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$T\bar{T}$ in JT Gravity and BF Gauge Theory
by Stephen Ebert, Christian Ferko, HaoYu Sun, Zhengdi Sun
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Submission summary
As Contributors:  Stephen Ebert · Christian Ferko · Zhengdi Sun 
Arxiv Link:  https://arxiv.org/abs/2205.07817v3 (pdf) 
Date accepted:  20220906 
Date submitted:  20220825 02:41 
Submitted by:  Ebert, Stephen 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
JT gravity has a firstorder formulation as a twodimensional BF theory, which can be viewed as the dimensional reduction of the ChernSimons description of $3d$ gravity. We consider $T\bar{T}$type deformations of the $(0+1)$dimensional dual to this $2d$ BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its $3d$ ChernSimons lift, are Wilson lines and loops. In the $3d$ ChernSimons setting, we study modifications to correlators involving boundaryanchored Wilson lines which are induced by a $T\bar{T}$ deformation on the $2d$ boundary; results are presented at both the classical level (using modified boundary conditions) and the quantummechanical level (using conformal perturbation theory). Finally, we calculate the analogous deformed Wilson line correlators in $2d$ BF theory below the Hagedorn temperature where the principal series dominates over the discrete series.
Published as SciPost Phys. 13, 096 (2022)
List of changes
1.) Added comments and clarifications to address the referees' remarks.
2.) Added references.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2022829 (Invited Report)
Report
I thank the authors for the clarifications to address my comments and questions. I am now happy to recommend publication.
Anonymous Report 2 on 2022826 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.07817v3, delivered 20220826, doi: 10.21468/SciPost.Report.5590
Report
I think I'm happy with the authors' reply to my previous questions and I strongly recommend the paper for publication.
I would however like to share the following comment regarding the response to my first question.
I think the equation $T_\mu^\mu(\lambda)=2\lambda T\bar{T}(\lambda)$ should strictly be understood perturbatively to arbitrary order $\lambda$. I guess the authors also agree with this statement.
What is on a better footing at "arbitrary $\lambda$" is the flow equation when expressed in terms of the partition function (Aharony et al or Cardy) or the energy eigenfunction (ZamolodchikovSmirnov). This can be written unambiguously at finite $\lambda$ because they don't involve the stress tensor (or any other local operators). Another quantity that is also well defined at arbitrary $\lambda$ is the kernel formula of Dubovsky et al for a generic local QFT (seed) or the HashimotoKutasov kernel for a seed CFT. This is like an integral form of the abovementioned flow equation that relates the theory at finite $\lambda$ with the seed. It would be interesting to see how one can construct the bulkboundary map using the kernel formula as the definition of the theory at finite $\lambda$, something like rewriting the kernel formula in terms of bulk variables and reading off the dictionary (but not in terms of expectation of the stress tensor operator).
Similar kernel formula exists in $d=1$ as well. Once understood in $d=2$, one should be able to understand in $d=1$.
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