Bounds on operator dimensions in 2D conformal field theories

A bstract We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension Δ 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 ∆ 1 : ∆ 2 ≤ + O (1). We obtain a similar bound on the conformal dimension Δ 3 , and present a method for deriving bounds on Δ n for any n , under slightly modified assumptions. For asymptotically large c tot and n ≲ exp( πc /12), we show that ∆ n ≤ + 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 / 4 G N .