General Static Spherically Symmetric Black Holes of Heterotic String on a Six Torus

TL;DR

This work constructs the most general static, spherically symmetric black-hole solutions in four-dimensional heterotic string theory compactified on a six-torus by applying a sequence of O(8,24) symmetry transformations to the Schwarzschild solution. Starting from a five-parameter generating solution (four charges plus non-extremality) with a zero Taub-NUT constraint, the authors deploy T-duality and S-duality to generate a full 56-charge family without altering the four-dimensional metric. They classify the global spacetime structure and thermodynamics, distinguishing non-extreme and extreme (BPS and non-BPS) regimes, and recover several known solutions as special cases. The results advance the no-hair paradigm in string-theoretic black holes and provide a framework to study moduli and coupling dependencies of black-hole thermodynamics in heterotic string theory.

Abstract

We present the most general static, spherically symmetric solutions of heterotic string compactified on a six-torus that conforms to the conjectured ``no-hair theorem'', by performing a subset of O(8,24) transformations, i.e., symmetry transformations of the effective three-dimensional action for stationary solutions, on the Schwarzschild solution. The explicit form of the generating solution is determined by six $SO(1,1)\subset O(8,24)$ boosts, with the zero Taub-NUT charge constraint imposing one constraint among two boost parameters. The non-nontrivial scalar fields are the axion-dilaton field and the moduli of the two-torus. The general solution, parameterized by {\it unconstrained} 28 magnetic and 28 electric charges and the ADM mass compatible with the Bogomol'nyi bound, is obtained by imposing on the generating solution $[SO(6)\times SO(22)]/[SO(4)\times SO(20)] \subset O(6,22)$ (T-duality) transformation and $SO(2)\subset SL(2,R)$ (S-duality) transformation, which do not affect the four-dimensional space-time. Depending on the range of boost parameters, the non-extreme solutions have the space-time of either Schwarzschild or Reissner-Nordstr\" om black hole, while extreme ones have either null (or naked) singularity, or the space-time of extreme Reissner-Nordstr\" om black hole.