CFT Constraints on the Weak Gravity Conjecture

The Weak Gravity Conjecture (WGC) is a swampland criterion of long standing: any consistent theory of quantum gravity must contain a charged particle whose charge-to-mass ratio exceeds that of an extremal black hole, so that gravity remains the weakest force. The AdS/CFT correspondence offers a calculable boundary handle on bulk gravity, and the imaginary parts of bulk quasinormal modes are read off the boundary as poles of a retarded Green's function. We show that the WGC follows from this boundary calculation in two settings that fall outside the Reissner--Nordstr\"om idealisation: static spherically symmetric black holes in dRGT massive gravity, and dyonic black holes in Einstein--ModMax non-linear electrodynamics. The chain runs from the metric and gauge field, through the charged Klein--Gordon equation, into a near-horizon scaling limit whose radial equation reduces to Whittaker form; the conformal weight $\nu_0$ then enters a damping-time inequality. For the dRGT black hole every massive-gravity parameter ($\alpha,\beta,m_g,h$) cancels out, leaving the universal saturation $q/(m r_+) \geq 1/\sqrt{2} \approx 0.707$. For the Einstein--ModMax black hole the duality-symmetric non-linearity parameter $\gamma$ survives, and yields $q/(m r_+) \geq e^{-\gamma/2}$, which reduces to the Reissner--Nordstr\"om bound $q/(m r_+) \geq 1$ in the Maxwell limit $\gamma \to 0$. Either result is of order unity, and the second weakens monotonically as the non-linearity grows. We then relax three of the simplifying assumptions of the dRGT derivation, namely exact extremality, minimal coupling, and the absence of higher-curvature terms. The cancellation breaks. Each correction reintroduces $m_g,\alpha,\beta$ into the bound through a controlled functional dependence, and we tabulate and plot the relaxed forms across parameter space.