Bounds on operator dimensions in 2d conformal field theories

We extend the work of Hellerman to derive an upper bound on the conformal dimension Δ₂ of the next-to-lowest nontrival primary operator in unitary, modular- invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c_tot > 2. The bound we find is of the same form as found by Hellerman 12 +O(1). We obtain a similar bound on the conformal dimension Δ₃, and present a method for deriving bounds on Δ_n for any n, under slightly modified assump-12 + O(1). This implies an asymptotic lower bound of order exp(πc_tot/12) on the number of primary operators of dimension ≤ c_tot/12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G_N.