$T\bar{T}/J\bar{T}$-deformed WZW models from Chern-Simons AdS${}_3$ gravity with mixed boundary conditions

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.

34 pages; v2: added section 3.2 and 4.4 to discuss the constraints, added references; v3: revised to match the published version