$ M \leq M_{Planck} / N^{1/2} $This statement is supported by
- a perturbative naturalness argument: having a smaller Planck mass than dictated by the inequality above would be a form of fine-tuning
- black hole physics: black holes with all the discrete charges must remember them which requires them to be heavier than the bound above
Disclaimer: I discourage all crackpots who routinely read "Not Even Wrong" and who have visited this weblog unintentionally to stop reading because the large numbers below could make them really sick.
Gia argues that if there are $10^{32}$ copies of the Standard Model particle species - which would include $10^{32}$ electron squirrel species -, then the gap between the weak scale and the Planck scale becomes natural because of his N-scaling. ;-) Recall that $10^{30}$ is called a quintillion in the U.K. but nonillion in the U.S.
Gia urges the readers to study the relation between his statements and the weak gravity conjecture. What I find perhaps even more interesting is that Dvali's inequality may be understood as a lower bound on the amount of discrete symmetry in a theory of quantum gravity. If the argument is refined, it could also be used to explain why pure AdS3 gravity has to have the monster group symmetry.
Oh, sorry, it's actually an upper bound on the amount of discrete symmetry. But surely it must be saturated in some sense. ;-)
Dvali also argues that any charge that leads to no long-range force must only be conserved modulo
$ N \leq (M_{Planck} / m)^2 $where $m$ is the mass of the unit charge. That should explain, in our non-Abelian monster setup, why the elements of the monster group have order up to 71 only. As you can see, I don't really believe that there are quintillions of sub-Planckian particle species in the real world but the conceptual, swampland implications of Gia's insights could turn out to be important.
