19 - Supersymmetric black holes from supergravity

Summary

The construction of classical solutions of GR, with or without matter, is always a difficult task. In the preceding chapters we have constructed and studied the (static) black-hole solutions of some simple theories by using educated ansatz for the symmetries and general form of the metric, solution-generating transformations and other strategies, but the complexity of the solutions of the axion–dilaton model (Section 16.2) and the effort necessary to find them suggests that we must use more powerful methods or restrict ourselves to simpler kinds of solutions if we want to do the same for more complicated theories.

A combination of these two possibilities arises for the supersymmetric solutions of supergravity theories. Considering only supergravity theories is, by itself, a restriction, albeit not a very strong one, in view of the large number of possible models. Their supersymmetric solutions are usually the simplest in their class but they still have interesting physics. In any case they can be used as a first approximation to more complicated and realistic solutions and, sometimes, the latter can be obtained as deformations of the former. While the results obtained using supersymmetric solutions as tractable, solvable, toy models should not be extrapolated lightly to the non-supersymmetric ones, it is clear that there is a great deal to be learned from them. For instance, it is in this class of solutions that the attractor mechanism that holds for all extremal (not necessarily supersymmetric) blackhole solutions was first discovered [531, 1157, 526, 527].

In Chapter 18 we studied a method that allows the complete characterization of those solutions to the point that, in many interesting cases, a recipe can be given to construct them all. These solutions include, depending on the choices of symmetries, functions, and integration constants, cosmic strings, pp waves and other interesting solutions, plus many other less interesting and typically singular solutions of difficult interpretation.