Supersymmetric complexity in the Sherrington-Kirkpatrick model.

By using a supersymmetric approach we compute the complexity of the metastable states in the Sherrington-Kirkpatrick spin-glass model. We prove that the supersymmetric complexity is exactly equal to the Legendre transform of the thermodynamic free energy, thus providing a recipe to find the complexity once the free energy is known. Our results suggest that the supersymmetry may be a useful tool for the calculation of the entropy of metastable states in generic glassy systems.