# $T\bar{T}$ in JT Gravity and BF Gauge Theory

### Submission summary

 Authors (as Contributors): Stephen Ebert · Christian Ferko · Zhengdi Sun
Submission information
Date accepted: 2022-09-06
Date submitted: 2022-08-25 02:41
Submitted by: Ebert, Stephen
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
Approach: Theoretical

### Abstract

JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons 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 $3d$ Chern-Simons lift, are Wilson lines and loops. In the $3d$ Chern-Simons setting, we study modifications to correlators involving boundary-anchored 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 quantum-mechanical 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)

### Submission & Refereeing History

Resubmission 2205.07817v3 on 25 August 2022
Submission 2205.07817v2 on 6 June 2022

## Reports on this Submission

### Report

I thank the authors for the clarifications to address my comments and questions. I am now happy to recommend publication.

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### Anonymous Report 2 on 2022-8-26 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2205.07817v3, delivered 2022-08-26, 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 (Zamolodchikov-Smirnov). 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 Hashimoto-Kutasov kernel for a seed CFT. This is like an integral form of the above-mentioned flow equation that relates the theory at finite $\lambda$ with the seed. It would be interesting to see how one can construct the bulk-boundary 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 $d=1$.

### Requested changes

No changes are necessary.

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