General static spherically symmetric black holes of the heterotic string on a six-torus Original Research Article

We present the most general static, spherically symmetric solutions of the 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) ⊂ 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 unconstrained 28 magnetic and 28 electric charges and the ADM mass compatible with the Bogomolʹnyi bound, is obtained by imposing on the generating solution [SO(6) × SO(22)]/[SO(4) × SO(20)] ⊂ O(6,22) (T-duality) transformation and SO(2) ⊂ 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 the Schwarzschild or the Reissner-Nordström black hole, while the extreme ones have either a null (or naked) singularity, or the space-time of the extreme Reissner-Nordström black hole.