Bootstrapping Pure Quantum Gravity in AdS3

TL;DR

This paper probes the existence of holomorphic extremal CFTs with central charges $c_L=c_R=24k$ as the CFT dual to pure gravity in AdS$_3$ by applying the numerical Virasoro bootstrap. Building on Zamolodchikov's recursive Virasoro blocks, the authors formulate crossing symmetry constraints as a semidefinite programming problem solved with SDPB, examining spectra with a gap $h_{gap}=k+1$ (and near-extremal variants). They find strong numerical evidence that extremal CFTs do not exist for $k\ge20$ and that near-extremal versions are obstructed for $k\ge4$, implying that pure gravity with holomorphic factorization may not be a consistent quantum theory at large central charge. The results motivate exploring beyond holomorphic factorization, including modular or supersymmetric bootstrap approaches and extensions to higher-dimensional AdS contexts.

Abstract

The three-dimensional pure quantum gravity with negative cosmological constant is supposed to be dual to the extremal conformal field theory of central charge $c=24k$ in two dimensions. We employ the conformal bootstrap method to analyze the extremal CFTs, and find numerical evidence for the non-existence of the extremal CFTs for sufficiently large central charge ($k \ge 20$). We also explore near-extremal CFTs, a small modification of extremal ones, and find similar evidence for their non-existence for large central charge. This indicates, under the assumption of holomorphic factorization, the pure gravity in the weakly curved AdS$_3$ do not exist as a consistent quantum theory.

Figures (3)

• Figure 1: This plot shows the stabilization of Taylor coefficients of the Virasoro conformal block for $k=30$. The vertical axis represents $\partial_n {\cal F}(c,h_\phi,h_\text{gap},1/2)_{p+1}/\partial_n {\cal F}(c,h_\phi,h_\text{gap},1/2)_{p}$ where the subscripts denote the numbers of iteration performed to obtain the Virasoro conformal block ${\cal F}(c,h_\phi,h_\text{gap},1/2)$.
• Figure 2: The SDPB solver finds optimal solutions of (\ref{['opt1']}) with the negative maximum value inside grey region. However, (\ref{['opt1']}) has no optimal solutions in exterior region. The red dotted line indicates the case of extremal CFT, i.e., $h_{gap} = k + 1$. In this analysis, $h_{gap}$ takes integer values.
• Figure 3: The SDPB solver finds optimal solutions of (\ref{['opt1']}) with the negative maximum value inside grey region. However, (\ref{['opt1']}) has no optimal solutions in exterior region. The red dotted line indicates the case of near-extremal CFT, i.e., $h_{gap} = k$. In this analysis, $h_{gap}$ takes integer values.