## Abstract

We extend the work of Hellerman 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 12 +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 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}.

Original language | English |
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Article number | 91 |

Journal | Journal of High Energy Physics |

Volume | 2014 |

Issue number | 5 |

DOIs | |

State | Published - May 2014 |

### Bibliographical note

Publisher Copyright:© The Authors.

## Keywords

- AdS-CFT Correspondence
- Conformal and W Symmetry
- Field Theories in Lower Dimensions

## ASJC Scopus subject areas

- Nuclear and High Energy Physics