Rotating black holes, global symmetry and first order formalism

In this paper we consider axisymmetric black holes in supergravity and address the general issue of defining a first order description for them. The natural setting where to formulate the problem is the De Donder-Weyl-Hamilton-Jacobi theory associated with the effective two-dimensional sigma-model action describing the axisymmetric solutions. We write the general form of the two functions S_m defining the first-order equations for the fields. It is invariant under the global symmetry group G_(3) of the sigma-model. We also discuss the general properties of the solutions with respect to these global symmetries, showing that they can be encoded in two constant matrices belonging to the Lie algebra of G_(3), one being the Noether matrix of the sigma model, while the other is non-zero only for rotating solutions. These two matrices allow a G_(3)-invariant characterization of the rotational properties of the solution and of the extremality condition. We also comment on extremal, under-rotating solutions from this point of view.