Bounds on operator dimensions in 2D conformal field theories

We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension Delta(2) 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 for Delta(1) : Delta(2) <= c(tot)/12 + O(1). We obtain a similar bound on the conformal dimension Delta(3), and present a method for deriving bounds on Delta(n) for any n, under slightly modified assumptions. For asymptotically large c(tot) and n less than or similar to exp(pi c/12), we show that Delta(n) <= c(tot)/12 + O(1). This implies an asymptotic lower bound of order exp(pi c(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) .