Quenched computation of the dependence of complexity on the free energy in the Sherrington-Kirkpatrick model

The quenched computation of the complexity in the Sherrington-Kirkpatrick model is presented. A modified full replica symmetry breaking ansatz is introduced in order to study the complexity dependence on the free energy. Such an ansatz corresponds to require Becchi-Rouet-Stora-Tyutin supersymmetry. The complexity computed this way is the Legendre transform of the free energy averaged over the quenched disorder. The stability analysis shows that this complexity is inconsistent at any free energy level but the equilibrium one. The further problem of building a physically well-defined solution not invariant under supersymmetry and predicting an extensive number of metastable states is also discussed.