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Solving 3d Gravity with Virasoro TQFT

by Scott Collier, Lorenz Eberhardt, Mengyang Zhang

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Submission summary

Authors (as registered SciPost users): Scott Collier · Lorenz Eberhardt
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-05-24 20:25
Submitted by: Eberhardt, Lorenz
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We propose a precise reformulation of 3d quantum gravity with negative cosmological constant in terms of a topological quantum field theory based on the quantization of the Teichm\"uller space of Riemann surfaces that we refer to as ``Virasoro TQFT.'' This TQFT is similar, but importantly not equivalent, to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory. This sharpens the folklore that 3d gravity is related to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory into a precise correspondence, and resolves some well-known issues with this lore at the quantum level. Our proposal is computationally very useful and provides a powerful tool for the further study of 3d gravity. In particular, we explain how together with standard TQFT surgery techniques this leads to a fully algorithmic procedure for the computation of the gravity partition function on a fixed topology exactly in the central charge. Mathematically, the relation leads to many nontrivial conjectures for hyperbolic 3-manifolds, Virasoro conformal blocks and crossing kernels.

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Reports on this Submission

Anonymous Report 2 on 2023-7-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2304.13650v2, delivered 2023-07-16, doi: 10.21468/SciPost.Report.7508


1. Largely solves an important open problem, namely how to compute amplitudes of 3d gravity on generic topologies.
2. Very pedagogically written. Seemingly deep insights are gained from rather simple to follow calculations (the calculations are simple because the authors succeed in solving difficult problems in very elegant manners).
3. Many new interesting technical results, for instance equation (1.1) and (2.21).


This is an overall great piece of work. The authors solve a major open problem by formally solving the path integral of AdS3 gravity on arbitrary topologies. In many cases, the solution is not just formal as finite concrete amplitudes can be obtained. This can be considered a breakthrough.

The paper is very pedagogically written and was an absolute joy to read.

The fact that AdS3 gravity is closely related with the type of TQFT the authors discuss is not by itself new. To me, the most important key new element (though there are certainly others) consists of the steps between (2.61) and (2.63) (see also appendix C) where the authors explain that dividing out by the bulk mapping class group is compensated for hyperbolic 3 manifolds by part of the sum over bulk geometries (which includes the boundary mapping class group). This may sound like a technicality, but without this a solution can not be found. In some sense this is very similar to the reason why Mirzakhani recursion works for AdS2 gravity, if one goes through the proof.

Another piece of progress worth mentioning is equation (2.21), a very general result derived in a beatiful manner.

Lastly let me mention it was clever to use Heegaard splitting on hyperbolic 2 surfaces in order to cut and glue the three manifolds, since the more commonly used surgery on tori is ill defined for this theory.

I do have several small questions.
1. I was confused by a statement in section 2.1 that the metric is Lorentzian, whereas we are computing Eulidean path integrals. Perhaps the authors can explain this a bit more?
2. Between (2.26) and (2.27) the authors prove (2.21). They are an argument that string amplitudes decompose as Feynman diagrams. Do I understand correctly that for this argument to work it is key that we integrate over the whole Teichmuller space in (2.5)? Ordinarily as far as I understand closed string amplitudes do not have such a simple decomposition, because for closed strings we integrate only over the moduli space of Riemann surfaces (unlike for open strings where there is such a simple decomposition). If this is true, I think it would be worth clarifying this. If this is not true, additional explanation would also be helpful.
3. Why is the Hilbert space not spanned by conformal blocks of $SL(2)_k$? One would think this would be more natural from the bulk point of view, since ordinarly we think of the Hamiltonian reduction from $SL(2)_k$ to Virasoro as arising only on the boundary (because of the asymptotic boundary conditions). The quantum numbers might be identical, so perhaps the basis are related in a simple manner and the Virasoro blocks are just technically easier to work with?

Requested changes

1. I would appreciate it if the authors could answer the above questions.
2. In the mapping class group paragraph in the introduction when the statement is made that the bulk mapping class group is a subset of the boundary mapping class group, it would help to add a reference or refer to appendix C and section 2.7. At that point in the text it seems as if this is a fact that the reader is supposed to be aware of, and which is absolutely key in the derivation.
3. In a note at the end of the introduction the authors suggest that by correctly giving people credit for their scientific contributions, we implicitly endorse their polotical convictions. I find this a dangerous precedent and believe SciPost should consider whether or not they want to back up such a view of how we value scientific contributions. With the ample use of Virasoro blocks (instead of $SL(2)_k$ blocks, see question 3 above) the name Virasoro TQFT is motivated enough.

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Author:  Lorenz Eberhardt  on 2023-07-20  [id 3825]

(in reply to Report 2 on 2023-07-16)

Dear referee,

We would like to comment on the 3rd requested change. We will respond to the rest of the scientific criticism once an editorial recommendation has been made.
The requested change misrepresents what we wrote in the paper and we fully agree that scientific contributions should be correctly attributed to their originator, regardless of politics. We however strongly feel that a person like O. Teichmuller does not deserve additional scientific honor for a recent development that he did not contribute to. In particular, this does not set a dangerous precedent for SciPost.
For reference, the relevant quote in our paper is ``We prefer the name “Virasoro TQFT” because we feel it better captures the physical meaning of the TQFT, and avoids the implicit endorsement of the political convictions of O. Teichmuller’’.

With best wishes

The authors

Anonymous Report 1 on 2023-7-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2304.13650v2, delivered 2023-07-15, doi: 10.21468/SciPost.Report.7507


Dear Editor,

In this work, the authors perform the computation of 3d pure gravity amplitudes on hyperbolic 3-manifolds by constructing an explicit formalism in terms of
Virasoro CFT. The main technical achievement is a prescription on how to deal with the mapping class group action as summarized in their equation (1.1), and how to decompose hyperbolic 3-manifolds in terms of computable building blocks.
In particular, it shows us how to combine (local) TQFT techniques with the (non-local) large diffeomorphisms on the surface.
3d gravity has been attracting much attention lately, and the current work addresses and resolves an important problem.

As such, I highly recommend the paper for publication.

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