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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 · Mengyang Zhang 
Submission information  

Preprint Link:  https://arxiv.org/abs/2304.13650v3 (pdf) 
Date accepted:  20230915 
Date submitted:  20230815 12:26 
Submitted by:  Eberhardt, Lorenz 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
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 Teichmüller space of Riemann surfaces that we refer to as "Virasoro TQFT." This TQFT is similar, but importantly not equivalent, to $\text{SL}(2,\mathbb{R})$ ChernSimons theory. This sharpens the folklore that 3d gravity is related to $\text{SL}(2,\mathbb{R})$ ChernSimons theory into a precise correspondence, and resolves some wellknown 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 3manifolds, Virasoro conformal blocks and crossing kernels.
Author comments upon resubmission
We would like to thank both referees for their careful reading of the manuscript and their feedback. Let us answer the excellent questions by referee 2:

Indeed, it is very important to start in Lorentzian signature, since the correct phase space can only be determined in Lorentzian signature. Starting with gravity in Euclidean signature would have led to the incorrect assertion that the phase space is given by (a subset of) the moduli space of flat $\mathrm{SL}(2,\mathbb{C}$ connections on the initial value surface. One can then proceed by quantizing the phase space to obtain the Hilbert space. Once one has determined the Hilbert space, one can switch to Euclidean signature, since essentially by definition, the Hilbert space of the theory does not care about the spacetime signature. In particular, the \emph{Euclidean} gravity partition function on $\Sigma \times \mathrm{S}^1$ can be obtained by tracing over the Hilbert space. We have added some comments along these lines at the beginning of Section 2.7.

Yes, it is correct that it is crucial that one integrates over all of Teichmuller space. The main reason is that Teichmuller space naturally factorizes as follows. Choosing a pair of pants decomposition of the surface, Teichmuller space can be globally parametrized by FenchelNielsen coordinates associated to the decomposition. Thus in spacetime, the 'amplitude' corresponding to the inner product can be computed by a single 'Feynmandiagram' corresponding to the pair of pants decomposition. As mentioned by the referee, such a statement would be very wrong for actual string amplitudes, which instead unify all channels in one diagram. We added similar explanations in Section 2.3.

The answer to this question is very interesting. Teichmuller space can also be quantized by certain $\mathrm{PSL}(2,\mathbb{R})_k$ conformal blocks (although this has never been made precise to our knowledge). Only some conformal blocks appear since only a part of the moduli space of flat $\mathrm{PSL}(2,\mathbb{R})$ connections was quantized. There is a correspondence between correlation functions of the $\mathrm{PSL}(2,\mathbb{R})_k$ WZW model and Liouville theory, which also implies a simple relationship between the conformal blocks (see hepth/0507114, equation 3.29 for $r=n2$ for the genus 0 case). In particular, this implies that the crossing properties of the $\mathrm{PSL}(2,\mathbb{R})_k$ blocks are equivalent and thus the Hilbert space obtained from quantization in terms of $\mathrm{PSL}(2,\mathbb{R})_k$ blocks is equivalent. From the point of view of quantization, whether one obtains current algebra blocks or Virasoro blocks depends on a choice of polarization (i.e. coordinates on which the wave function depends). In the usual quantization known from ChernSimons theory, the wavefunction depends homomorphically on the gauge connection, i.e. is a "function" of $A^a_z$ for $a=3,+,$ (it is actually a holomorphic section of a certain line bundle). To obtain Virasoro conformal blocks, one has to use the polarization described in 10.1016/05503213(90)90510K. The fact that they lead to the same mapping class group representations, means that the two quantizations are equivalent. In particular, we could have also taken the quantization in terms of $\mathrm{PSL}(2,\mathbb{R})_k$ blocks; for the computation of partition functions on closed manifolds this gives on the nose the same result. In the presence of boundaries, the partition function takes values in $\mathrm{PSL}(2,\mathbb{R})_k$ blocks, to get the correct answer one has to implement Hamiltonian reduction which is the aforementioned connection to Virasoro conformal blocks. For the purpose of AdS/CFT, it is natural and simpler to directly use the quantization in terms of Virasoro conformal blocks. Some of us plan to return to these issues in a future publication.
List of changes
1. See the comments above.
2. Good point, we implemented this in the new version.
3. We changed the statement to 'We prefer the name “Virasoro TQFT” because we feel it better captures the physical meaning of the TQFT, and we strongly feel that a person like O. Teichmüller does not deserve additional scientific honor for a recent development that he did not contribute to'. This should make it impossible to misinterpret the statement.
Published as SciPost Phys. 15, 151 (2023)