TAP complexity, the cavity method and supersymmetry

We compute the Bray and Moore (BM) TAP complexity for the Sherrington-Kirkpatrick model through the cavity method, showing that some essential modifications are needed with respect to the standard formulation of the method. This allows us to understand various features recently discovered and to unveil at last the physical meaning of the parameters of the BM theory. We also reconsider the supersymmetric (SUSY) formulation of the problem finding that the BM solution satisfies some proper SUSY Ward identities that are different from the standard ones. The SUSY relationships encode the physical meaning of the parameters obtained through the cavity method. The problem of the vanishing prefactor is addressed, showing how it can be avoided.