Weak gravity conjecture in the AdS/CFT correspondence

We study implications of the weak gravity conjecture in the AdS/CFT correspondence. Unlike in Minkowski spacetime, Anti-de Sitter (AdS) spacetime has a physical length scale, so that the conjecture must be generalized with an additional parameter. We discuss possible generalizations and translate them into the language of dual conformal field theories (CFTs), which take the form of inequalities involving the dimension and charge of an operator as well as the current and energy-momentum tensor central charges. We then test these inequalities against various CFTs to see if they are universally obeyed by all the CFTs. We find that certain CFTs, such as supersymmetric QCDs, do not satisfy them even in the large N limit. This does not contradict the conjecture in AdS spacetime because the theories violating them are either unlikely or unclear to have weakly coupled gravitational descriptions, but it suggests that the CFT inequalities obtained here by naive translations do not apply beyond the regime in which weakly coupled gravitational descriptions are available.