS string states in $N=4$ heterotic theories, extending the proposal to CHL compactifications. By defining a holomorphy-corrected black hole entropy and a duality-preserving statistical entropy, the author shows that the black hole entropy $S_{BH}$ and the statistical entropy $\tilde{S}_{stat}$ agree for toroidal and CHL models up to non-perturbative corrections, with the holomorphic anomaly coefficient $K$ related to the unbroken left-moving gauge rank via $K = 24 - k$. The leading term matches as $4\pi\sqrt{N}$ and subleading logarithmic terms depend on the orbifold data, while non-perturbative corrections are of order $e^{-\\pi\\sqrt{N}}$, indicating a robust but nuanced correspondence. These results reinforce a unified, T-duality–invariant microstate counting approach and suggest avenues for generalization to more complex $N=4$ compactifications and charge configurations.

Abstract

In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.