Abstract
In this work we consider AdS$_3$ gravitational theory with certain mixed boundary conditions at spatial infinity. Using the Chern-Simons formalism of AdS$_3$ gravity, we find that these boundary conditions lead to non-trivial boundary terms, which, in turn, produce exactly the spectrum of the $T\bar{T}/J\bar{T}$-deformed CFTs. We then follow the procedure for constructing asymptotic boundary dynamics of AdS$_3$ to derive the constrained $T\bar{T}$-deformed WZW model from Chern-Simons gravity. The resulting theory turns out to be the $T\bar{T}$-deformed Alekseev-Shatashvili action after disentangling the constraints. Furthermore, by adding a $U(1)$ gauge field associated to the current $J$, we obtain one type of the $J\bar T$-deformed WZW model, and show that its action can be constructed from the gravity side. These results provide a check on the correspondence between the $T\bar{T}/J\bar{T}$-deformed CFTs and the deformations of boundary conditions of AdS$_3$, the latter of which may be regarded as coordinate transformations.
Highlights
Over the past few years, we have seen a surge of interest in deformed 2D conformal field theories (CFTs) [1,2,3,4,5,6,7,8,9,10]
Using the Chern-Simons formalism of AdS3 gravity, we find that these mixed boundary conditions lead to nontrivial boundary terms, which, in turn, produce exactly the spectrum of the TT =JT -deformed conformal field theories (CFTs)
We show that the asymptotic dynamics of AdS3 gravity with the mixed boundary conditions are described by the deformed conformal theories
Summary
Over the past few years, we have seen a surge of interest in deformed 2D conformal field theories (CFTs) [1,2,3,4,5,6,7,8,9,10]. In the context of AdS3=CFT2, the boundary dynamics of AdS3 gravity with the Brown-Henneaux boundary condition turns out to be a SLð2; RÞ WZW model This result can be derived through the Chern-Simons form of AdS3 gravity. It has been shown that the Chern-Simons AdS3 gravity at quantum level is equivalent to the Alekseev-Shatashvili quantization of coadjoint orbit DiffðS1Þ=PSLð2; RÞ of the Virasoro group [41] These considerations may be extended to the case of TTand JTdeformation. We show that the asymptotic dynamics of AdS3 gravity with the mixed boundary conditions are described by the deformed conformal theories. In Sec III, the boundary dynamics of AdS3 with mixed boundary conditions turns out to be the constrained TT -deformed WZW.
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