Reparametrization invariance: a gauge-like symmetry of ultrametrically organised states

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit of infinite step replica symmetry breaking and n → 0 is needed to derive this continuous gauge-like symmetry from the discrete permutation invariance of the n replicas. Goldstone’s theorem and Ward identities can be deduced from the disappearence of the second (and higher order) variation of the longitudinal free energy. We recall also how these and other exact statements follow from permutation symmetry after introducing the concept of ”infinitesimal” permutations.