We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the light-cone limit. By recursively solving the modular crossing equations associated with different $PSL(2,\mathbb{Z})$ elements, we identify the universal contribution to the density of large spin states from the vacuum in the dual channel. Our result takes the form of a sum over $PSL(2,\mathbb{Z})$ elements, whose leading term generalizes the usual Cardy formula to a wider regime. Rather curiously, the contribution to the density of states from the vacuum becomes negative in a specific limit, which can be canceled by that from a nonvacuum Virasoro primary whose twist is no bigger than $\frac{c-1}{16}$. This suggests a new upper bound of $\frac{c-1}{16}$ on the twist gap in any $c>1$ compact, unitary conformal field theory with a vacuum, which would in particular imply that pure ${\mathrm{AdS}}_3$ gravity does not exist. We confirm this negative density of states in the pure gravity partition function by Maloney, Witten, and Keller. We generalize our discussion to theories with $\mathcal{N}=(1,1)$ supersymmetry and find similar results.

• Received 7 July 2019

DOI:https://doi.org/10.1103/PhysRevD.100.066029

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Published by the American Physical Society