Entropy and supersymmetry of $D$ dimensional extremal electric black holes versus string states

TL;DR

This paper extends the black hole–string state correspondence to $D=(10-p)$ dimensions with $4\le D\le9$ for electrically charged, extremal black holes in tree-level heterotic string theory on a $p$-torus. Using $O(16+p,1)\otimes O(p,1)$ solution-generating transformations, the authors construct a one-parameter extremal class of backgrounds, type $\mathcal{R}$, which are shown to be supersymmetric (half of the original supersymmetry) in all $D=4\ldots9$. At the stringy stretched horizon, the extremal black-hole entropy scales with the charges as $\Sigma_{S.H.} \propto \sqrt{\vec{Q}_R^2 - \vec{Q}_L^2}$ and matches, up to a constant of order unity independent of the string coupling, the logarithm of the degeneracy of elementary string states with the corresponding quantum numbers, supporting a robust black hole–string state identification beyond $D=4$. The analysis also clarifies the role of right-moving charges in preserving supersymmetry and provides a framework for comparing black-hole thermodynamics with string-state counting in higher dimensions.

Abstract

Following the work of Sen, we consider the correspondence between extremal black holes and string states in the context of the entropy. We obtain and study properties of electrically charged black hole backgrounds of tree level heterotic string theory compactified on a $p$ dimensional torus, for $D=(10-p)=4 \ldots 9$. We study in particular a one--parameter extremal class of these black holes, the members of which are shown to be supersymmetric. We find that the entropy of such an extremal black hole, when calculated at the stringy stretched horizon, scales in such a way that it can be identified with the entropy of the elementary string state with the corresponding quantum numbers.