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

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.

• The paper constructs a normalizable measure on the space of 2D CFTs to probe the duality with pure 3D gravity.
• It uses the Zamolodchikov metric and analyzes permutation orbifolds to reveal doubly non-perturbative density behaviors.
• The study finds that holographic CFTs with large spectral gaps are statistically atypical within this ensemble.

A Measure on the Space of CFTs and Pure 3D Gravity: An Expert Analysis

Introduction and Motivation

This work addresses a central question in the study of AdS$_3$/CFT$_2$ duality: can pure three-dimensional gravity in AdS$_3$ be understood as dual to an ensemble average over two-dimensional conformal field theories (CFTs)? Motivated by the ensemble interpretation of JT gravity in AdS$_2$, the authors construct a normalizable, "maximum ignorance" measure on the space of 2D CFTs and investigate whether the average over this ensemble reproduces the features expected of pure 3D gravity. The analysis also yields new structural results about the landscape of CFTs, including the density of CFTs in central charge and the sparsity of holographic CFTs within this space.

Construction of a Measure on the Space of CFTs

The authors define a probability measure on the space of 2D CFTs, parameterized by a central charge window $[c_0-\epsilon_c, c_0+\epsilon_c]$ and a minimal scaling dimension $\Delta_{\min}$ for primary operators. The measure is constructed as follows:

• Isolated CFTs are assigned equal weight.
• Conformal manifolds (families of CFTs connected by exactly marginal deformations) are integrated over with the Zamolodchikov metric, normalized by the total volume of the manifold.
• The measure is normalized to unity over the space.

This construction is justified as a "maximum ignorance" ensemble, treating all CFTs democratically and using the most natural metric for continuous families. The normalizability of the measure relies on a conjecture (Kontsevich-Soibelman) that the number of CFTs below a given central charge and above a fixed gap is finite, with divergences only arising from singularities in the conformal manifold.

Structural Results on the Space of CFTs

The analysis of the measure leads to several new results about the structure of the space of 2D CFTs:

• Density of CFTs in Central Charge: The set of central charges supporting an infinite number of CFTs (accumulation points) is extremely dense. The maximal spacing between such points at large $c$ is doubly exponentially small, $\delta c \sim e^{-e^{c}}$. This is the first explicit appearance of such doubly non-perturbative behavior in $c$ in the CFT landscape.
• Accumulation Points of Irrational CFTs: There exist accumulation points of conformal manifolds containing irrational CFTs, e.g., at $c=9$ via symmetric orbifolds of $\mathcal{N}=(2,2)$ minimal models.
• Unbounded Dimensionality of Conformal Manifolds: There are conformal manifolds of bounded central charge but unbounded dimension, constructed via products of minimal models.
• Sparsity of Holographic CFTs: CFTs with a large gap in the spectrum (of order $c$) are highly atypical in the ensemble. In particular, permutation orbifolds with a good large $N$ limit and a single stress tensor are exponentially rare among all subgroups of $S_N$.

Implications for Pure 3D Gravity

The central test is whether the average over the constructed ensemble of CFTs yields a theory with the properties expected of pure AdS$_3$ gravity—namely, a large gap to the first nontrivial primary and no extra conserved currents. The analysis shows:

• Negative Result: In the limit where the regulator $\Delta_{\min} \to 0$, the ensemble average is dominated by tensor products of low-$c$ CFTs (e.g., minimal models), which proliferate and wash out any pure gravity-like features. The average gap in the spectrum vanishes, and the typical theory is not holographic.
• Atypicality of Holographic CFTs: Theories with a large gap and minimal chiral algebra are not typical in the space of all CFTs. This is reinforced by the combinatorial analysis of permutation orbifolds, where the fraction of subgroups of $S_N$ yielding holographic-like theories decays as $e^{-N^2}$.

Permutation Orbifolds and Group-Theoretic Sparsity

Permutation orbifolds provide a tractable subspace of the CFT landscape where holographic properties can be analyzed via group theory. The key findings are:

• Transitivity and Oligomorphicity: Holographic properties (finite number of low-dimension operators, unique stress tensor) correspond to transitive and oligomorphic subgroups of $S_N$.
• Exponential Suppression: The number of such subgroups is exponentially suppressed compared to the total number of subgroups, confirming the sparsity of holographic CFTs in this construction.
• Lower and Upper Bounds: Explicit constructions yield a lower bound on the number of oligomorphic, transitive subgroups scaling as $(\log N)^{l-1}$ for any fixed $l$, while the upper bound from transitivity alone is $e^{b N^2/\sqrt{\log N}}$.

Discussion and Open Problems

The authors discuss several alternative regularizations and their limitations:

• Twist Gap: Imposing a twist gap (minimum $h$ or $\bar{h}$) is a stronger condition that could, in principle, select for pure gravity-like theories, but this is not a "maximum ignorance" ensemble and is not well understood.
• Chemical Potentials: Introducing chemical potentials for the gap or central charge could regularize the measure, but does not resolve the dominance of non-holographic theories in the average.
• Double Scaling Limits: A double scaling limit where $\Delta_{\min} \to 0$ as $c \to \infty$ could, in principle, yield a gravity-like average, but this is not realized in the current construction and is at odds with the finite-$c$ definition of pure gravity.

The work highlights the need for a deeper understanding of the space of CFTs with a twist gap and the development of new techniques to characterize typicality in the CFT landscape.

Conclusion

This paper provides a rigorous construction of a probability measure on the space of 2D CFTs and demonstrates that, under this measure, pure 3D gravity is not realized as the ensemble average. The analysis reveals that CFTs with large spectral gaps and minimal chiral algebra are highly atypical, both in the full space of CFTs and in explicit constructions such as permutation orbifolds. The results underscore the structural complexity of the CFT landscape and the challenges in realizing holographic duals as typical members of this space. Future progress will require new insights into the geometry of CFT moduli space, the role of twist gaps, and the statistical properties of operator spectra in large-$c$ limits.