Extremality Versus Supersymmetry in Stringy Black Holes

TL;DR

The paper analyzes general black-hole solutions in the low-energy string effective action across arbitrary dimensions using a unified metric with multiple harmonic functions and an extremality parameter $r_0$. It derives mass, temperature, and entropy expressions, showing that extremality corresponds to $r_0\to0$ with $m=\sum_i r_i|q_i|$ and that there exist $2^{(n-1)}$ Bogomol'nyi-like conditions; crucially, many extremal solutions are not supersymmetric, particularly in $N=8$ supergravity. The results reveal a rich landscape where extremality does not imply BPS saturation, including explicit examples where extreme black holes saturate neither SUSY bounds nor central charges. The findings motivate the idea of an underlying higher-dimensional structure (potentially twelve-dimensional) and hint at a deeper role for central charges beyond conventional supersymmetry bounds, with potential implications for dualities and microstate counting. Overall, the work clarifies the nuanced relationship between extremality and supersymmetry in stringy black holes and points to intriguing connections with higher-dimensional theories.

Abstract

We study general black-hole solutions of the low-energy string effective action in arbitrary dimensions using a general metric that can describe them all in a unified way both in the extreme and non-extreme cases. We calculate the mass, temperature and entropy and study which relations amongst the charges and the mass lead to extremality. We find that the temperature always vanishes in the extreme limit and we find that, for a set of n charges (no further reducible by duality) there are 2^{(n-1)} combinations of the charges that imply extremality. Not all of these combinations can be central charge eigenvalues and, thus, there are in general extreme black holes which are not supersymmetric (or ``BPS-saturated''). In the N=8 supergravity case we argue that the existence of roughly as many supersymmetric and non-supersymmetric extreme black holes suggests the existence of an underlying twelve-dimensional structure.