Three-dimensional quantum gravity according to ST modular bootstrap

TL;DR

This paper uses modular bootstrap and the Cardy formula to show that pure 3D gravity holography cannot be described by a 2D CFT without additional low-energy primaries, implying constraints on dual theories.

Contribution

It combines large-c ST modular bootstrap with the Cardy formula to derive no-go results for pure 3D gravity duals as 2D CFTs without extra primaries.

Findings

Pure 3D gravity duals cannot be 2D CFTs without additional primaries.

The spectrum must include primaries below the BTZ threshold.

The lowest non-trivial primary has odd spin.

Abstract

We combine the large-$c$ ST modular bootstrap equations with the Cardy formula for the asymptotic growth of the density of states to prove that any $2d$ unitary, compact, conformal field theory (CFT) with no higher spin conserved currents leads to conflicting inequalities whenever the entire spectrum of non-trivial primaries lies above the BTZ threshold. As a consequence, the holographic dual of $3d$ pure gravity, if it exists, cannot be a $2d$ CFT. Consistent solutions of ST bootstrap equations require additional primaries lying below the BTZ threshold. The lowest non-trivial primary necessarily has odd spin