Holographic integration of $T \bar{T}$ and $J \bar{T}$ via $O(d,d)$

TL;DR

We address the holographic description of single-trace $T\bar{T}$ and $J\bar{T}$ deformations in AdS$_3$/CFT$_2$, showing their finite forms are encoded by $O(d,d)$ transformations and correspond to integrated Exactly Marginal Deformations on the worldsheet. By unifying TsT, Yang-Baxter, and Chaudhuri–Schwartz-type marginality within the $O(d,d)$ framework and employing generalized Killing vectors, we derive explicit expressions for the deformed actions and identify the CYBE as the condition ensuring exact marginality for the integrated deformations. The paper provides concrete AdS$_3\times S^3\times T^4$ examples, including pure $T\bar{T}$, pure $J\bar{T}$, their combination, and a $K\bar{T}$-type deformation, illustrating how finite deformations arise from $O(d,d)$ transformations (often as TsT plus shifts) and how worldsheet solvability persists. Overall, the results extend known worldsheet descriptions to non-Abelian current algebras, offer a practical route to finite deformations, and open avenues for applying the same framework to broader AdS/CFT contexts and generalized supergravity settings.

Abstract

Prompted by the recent developments in integrable single trace $T \bar{T}$ and $J \bar{T}$ deformations of 2d CFTs, we analyse such deformations in the context of $AdS_3/CFT_2$ from the dual string worldsheet CFT viewpoint. We observe that the finite form of these deformations can be recast as $O(d,d)$ transformations, which are an integrated form of the corresponding Exactly Marginal Deformations (EMD) in the worldsheet Wess-Zumino-Witten (WZW) model, thereby generalising the Yang-Baxter class that includes TsT. Furthermore, the equivalence between $O(d,d)$ transformations and marginal deformations of WZW models, proposed by Hassan and Sen for Abelian chiral currents, can be extended to non-Abelian chiral currents to recover a well-known constraint on EMD in the worldsheet CFT. We also argue that such EMD are also solvable from the worldsheet theory viewpoint.

Figures (1)

• Figure 1: Triangle of integrable deformations. The top corner denotes the integrable, irrelevant deformations of a 2d CFT, like single trace $T{\bar{T}}$ or $J{\bar{T}}$ deformations. The bottom left corner shows the general $J{\bar{J}}$ deformations, which are marginal deformations of the string worldsheet theory on $AdS_3\times S^3$ background. These marginal deformations are dual to the irrelevant deformations of the 2d CFT. It has been shown in the literature that these deformations of WZW model are also solvable. As we argued here $O(d,d)$ transformations allow us to find finite deformations in this class. These finite deformations can then be traced back in the 2d CFT side. The bottom right corner shows the gravity background obtained from the same $O(d,d)$ transformation. We should mention here that the string theory WZW model have another class of integrable deformations, the YB deformations that are dual to "noncommutative" 2d CFT. These deformations, too, are generated by another class of $O(d,d)$ transformations, which in especial cases reduce to TsT.