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The Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry is a powerful tool for thecalculation of the complexity of metastable states in glassy systems, and it isparticularly useful to uncover the relationships between complexity andstandard thermodynamics. In this work we compute the Thouless-Anderson-Palmercomplexity of the Sherrington-Kirkpatrick model at the quenched level, by usingthe BRST supersymmetry. We show that the complexity calculated at K steps ofreplica symmetry breaking is strictly related to the static free energy at K+1steps of replica symmetry breaking. The supersymmetry therefore provides aprescription to obtain the complexity of the TAP states from the standardstatic free energy, even in models which are solved by more than one step ofreplica symmetry breaking. This recipe states that the complexity is given bythe Legendre transform of the static free energy, where the Legendre parameteris the largest replica symmetry breaking point of the overlap matrix.Comment: 14 page

The role of the Becchi–Rouet–Stora–Tyutin supersymmetry in the calculation of the complexity for the Sherrington–Kirkpatrick model