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3d gravity from Virasoro TQFT: Holography, wormholes and knots
by Scott Collier, Lorenz Eberhardt, Mengyang Zhang
Submission summary
Authors (as registered SciPost users):  Scott Collier · Lorenz Eberhardt · Mengyang Zhang 
Submission information  

Preprint Link:  scipost_202410_00001v1 (pdf) 
Date accepted:  20241022 
Date submitted:  20241001 06:30 
Submitted by:  Collier, Scott 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We further develop the description of threedimensional quantum gravity with negative cosmological constant in terms of Virasoro TQFT formulated in our previous paper arXiv:2304.13650. We compare the partition functions computed in the Virasoro TQFT formalism to the semiclassical evaluation of Euclidean gravity partition functions. This matching is highly nontrivial, but can be checked directly in some examples. We then showcase the formalism in action, by computing the gravity partition functions of many relevant topologies. For holographic applications, we focus on the partition functions of Euclidean multiboundary wormholes with threepunctured spheres as boundaries. This precisely quantifies the higher moments of the structure constants in the proposed ensemble boundary dual and subjects the proposal to thorough checks. Finally, we investigate in detail the example of the figure eight knot complement as a hyperbolic 3manifold. We show that the Virasoro TQFT partition function is identical to the partition function computed in Teichm\"uller theory, thus giving strong evidence for the equivalence of these TQFTs. We also show how to produce a large class of manifolds via Dehn surgery on the figure eight knot.
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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
We would like to thank the referee for their careful reading of the manuscript and for their feedback on the paper. Here we respond to their questions and comments:  Thank you, we have added footnote 3 clarifying our notation for $\Sigma_{g,n}$.  Yes, the existence of mutations means that the Virasoro TQFT partition function is not always capable of distinguishing topologically distinct hyperbolic threemanifolds. The question you raise has to do with the sum over topologies in the gravitational path integral, which we mostly do not address in this paper. In principle it is expected that, given some fixed boundary conditions, one should (at least) sum over all hyperbolic threemanifolds in order for the boundary dual to solve the CFT crossing equations, but this question is beyond the scope of this work. Although the VTQFT partition function of some hyperbolic threemanifolds may coincide, let us note that this does not lead to the vacuum being counted more than once, since such manifolds give contributions supported entirely above the black hole threshold.  In situations where the boundary surfaces have moduli, the answer to the question of which hyperbolic threemanifold consistent with those boundary conditions gives the dominant contribution in the semiclassical limit will in general depend on the moduli; there will be phase transitions as the moduli are varied. In the simpler situation like the ones the referee mentions where the boundary surfaces do not have moduli, the answer to this question depends on a careful analysis of the Virasoro TQFT partition functions in the semiclassical limit. We are not aware of a general heuristic that determines the dominant topology.  The question of the holographic interpretation of the figureeight knot complement (and indeed of hyperbolic knot complements in general) is an interesting one, since knot complements do not a priori have asymptotic boundaries on which the dual CFT can live. However, such links play an important role as building blocks of more general threemanifolds (including those with asymptotic boundaries), since a general threemanifold may be obtained by performing Dehn surgery on such links embedded in $S^3$. They will hence play a central role in implementing the sum over topologies in the gravity path integral. We worked through the example of Dehn surgery on the figureeight knot in section 4.5, which furnishes a family of hyperbolic threemanifolds whose volume accumulates to that of the figureeight knot complement (although these examples do not have asymptotic boundaries).
List of changes
 We have added footnote 3 clarifying our notation for $\Sigma_{g,n}$.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report
The authors further develop the description of threedimensional quantum gravity with negative cosmological constant in terms of Virasoro TQFT that they have formulated in previous papers, by comparing partition functions in explicit examples. The paper is clearly written and scientifically sound.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
See previous report.
Weaknesses
See previous report.
Report
The authors have convincingly answered my questions, several of which go beyond the scope of their work, and were more related to my personal curiosity than to obstructions towards publication. I am happy to recommend the paper for publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)