SciPost Submission Page
A measure on the space of CFTs and pure 3D gravity
by Alexandre Belin, Alexander Maloney, Florian Seefeld
Submission summary
| Authors (as registered SciPost users): | Florian Seefeld |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2509.04554v2 (pdf) |
| Date submitted: | Jan. 5, 2026, 2:11 p.m. |
| Submitted by: | Florian Seefeld |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #1 by Antonio Antunes (Referee 1) on 2026-1-12 (Invited Report)
Strengths
1- Very clear context, goal, approach and results 2 - Two well-developed technical examples (regarding central charge spacing and permutation orbifolds) extensively detailed in the appendices 3 - The landscape results are exciting and thought-provoking
Weaknesses
1- The main object of study (maximum ignorance measure in the space of CFTs) leads to a negative result regarding the goal of constructing a dual to pure gravity in AdS3 2 - Many results concern decoupled theories with multiple stress tensors and do not explore other well known constructions of Rational CFTs
Report
AdS3/CFT2 is an arena where both approaches have a clear overlap: Microscopic examples such as the celebrated D1-D5 system are head-to-head with modern constructions of 2d CFT ensembles, examples of which have been built on the basis of a statistical solution to the crossing equation moment by moment, or via an exactly crossing symmetric tensor model encoding the sum over bulk topologies.
In the text, the authors propose a measure of "maximal ignorance" in the space of CFTs as a new candidate for an ensemble which can be averaged over to produce pure gravity in AdS3. While it is shown that this hope is not realized, the authors present many interesting results which follow from, or are inspired by, this construction.
The measure is defined by a reference value of the central charge, along with an associated interval centered around the reference value, and must be further endowed with a lower cutoff in scaling dimensions to ensure normalizability. Given this interval, the measure counts with uniform weight isolated CFTs and each connected conformal manifold weighted by the natural Zamolodchikov norm (and normalized by the associated Zamolodhickov volume). Compared to previous proposals, this measure has the advantage of operating directly in the space of QFTs instead of working with approximate proxies. Furthermore, being a "maximal ignorance" uniform measure, it is completely agnostic on potential features of a pure gravity theory one might want to ad-hoc engineer. This however come with a price to pay: the proliferation of CFTs obtained by tensor products completely contaminates averages when the cutoff in scaling dimension is taken to be small, meaning that averages over this ensemble are not a viable way to obtain theories of quantum gravity.
Nonetheless, the maximum ignorance ensemble led the authors to several interesting results on the space of CFTs showing: doubly exponentially small spacing between accumulation points, rigorous existence of accumulation points of irrational CFTs and existence of conformal manifolds of unbounded dimensionality but bounded central charge.
Finally, the authors explore the class of permutation orbifold CFTs where the "holographicness" of the theory is determined purely group theoretically and is encoded in the finiteness of the spectrum at a given scaling dimension (mathematically oligomorphicity of the subgroup) and uniqueness of the stress tensor (mathematically transitivity of the subgroup). The authors estimate the density of subgroups satisfying these properties, arguing that holographic theories are a super-exponentially small fraction of permutation orbifolds in the maximum ignorance measure, once again confirming the negative result: Uniform averages over CFTs do not produce holographic theories.
Overall, while the main goal of building a candidate for a measure in the space of CFTs which produces holographic theories under average was not achieved, the notion of a maximum ignorance ensemble leads to many new results and observations on the space of CFTs, and is complemented by two solid technical analyses on the spacing of tensor product CFTs and the density of "holographic subgroups"in permutation orbifolds.
Requested changes
While the results are solid and I recommend the paper for publication, there is one "main" concern (A) I would like to expose, along with a few minor concerns (1-).
Regarding my main concern: A) As the authors emphasize, the acummulation point $m\to \infty$ of minimal models leads to divergent volumes in the uniform measure if no scaling dimension cutoff is introduced. The cutoff then saves the day even in accumulation points involving tensor products of minimal models. However, there are more general standard accumulation points of CFTs even with a unique stress tensor. For example the diagonal coset models $(G_k\times G_\ell) / G_{k+\ell}$ in the $\ell\to \infty$ limit lead to accumulation points near the central charge of every WZW model $G_k$ and possess a unique stress tensor and no spin 1 currents. There are many other coset constructions (for example generalized parafermions where the maximal abelian subalgebra is gauged) of this type. Is it clear that they are all accompanied by light scalar operators that save the day, making volumes finite? It seems plausible that this is true, and the authors transfer the burden of proof to the conjecture of Kontsevich-Soibelman/Douglas-Acharya, but perhaps some explicit checks are merited. The use of the technology developed in Appendix A might also be interesting in this regard.
Regarding the remaining minor concerns: 1) In the beginning of page 6 where the authors discuss the potential divergences of the uniform ensemble they omit the accumulation points of isolated CFTs which they later discuss. I believe this point should be mentioned already at this stage, or at least a slight rephrasing should be made. 2) The first paragraph in section 2.1 is written in a somewhat cavalier fashion. While the authors are obviously aware of the ADE classification of minimal models alluded to in footnote 10, I believe a citation to the original work is merited. Furthermore the authors are implicitly making use of the classification of bosonic CFTs even though this is not explicitly mentioned. (Para)fermionic CFTs with $c<1$ are built similarly but do not fall into the ADE classification. The authors also omit the existence of the three isolated "Platonic" $c=1$ CFTs which would make an interesting illustration of a value of $c$ where both isolated points and continuous branches exist. 3) The landscape result 2 as a rigorous statement is, to my knowledge, new but if the authors are to believe the conjecture of reference [48], it is also clear that such accumulation points of irrational models naturally appear there. 4) Also regarding reference [48] the authors mention it as providing candidate theories with a positive twist gap. However, strictly speaking the authors are only providing evidence for the chiral algebra being Virasoro, i.e. it is not clear whether the twist approaches zero or a finite value at large spin. Is there anything interesting to say about the scenario where the twist accumulates to zero at large spin? 5) Finally, the citations regarding explicit constructions of compact unitary CFTs with $c>1$ and only Virasoro symmetry (henceforth "generic") could be more complete. Reference [48] is heavily inspired by the much earlier work of arXiv:cond-mat/9812227 (and references therein) which while unaware of the irrationality, first constructed a plausibly "generic" CFT. Following [48] there has been a recent revival of this topic, including arXiv:2405.19416, arXiv:2412.21107, arXiv:2507.14280, arXiv:2512.23664, but this sample is biased towards the work of the reviewer.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Florian Seefeld on 2026-01-08 [id 6216]
Dear referees,
Please note that there is a typo in equation 1.5 of the preprint. The right parenthesis inside the integration is supposed to be part of the subscript, so the correct tex code for this equation is
\[
\overline{ \cdot}\equiv \int_{C(c_0,\epsilon_c)} d\mu \cdot \, ,
\]
Best,
the authors