Two dimensional conformal field theories (CFTs) have a very powerful property called modular invariance, which relates the high and low temperature limits of the theory. This can give nontrivial relations between the low-energy spectrum of the theory and the high-energy spectrum. By making very basic assumptions (e.g. unitarity), we can make sharp statements about the high-energy spectrum this way.
In this paper, we consider any 2d CFT with a U(1) current. Each such theory has a`central charge' c that (roughly) counts the number of degrees of freedom. Using modular invariance, we show that there must exist states charged under the U(1) with energy at most c/6+1/3 above the vacuum. We also show that a state must exist with charge-to-dimension ratio above a certain bound that scales with c. In theories with supersymmetry, we show that a stronger bound must be satisfied, and we conjecture a stronger bound for all theories.
These results can be applied to the special, large c 2d CFTs dual to anti de Sitter gravity in three dimensions. Our results translate into a bound on the mass of the lightest charged state, and a bound exhibiting a state with large charge-to-mass ratio. These bounds are related to the "weak gravity conjecture".