# Lightcone Modular Bootstrap and Pure Gravity

## TL;DR

This paper uses the lightcone modular bootstrap to analyze the large spin spectrum in 2D CFTs, revealing a potential upper bound on the twist gap and implications for pure AdS$_3$ gravity.

## Contribution

It introduces a recursive method to solve modular crossing equations, generalizes the Cardy formula, and suggests a new twist gap bound in 2D CFTs with implications for pure gravity.

## Key findings

- Universal large spin density from the vacuum identified.
- Negative vacuum contribution can be canceled by non-vacuum primaries.
- Proposes an upper bound of (c-1)/16 on the twist gap in unitary CFTs.

## Abstract

We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the lightcone limit. By recursively solving the modular crossing equations associated to 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 non-vacuum Virasoro primary whose twist is no bigger than $c-1\over16$. This suggests a new upper bound of $c-1\over 16$ on the twist gap in any $c>1$ compact, unitary conformal field theory with a vacuum, which would in particular imply that pure 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.