Elementary particle

Partition function (quantum field theory)

Quantum mechanics

Modular invariance

Cone (formal languages)

Theoretical physics

Algorithm

Spin (aerodynamics)

Mathematical physics

Nuclear physics

Particle physics

Operating system

Computer science

Limit (mathematics)

Vacuum state

Classical mechanics

Modular design

Thermodynamics

Physics

Bootstrap model

Mathematics

Light cone

Mathematical analysis

Geometry

Conformal field theory

Conformal map

Twist

Particle decay

Physical review. D/Physical review. D.

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.

Light-cone modular bootstrap and pure gravity