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Weak Gravity Conjecture, Central Charges and $η/s$

We correlate the weak gravity conjecture (WGC), the KSS conjecture with chemical potential at extremality and the central charges by going through a particular example in five dimensional AdS spacetime with two unknown coefficients $c_1, c_2$, assuming WGC exists in AdS spacetime. The result that follows from this example suggests that WGC makes the KSS conjecture to hold in the extremal limit but only when one of the coefficient vanishes ($c_1=0, c_2\neq 0$ or $c_2=0, c_1\neq 0$) and when both the coefficients are non zero it can respect and/or violate the KSS conjecture depending on the choice to $c_1$ at extremality, even though $η/s$ do not depend on $c_1$ at extremality. Moreover, WGC is not fully compatible with the calculation of central charges even though the bounds on coefficient $c_1$ that follows from demanding WGC stays within the bounds that central charges predict. As usual, the KSS conjecture is violated, of course, in the non-extremal limit.