Lightcone Modular Bootstrap and Pure Gravity

TL;DR

The article develops a lightcone modular bootstrap framework for 2d CFTs with a finite twist gap and uses a generalized PSL(2, Z) crossing to derive a universal vacuum contribution to the density of large spin states, extending Cardy-like growth beyond the traditional regime. It identifies a nonanalytic, spin-dependent structure governed by Kloosterman sums and shows that the vacuum contribution can become negative in a certain double limit, implying a bound on the twist gap of at most (c 1)/16 for unitary theories with a vacuum. This bound would rule out strict pure AdS3 gravity and is corroborated by matching calculations in the Maloney-Witten-Keller partition function, including in a double limit where the universal vacuum density is recovered. The work further generalizes to N = 1,1 supersymmetry, yielding analogous results with supersymmetric modular kernels and a similar twist-gap constraint. Altogether, the results illuminate a deep link between modular invariance, high spin spectra, and holography in AdS3, challenging the existence of a pure gravity theory and pointing to a universal structure in irrational 2d 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.