A measure on the space of CFTs and pure 3D gravity

TL;DR

This work defines a normalizable, maximum-ignorance measure on the space of two-dimensional CFTs, parameterized by a central-charge center $c_0$, a width $\epsilon_c$, and a spectral-gap regulator $\Delta_{\min}$, and uses it to probe whether pure gravity in $AdS_3$ can emerge as the ensemble average. The authors show that, once $\Delta_{\min}$ is removed, the average is dominated by products of low-$c$ theories (e.g., minimal models), preventing a gravity-like spectrum; this indicates pure gravity and holographic CFTs are atypical in the full space of CFTs. They derive new results on the density and accumulation points of CFTs, including doubly-exponential spacing in central charge and accumulation points at $c=9$ in certain irrational CFTs, as well as the existence of conformal manifolds of bounded central charge with unbounded dimension. In the permutation-orbifold sector, they show holographic-like subgroups are exponentially rare among all subgroups of $S_N$, supporting the view that holographic CFTs are atypical. The work highlights both regulation challenges and possible pathways (e.g., double scaling or twist-gap regulators) toward connecting ensemble averages to bulk gravity, outlining crucial open questions about the space of CFTs and their holographic realizations.

Abstract

We define a normalizable measure on the space of two-dimensional conformal field theories, which we interpret as a maximum ignorance ensemble. We test whether pure quantum gravity in AdS$_3$ is dual to the average over this ensemble. We find a negative answer, which implies that CFTs with a primary gap of order the central charge are highly atypical in our ensemble. We provide evidence that more generally, holographic CFTs are atypical in the space of all CFTs by finding similar results for permutation orbifolds: subgroups of $S_N$ with a good large $N$ limit are very sparse in the space of all subgroups. Along the way, we derive several new results on the space of CFTs. Notably we derive an upper bound on the spacing in central charge between CFTs, which is doubly exponentially small in the large central charge limit.