For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle φ can be used to analyze the Many-Body-Localization Transition. The sensitivity of the energy levels En(φ) is measured by the level curvature Kn = E ′′ n (0), or more precisely by the Thouless dimensionless curvature kn = Kn/∆n, where ∆n is the level spacing that decays exponentially with the size L of the system. For instance ∆n ∝ 2 −L in the middle of the spectrum of quantum spin chains of L spins, while the Drude weight Dn = LKn studied recently by M. Filippone, P.W. Brouwer, J. Eisert and F. von Oppen [arXiv:1606.07291v1] involves a different rescaling. The sensitivity of the eigenstates |ψn(φ) > is characterized by the susceptibility χn = −F ′′ n (0) of the fidelity Fn = | < ψn(0)|ψn(φ) > |. Both observables are distributed with probability distributions displaying power-law tails2 , where β is the level repulsion index taking the values β GOE = 1 in the ergodic phase and β loc = 0 in the localized phase. The amplitudes A β and B β of these two heavy tails are given by some moments of the off-diagonal matrix element of the local current operator between two nearby energy levels, whose probability distribution has been proposed as a criterion for the Many-Body-Localization transition by M. Serbyn, Z. Papic and D.A. Abanin [Phys. Rev. X 5, 041047 (2015)].