Complexity in the Sherrington-Kirkpatrick model in the annealed approximation

A careful critical analysis of the complexity, at the annealed level, of the Sherrington-Kirkpatrick model has been performed. The complexity functional is proved to be always invariant under the Becchi-Rouet-Stora-Tyutin supersymmetry, disregarding the formulation used to define it. We consider two different saddle points of such functional, one satisfying the supersymmetry [A. Cavagna , J. Phys. A 36, 1175 (2003)] and the other one breaking it [A. J. Bray and M. A. Moore, J. Phys. C 13, L469 (1980)]. We review the previews studies on the subject, linking different perspectives and pointing out some inadequacies and even inconsistencies in both solutions.