Extremal solutions of the S3 model and nilpotent orbits of G2(2)

We study extremal black hole solutions of the S-3 model (obtained by setting S-T-U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S 3 model exhibits the pseudo-Riemannian coset structure G/(K) over tilde with G = G(2(2)) and (K) over tilde = SO0(2; 2). We study nilpotent (K) over tilde -orbits of G(2(2)) corresponding to non-rotating single-center extremal solutions. We find six such distinct (K) over tilde -orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in supersymmetric orbits when uplifted to five-dimensional minimal supergravity have single-center Gibbons-Hawking space as their four-dimensional Euclidean hyper-Kahler base space. We construct hitherto unknown extremal (supersymmetric as well as non-supersymmetric) pressureless black strings of minimal five-dimensional supergravity and briefly discuss their relation to black rings.