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

TL;DR

This paper derives the most general static, spherically symmetric black hole solutions in heterotic string theory compactified on a six-torus, incorporating various duality transformations and charge configurations.

Contribution

It provides explicit forms of the most general solutions with unconstrained charges, extending previous work by including scalar fields and dualities in heterotic string compactifications.

Findings

Solutions include Schwarzschild and Reissner-Nordström black holes.

Extremal solutions can have null or naked singularities.

The solutions are parameterized by 28 electric and 28 magnetic charges.

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.