Entanglement Entropy for $T\bar{T}$, $J\bar{T}$, $T\bar{J}$ deformed holographic CFT

Abstract

We derive the geodesic equation for determining the Ryu-Takayanagi surface in $Ad{S}_3$ deformed by single trace $\mu T\bar{T}+{\epsilon }_{+}J\bar{T}+{\epsilon }_{-}T\bar{J}$ deformation for generic values of $(\mu ,{\epsilon }_{+},{\epsilon }_{-})$ for which the background is free of singularities. For generic values of ${\epsilon }_{\pm }$, Lorentz invariance is broken, and the Ryu-Takayanagi surface embeds non-trivially in time as well as spatial coordinates. We solve the geodesic equation and characterize the UV and IR behavior of the entanglement entropy and the Casini-Huerta $c$-function. We comment on various features of these observables in the $(\mu ,{\epsilon }_{+},{\epsilon }_{-})$ parameter space. We discuss the matching at leading order in small $(\mu ,{\epsilon }_{+},{\epsilon }_{-})$ expansion of the entanglement entropy between the single trace deformed holographic system and a class of double trace deformed theories where a strictly field theoretic analysis is possible. We also comment on expectation value of a large rectangular Wilson loop-like observable.

29 pages, 7 figures

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