Solving 3d Gravity with Virasoro TQFT

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:

1. 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.

2. 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 Fenchel-Nielsen coordinates associated to the decomposition. Thus in spacetime, the 'amplitude' corresponding to the inner product can be computed by a single 'Feynman-diagram' 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.

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 hep-th/0507114, equation 3.29 for $r=n-2$ 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 Chern-Simons 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 10.1016/0550-3213(90)90510-K. 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.

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.